Trvv/Zp (Res(x.c?iog(Col(t/)))) mod p".
Comme Wn-i{x^ri) n'est autre que la reduction mod p" de ^""^(x), et comme pour tout A G VF, TTw/ip{^^~^{^)) = ^Tr^y/z (A), cela resulte la proposition 2.3.4. A 3 . Le cas des corps locaux de caracteristique 0 Dans tout ce paragraphe, K est une cloture algebrique fixee de A^oPour toute extension L de A"o contenue dans A", on pose GL = Gcd{K/L). On note C le complete de A pour la topologie p-adique usuelle (sur lequel GKQ opere par continuite). 3.1.
Le corps Fr R et quelques-uns de ses sous-amieaux
3.1.1. Dans ce numero, KQ = Ko(^) et on note A^co C K la Zpextension cyclotomique de A'o si p :^ 2 et A"00 = UA'o( V^) si p = 2. Pour toute extension algebrique L de A^o contenue dans A', on pose HL = Gl n GK^^ ^L = GL/HL ((l^i est done un groupe isomorphe a Zp si L/KQ est finie, sauf peut-etre si p = 2, auquel cas il pent aussi etre isomorphe a Z2 x (Z/2Z)).
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269
3.1.2. Rappelons la definition de Tanneau R introduit dans [Fo77] (cf., par ex., [Fo83a], n° 3.6). En t a n t qu'ensemble, R (resp. Fr R) est forme des families x = {x^^^)nei d'elements de Oc (resp. C) verifiant (^^(n+iy _ ^(n) pQ^j. ^Q^^ n £ l . On definit sur R Taddition et la multiplication par les formules
(x2/)(") = ^('^)yW
et
(x + y)^''^
lim (x^+"^)-f y ( " + ^ ) ) mi-»-co \
.
/
Cela munit R d'une structure d'anneau. Choisissons un ideal a de Oc strictement contenu dans I'ideal maximal et contenant p. L'application, qui k X £ R associe la suite des reductions mod a des x^^\ pour n G N, definit un isomorphisme canonique R c=i lim .proj.nGNO^/^ (chaque application de transition etant I'elevation a la puissance p) que nous utilisons pour identifier ces deux anneaux (en particulier cette limite projective ne depend pas du choix de a; le choix habituel est I'ideal engendre par p). Soit k le corps residuel de Oc, qui est done une cloture algebrique de k. Pour tout a G ^, on note [a] son representant de Teichmiiller dans W{k) C Oc' L'application, qui a a G ^ associe {[a^ ])n£ii est un homomorphisme d'anneaux et nous permet de considerer R comme une ^-algebre (done a fortiori comme une fc-algebre). L'anneau R est un anneau parfait de caracteristique p, integre, et Fr R s'identifie bien a son corps des fractions. Si Vp est la valuation de C normalisee par Vp{p) = 1, on pose v{x) — Vp{x^^^)^ pour tout X G Fr i?. L'application v est une valuation de Fr R^ qui est complet, son anneau des entiers est R et son corps residuel est k. Le groupe GKQ opere par fonctorialite sur Fr R. 3.1.3. On choisit un generateur du "Zp(l) multiplicatif," i.e., un element e = (6^^))nGN G R tel que 6^^) = 1 et e<^^) ^^ 1. C'est un element du Q^espace vectoriel des unites de l'anneau R congrues a 1 modulo son ideal maximal. On pose alors
aGFp
(ou, rappelons-le, [a] G Zp est le representant de Teichmiiller de a). 3.1.4. On note S(KQ) (resp. E(KQ)) I'adherence dans R (resp. dans Fr R) de la sous-A:-algebre de R engendree par e (resp. de son corps des fractions).
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JEAN-MARC FONTAINE
On voit que E(KQ) est un corps value complet, a valuation discrete, de corps residuel Ar, que S(KQ) est Tanneau de ses entiers et que e — 1 en est une uniformisante. En particulier e appartient au Zp-module des unites de S{KQ) congrues a 1 modulo Tideal maximal, et E{KQ) est stable par GKQ (si X ' GKQ -^ 2* designe le caractere cyclotomique, on a g(€) = 6^(^)) et independant du choix de e. Le groupe fini F ; = G31{K'Q/KO)= Ga\{KooKo/^OO) opeiesm E{K'Q), avec action triviale sur le corps residuel. On note EQ le corps fixe et So I'anneau de ses entiers. L'extension E(KQ)/EO est une extension cyclique de degre p—l totalement ramifiee. On voit que TTQ = 1 + tj:E{K' )/EQ{^) € £"0. Un calcul facile montre que i;(7ro) = P = {p — l).v{€ — 1), i.e., que TTQ est une uniformisante de EQ. En particulier, on a montre: 3.1.5. Proposition. Avec les notations qui precedent, soit So (resp. Eo) radherence dans R (resp. dans Fr R) de la sous-k-algebre de R engendree par TTQ (resp. de son corps des fractions). Alors So et EQ sont independants du choix de 6, stables par GKQ et fixes par HKQ = Gai(A'/A'oo)- On a ^(TTQ) = p, Eo est un corps value complet, a valuation discrete, a corps residuel k; Vanneau So est Vanneau de ses entiers et TTQ en est une uniformisante. 3.1.6. T h e o r e m e . Le corps Fr R est algebriquement clos. La fermeture algebrique Eo et la fermeture separable E^^ de Eo dans FTR sont denses dans FvR et stables par GKQ- L^action induite de HKQ sur El ^^ est Eo-lineaire et identifie HKQ a Ga\{El^^/Eo). Pour tout sousgroup e ferme H de HKO ? (Fr R)^ est le complete de la cloture radicielle de (EQ^^)^ (en particulier, (Fr R) ^0 est le complete de la cloture radicielle de Eo = k{{7to))). 3.1.7 P r e u v e . Posons ( = e(^) et, pour tout n, K^ = /^(e^")). Soit An le quotient de Tannneau des entiers de Kn par Tideal a„ == (^ — l)OKn • On voit que pour tout n > 2, tout g E Ga\{Kn/Kn-i) et tout x G OKU^ {g — l)x E an. Comme par ailleurs p E. Gn^ I'application qui k x ^ Oxn associe I'image dans An-i de ^OTK^/X^_^{X) est un homomorphisme d'anneaux dont le noyau contient an- Par passage au quotient, on obtient done un homomorphisme d'anneaux An —> ^ n - i - L'anneau A = lim.proj.An est Tanneau des entiers du corps des normes de Pextension {UKn)/I< introduit dans [FW79] et etudie dans [Wi83]. Par ailleurs, I'inclusion de Oxn dans Oj^ peimet d'identifier chacun des An a un sous anneau de Oj^/iC ~ ^)^Jc ^ ^c/a en notant a I'ideal de Oc engendre par C — 1; via cette identification, Tapplication de transition de An dans An-i est juste Televation a la puissance p. Autrement dit
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271
I'anneau A s'identifie a un sous-anneau de R. II est facile de voir qu'alors A = k[[€ — 1]] = S(KQ), done que le corps des normes de I'extension (UKn)/K est E{K'Q). II resulte alors du theoreme 3.1.2 de [Wi83] que EQ est le corps des normes de Textension. Le theoreme resulte alors de [op. cit. , cor. 4.3.4]. 3.1.8. Pour toute extension finie K de A'o contenue dans K, on note p^^' le degre de 1 'extension Koo H K/KQ, on pose
et on note S{K) I'anneau des entiers de E{K) (lorsque A' = A^Q, les notations sont compatibles avec celles du n° 3.1.3). C'est une extension finie de EQ. Si A C L, E{K) C E{L) et [E{L) : J^(A)] = [L : A ] , le degre d'inseparabilite etant p^^~^^. Si I'extension L/K est galoisienne, E(L)/E{K) est normale et le groupe de ses automorphismes s'identifie a Gal(AAoo/AVi^oo) = HK/HL.
_
Enfin, si L est une extension algebrique de KQ contenue dans A , on note E{L) la reunion des E{K) pour A parcourant les extensions finies de Ao contenues dans L. 3.2.
Le corps W-j^{Fv R) et quelques-uns
de ses
sous-anneaux
3.2.1. Les anneaux W{R) et W{Fi R) spnt_des Py(fc)-algebres. Si A est une extension finie de K contenue dans A , ou I'anneau de ses entiers, et si k' est le corps residuel de cette extension, on pose WA{R) — A^w(k')^^{R) et WA(FT R) = A(^w(k')W{Fi R). Si L est une extension algebrique de A'o contenue dans A , on note Wi,(Fr R) la limite inductive des WK{FT R)^ pour K parcourant les extensions finies de A^o contenues dans L. On voit que WKQ(FT R) = W{FT R)[l/p] est un corps value complet de caracteristique 0, absolument non ramifie, de corps residuel Fr R, dont I'anneau des entiers est W(FT R). Le fermeture algebrique de A Q dans WK^{FI R) s'identifie a I'extension maximale non ramifiee de A^o contenue dans K. On en deduit que pour toute extension algebrique L de A' contenue dans A', WL{FV R) est un corps, extension algebrique de WKQ (FTR) dont I'anneau des entiers est
Wo^i^T R). Le Frobenius (p agit sur Fr R (par x h-> x^) et par fonctorialite sur WKO{FV R). Les sous-anneaux W{R), W{FT R) et WKoiR) = W(R)[l/p] sont stables par (p. L'action naturelle de GKQ sur A et sur WKQ{FT R) s'etend en une action sur W^^(Fr R). Un grand nombre de ses sous-anneaux sont stables par GKQ] si A C i C A , WL(Fr R) est stable p a r GL (et aussi p a r GKQ si
272 L/KQ
JEAN-MARC FONTAINE est galoisienne).
3.2.2. Dans W{R), [e] appartient au Zp-module des unites congrues a 1 modulo rideal maximal. On pose TT^ = [e] — 1 G W{R). Pour tout n E N, Wn{R) (resp. W„(Fr R)) s'identifie, en t a n t qu' ensemble a R^ (resp. (Fr R)^) et on le munit de la topologie produit (avec la topologie definie par la valuation sur R et Fr R). O n munit W{R) et W{Yi R) de la topologie de la limite projective et WKQ{F^ R) = \J p~'^W{Fv R) de la topologie de la limite inductive. Notons S{KQ) (resp S(KQ)) Tadherence dans W(R) (resp. dans WKQ(P^ R)) du sous-anneau (resp. du sous corps) engendre par W et II est immediat que la reduction mod p de S{KQ) s'identifie a S{KQ) et que S{KQ) s'identifie a Tanneau des series formelles en la variable TT^ a coeflftcients dans W. Le corps S{KQ) est un corps value complet, absolument non ramifie, dont le corps residuel s'identifie a E{KQ) et I'anneau de ses entiers est le separe complete de W{{'K^)) pour la topologie p-adique. Si ^ G GKQ ) on a g-K^ — {^-\- TT^)^^^^ — 1; on a aussi (fw^ — {\-\- TT^Y — 1.
On en deduit que S{KQ) et S{KQ) sont independants du choix de e, stables par GKQ et (f. On pose So = ( 5 ( / i ^ ) ) ^ ^ o et So = {S{K'o)f^o . En particulier est une extension cyclique, de degre p—\. On pose
S{K'Q)/SO
a€Fp
On voit que TTQ = ^^e{K')/So{'^^) ^^ ^^ I'image de TTQ dans R n'est autre que TTQ. La propostion suivante est alors immediate: 3.2.3.
P r o p o s i t i o n . Soit TTQ G W{R)
defini par
aefp Alors Fadherence So de la sous-W-algebre de W{R) engendree par s'identifie a Fanneau des series formelles en TTQ a coefficients dans Le separe complete de W{{7ro)) pour la topologie p-adique s'identifie Vanneau des entiers OSQ d'un sous-corps ferme So de WKQ{F^ R), est un corps absolument non ramifie dont le corps residuel est EQ. anneaux So, OSQ, Gt So sont des sous-anneaux de WKQ(FI^ R), stables (f et GKOJ independants du choix de e.
TTQ W. a Qui Les par
3.2.4. Remarque. Soit i le generateur de Zp(l) associe a e (i.e., t est e mais on pense a lui additivement !). Soit Ko{{t)) le corps des fractions du separe complete de I'anneau /io '^ip SymzpZp(l) pour la topologie
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
273
^-adique (qui est in depend ante du choix de t). Ce corps est muni d'une action naturelle de GRQ et de (p (on pose
On a ^0 C Ko{{t)f''^
- I .
= Ko{{iP-^)). On voit que
7ro-(p-l).
J2
^"/^'-
n>0, ( p - l ) l n
On remarquera que SQ et Oso "refusent" de se plonger dans Ko{{i)). 3.3.
Les (f-T-modules
3.3.1. On reprend les hypotheses et notations du n° 1.1 et on suppose en outre que l'anneau A est muni d'une action d'un groupe F, compatible avec la structure d'anneau et commutant a Tendomorphisme cr. On appelle (f-T-module sur A la donnee d'un <^-module sur A muni d'une action semi-lineaire de F commutant a Faction de (p. Ces ^-F-modules sur A forment, de maniere evidente, une categorie abelienne T^MA, munie d'un produit tensoriel. Le foncteur "oubli de Faction de F" est un (8)-foncteur exact et fidele de T^M^ dans ^M^. Si F'^F est un homomorphisme de groupes, F' agit par transport de structure sur A et on a un (g)-foncteur exact et fidele " transport de structure" : T^MA -^ T'^MA ; le foncteur "oubli de Faction de F" est le cas particulier correspondant a choisir F' = {1}. 3.3.2. Supposons maintenant A et F munis d'une topologie pour laquelle ils sont separes et complets et que F opere continument sur A. Supposons egalement A cr-plat et noetherien. Par definition, un cp-T-moduk eiale sur A est un <^-F-module M sur A, dont le <^-module sous-jacent est etale et sur lequel Faction de F est continue (il n'y a pas d'ambiguite sur la topologie de M car un <^-module etale est en particulier de type fini sur A). Beaucoup des discours que I'on pent tenir sur les (^-modules etales s'etendent aux <^-F-modules etales. En particulier ceux-ci forment une (8)-categorie abelienne que nous notons T^M^^-
274 3.4.
JEAN-MARC FONTAINE Representations p-adiques de GK
3.4.1. On fixe une extension finie K de A'o contenue dans K. On ecrit E, S, G, H, r au lieu de ^(A^, 5 ( A ) , GK. HK, TK s'il n'y a pas de risque de confusion. On note £^ — ^"^ I'adherence, dans WKQ{^^ -R), de I'extension maximale non ramifiee de S contenue dans ce corps. L'anneau de ses entiers est I'adherence de I'henselise strict de Os dans W{FY R). Le groupe H s'identifie au groupe GE — Gal{E^^^/E). On pose
SoiK) = (S""")" , SiK) = ¥'"'•" (^o(A')) = (^"'•'^
{£'"'))"
(lorsque K = A'Q, cette notation est compatible avec celle du n° 3.2.2); on note (!?5Q(A) (resp. Os{K)) Tanneau des entiers de So{K) (resp. 8{K)). On a ^o(A^o) = S{I
De{V)=(o^^,®z^v)"
,
est muni d'une action semi-lineaire continue de G/H — F, i.e., a une structure naturelle de
Ve{M)=(o-^,®OeM)^^^ est stable par G. Le resultat suivant est immediat: 3.4.3. Theoreme. Le foncteur De induit une <S>-equivalence entre les categories Hepj (G) etT^M^ . Le foncteur Ve est un quasi-inverse. 3.4.4. Remarques. (a) Bien sur, dans tout ce qui precede, on pent travailler avec £Q{K) a la place &e £ — S{K). Si r = r ^ , le ^-F-module etaleD£:(F) sur Oe est canoniquement isomorphe kDeQ(^K){^). ^^ comme O^-module via la restriction des scalaires (p~^ : OeQ(K) —^ Oe{K) (via I'isomorphisme A0x i-^ (p^(X).(p^(x)^ si A E ^~^{Oe{K)) et x G De(K){^))' (b) Soient AQ C K C L C A", avec L et A' extensions finies de AQ. Si V est une representation Zp-adique de GR, on a
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
275
Inversement, si ron suppose en outre L/K galoisienne, D£^L^(V) est muni d'une action semi-lineaire de H(L/K) := HK/HL. Si Ton pose r = rt-VK, on a {EL)"^^I^) =
est un quasi-inverse. II est parfois commode d'utiliser le foncteur contravariant VJ et son quasi-inverse D^^ definis par V*e{D) = VeiD*) = Hom*^ [D,£„r) et
D*,(V)=D£{V*)=
Home^ ( l / , ^ n r ) .
3.4.5. L'operateur de monodromie: Soient V une representation padique de GK, MQ = D£Q(V) et M = Ds{V). Comme on Ta vu au n® 2.1.5., il existe sur MQ une unique connexion V : Mo -^ Mo^Qo^
^r..
telle que V o <^ = ^ o V., et il est clair que V commute a Taction de F. D'autre part, QQ^ .^. est un (^^^(x)-module libre de rang 1 de base ^iog(H)- Autrement dit Fapplication a®t
-^ ad\og{[€])
(ou t est le generateur de Zp(l) associe a e comme au n° 3.2.4) definit un isomorphisme canonique de 05Q(;^)-modules
Le choix de e permet alors de definir un operateur (fonctoriel)
276
JEAN-MARC FONTAINE
N : Mo -^ Mo, en posant V x = Nx.d\og{[€]). On voit que A^ est M^-lineaire, verifie N(p = p(pN,
Ng = x{9)9N,
pour tout g
eG
et N{Xx) = X.Nx + NX.x, si X e Os^, x e MQ (et ou, dX = A^A.diog([e]). On pent faire la meme chose pour M , a condition de remplacer 0 5 Q ( K ) par Osi^K) et rfiog(H) par p"''^rfiog(H). II semble que, dans le cas d'une representation p-adique potentiellement semi-stable ([Bu88]), N soit lie a Toperateur de monodromie qui opere sur "la realisation de de Rham" de cette representation (cf. 11°). B . R e p r e s e n t a t i o n s jp-adiques d e h a u t e u r Bl. 1.1.
Le cas d'egale caracterisiique:
finie
les (p-modules de hauteur
finie
Conventions
1.1.1. Dans tout ce paragraphe, on note S un anneau local regulier de dimension 2, complet, de caracteristique 0, a corps residuel parfait k de caracteristique p, muni d'un Frobenius, i.e., d'un endomorphisme y?(zr: a) : S ^^ S verifiant (px = a:P(mod p). Si ms designe Tideal maximal de 5,1'existence d'un Frobenius implique que p ^ m | . Si Ton pose W = W{k) et si Ton choisit w G rns tel que ms = (p, TT), Tanneau S s'identifie a Tanneau VF|7r| des series formelles en TT a coefficients dans W. Le Frobenius est determine par (pw (qui pent etre a priori n'importe quelle serie formelle congrue a TT^ mod p): on a
1, on pose Os n = Oe/p^Os — '5'(p)/p"5'(p) et
5„ = 5/p"5. Le corps E est un corps complet pour une valuation discrete, absolument non ramifie, dont le corps residuel est £" = Fr 5 i == ^((TT)), si ^ designe I'image de TT dans S\,
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX 1.2. 1.2.1.
277
Modules de type fini sur S Les ideaux premiers de hauteur 1 de 5 sont tous principaux. II
y a
- d'une part I'ideal engendre par p, - d'autre part, les ideaux de la forme (P(7r)), ou P est un polynome, a coefficients dans W, irreductible unit aire, dont tous les coefficients autre que celui du terme de plus haut degre sont divisibles par p. Nous disons qu'un 5-module est elementaire indecomposable s'il est libre de rang un ou isomorphe a S/p^, ou p est un ideal premier de hauteur 1 de 5 et n un entier > 1. Un 5-module est dit elementaire s'il est somme directe d'un nombre fini de modules elementaires indecomposables. Dans une telle decomposition, la multiplicite de chaque facteur indecomposable est bien determinee. 1.2.2. Proposition. Si N est un S-module de type fini, il existe un S-module elementaire M et une application S- lineaire r}:N-^M, dont le noyau et le conoyau sont des S-modules de longueur finie (on dit souvent que A^ est "quasi-isomorphe" a M). De plus, M est unique a isomorphisme pres. P r e u v e . C'est un resultat bien connu, qui est a la base de la theorie d'lwasawa (cf., par exemple, [Se58] ou [La78], Thm. 3.1). 1.2.3. Corollaire. Soit N un S-module de type Sni sans torsion. II existe des S-modules libres de type fini L et M et des applications Slineaires injectives L-^
N
et
N -^ M
dont le conoyau est annule par une puissance de p. P r e u v e . Soit 7]: N -^ M
une application 5-lineaire, a noyau et conoyau de longueur finie, de A^ dans un 5-module elementaire M. On pent ecrire M = M' 0 Mp.tor, ou M' est un module elementaire sans p-torsion. Comme A^ est sans torsion, T] est injective, de meme que son compose avec la projection de M sur M', ce qui montre que M = M', i.e., que M est sans p-torsion. Soit r un entier tel p^ annule Coker 77. L'application, qui a x E M associe I'unique y E N tel que 'r]{y) — p^x^ definit un isomorphisme de M sur un sous-module L de N. Mais alors L et M sont a la fois elementaires et sans torsion, done libres.
278
JEAN-MARC FONTAINE
1.2.4. Proposition. Soil N un S-module de type fini sans torsion. Alors Af — Os <S>s N est un Os-module libre de rang fini, Vapplication de N dans J\f, qui a x associe I ^ x est injective. Si Von s'en sert pour identifier N a un sous-S-module de Af, N[l/p] HAf es t un S-module libre de rang fini. Preuve. Soient L -^ N —^ M comme dans le corollaire 1.2.3. On a un diagramme commutatif L
>
Oe 0 5 L
N
y Oe Os A^
y
M
> Oe 0 5 M
ou les fleches horizontales du haut sent injectives par definition, celles du bas par platitude et les fleches verticales extremes parce que L et M sont libres. On en deduit I'injectivite de la fleche verticale du milieu et le fait que Os 0 5 L, O^r 0 5 M et done aussi Oe 0 5 N sont des C7£:-modules libres de rang fini egal au rang d de M sur S. Si Ton utilise ces injections pour identifier tons ces modules a des sous-5-modules de Ai = Oe 0 5 M, on voit que A^o = N[l/p]nAf
= M[l/p]nAfc
M[l/p] n A^ = M
est un 5-module de type fini. Par construction, Papplication No/pNo -^ N/pM est injective et NQ/PNQ est un C7^-module de type fini sans torsion, done libre. Si e i , e 2 , . . . , e^ sont des elements de NQ qui relevent une base de No/pNo, les e^ engendrent NQ et sont lineairement independants. lis constituent done une base de A^o qui est bien libre. 1.2.5. On dit qu'un 5-module M est sans p'-torsion si, pour tout x E M tel que Ann (x) ^ 0, il existe m G N tel que Ann (x) = p^S. Si M est un 5-module de p-torsion, M est sans p'-torsion si et seulement s'il est sans 7r-torsion. Un 5-module elementaire M est sans p'-torsion si et seulement s'il est de la forme M~5''e(®„gM(5„/"), ou d et les d^ sont des entiers presque tous nuls. 1.2.6. Remarquons que si M est un 5-module de type fini sans p'-torsion et tue par p, c'est un module de type fini, sans torsion sur Si — k[TT\\ il est done libre sur 5i, done elementaire. La proposition suivante permet alors de ramener, par devissage, un certain nombre de questions sur les S'-modules de type fini sans p'-torsion au cas ou ceux-ci sont element aires.
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
279
1.2.1. Proposition. Soit M un S-module de type fini, sans p'-torsion. Pour tout i G N, soit Mi = {xeM\3reN
tel que if x G p*M} .
Alors Mi est independent du choix de TT, chaque quotient Mi/Mi^i est un Si-moduIe libre et, pour i sufRsamment grand, Mi est un S-module libre. Preuve. Si TT' est un autre element de S tel que ms = (p,TT'), on pent, quitte a multiplier TT par une unite supposer que TT' = 7r(mod p). On a done w'^ = 7r^(mod p*) des que p*~^ divise r et Mi est bien independant du choix de TT. Chaque Mi/Mi^i est tue par p et sans 7r-torsion. C'est done bien un 5i-module libre. Soit T) : M —^ L une applieation lineaire de M dans un 5-module libre, telle que le noyau et le eonoyau sont de p-torsion. Cette applieation permet d'identifier M[l/p] a L[l/p]. Par ailleurs, si Ton definit les Li eomme les M^, on a Li — p^L. Notons Mi (resp Li) C M[\/p] Tensemble des p~* 0 x, pour x G Mi (resp. Li). On a Li = L et les Mi forment une suite eroissante de sous-5-modules de L, qui est done stationnaire. Si Ton ehoisit i tel que p*M est sans p-torsion et Mi = M^^i, on a Mi^i — pMi, done Mi/pMi est hbre sur Si et Mi est libre sur S. 1.2.8. Remarque. Si M est un S'-module de type fini sans p'-torsion, il en est evidemment de meme de Ker p* et de M/Ker p*. En partieulier, les Ker pYKer p*"^^ sont elementaires. 1.2.9.
Comme a : S -^ S est fidelement plat, M ^ M^ =S<S)s M a
est un foneteur exaet et fidele de la eategorie des 5-modules de type fini dans elle-meme. Proposition. Si M est un S-module de type Rni, sans p'-torsion, alors Ma est aussi sans p'-torsion. P r e u v e . Par devissage, la proposition 1.2.7 nous ramene au eas ou M est element aire; il suffit de le verifier lorsque M est element aire indecomposable, auquel eas e'est trivial. 1.3.
Les (p-modules p-eiales
1.3.1. Nous disons qu'un v^-module M sur S est p-eiale si c'est un 5module de type fini, sans p'-torsion , et si le (^-module M(p) = 5(p) 0 5 M est etale. On note ^ M J ^ la sous-categorie pleine de la eategorie des <^-modules sur S dont les objets sont eeux qui sont p-etales. C'est une
280
JEAN-MARC FONTAINE
categoric additive, Z^-lineaire, qui n'est pas abelienne. Le foncteur M H-^ M(p) est un foncteur additif, exact et fidele de ^ A f | dans ^Af|* . 1.3.2. Remarque. Soit ^Ms^p la categoric suivante: - un objet est un couple {N, M) forme d'un ^-module etale TV sur Sfp) et d'un "5-reseau" M de A'', i.e., d'un sous-5-module de type fini M de N tel que I'application naturelle 5(p) 0 5 M -^ N est un isomorphisme (il revient au meme de se donner un 5-module de type fini M sans p ' torsion et une structure de ^-module sur N = ^(p))] - un morphisme de {Ni,Mi) dans {N2,M2) est un morphisme de (pmodules a : Ni -^ N2 tel que a{Mi) C M2. La categoric ^Mi s'identific alors a la sous-categoric plcine de ^Ms,p formee des (AT, M ) tels que
M est
p-etale;
(ii) il existe q E S, non divisible par p, qui annule (iii) rapplication $ : M^ -^ M est injective. Alors (i) <^ (ii) => (iii). Si M est de p-torsion, equivalentes.
M/^{Ma);
les trois assertions
sont
P r e u v e . D'apres la proposition 1.2.9, M^ est aussi sans p'-torsion et dans le carre commutatif
M,
. (M,),„ = («„))„
1
1
M
>
Mf^p^
les fleches horizontales sont des injections. Ceci nous permet d'identifier M (resp. Ma) a un sous-5-module de M(p) (resp. de (M(^p^)a)L'implication (ii) => (i) resulte de ce que, si q E S n'est pas divisible par p, alors q est inversible dans 5(p). Inversement, si M est p-etalc et si xi,X2,' • • ,Xd engendrent M comme 5-module, il existe yi, y2, • • • ,yd G (M(p))^ tels que ^yj = Xj. Mais il existe q £ S — pS tel que qi/j G M^^ pour tout j . Done ^(M^j) contient le sous-5-module engendre par les ^{qi/j) = qxj et M/<^{M(j) est annule par q. Si M est p-etale, I'injectivite de <^ : {M(j,^)(j —> M(^p) implique celle de Ma -^ M et (i) => (iii). Si M est de p-torsion, et si ^ : Ma -^ M est injective, il en est de meme de (M(p))<7 —^ ^(p) Q^i ^st done bijective puisque source et but sont des S'(p)-modules de meme longueur finie; on a done bien (iii) => (i) dans ce cas. 1.3.4. Si M est un (p-mod\i\e p-etale sur 5 , on appelle hauteur on note h{M) I'ideal Ann ( M / ^ ( M ^ ) ) .
de M et
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX 1.3.5.
281
Proposition. Soit 0 - ^ M' -> M ^ M " - ^ 0
une suite exacte de (p-modules sur S dont les S-modules sous-jacents sont de type £ni et sans p'-torsion. Alois M est p-etale si et seulement si M' et M" le sont. S'il en est ainsi, h{M') D h{M), h{M") D h{M) et la suite 0 -> M 7 ^ ( M ; ) -> M/^ (M^) -^ M"l^
(M'J) -^ 0
est exacte. Preuve. Avec des notations evidentes, on a un diagramnie commutatif 0
X
0
X
1
M„
M'l
i
M
M"
1
A
K"
1
i
0
0
0
dont toutes les lignes et les colonnes sont exactes. Si M' et M" sont p-etales, il existe q', q" E S — pS tels que q' annule A' et q" annule A''; alors q'q" annule A et M est p-etale. Supposons M p-etale. II existe q E S — pS qui annule A; il annule a fortiori A" et M " est p-etale. D'apres la proposition precedente, ^ : M'J —^ M" est injective, done X = 0. Le result at est alors evident. 1.3.6. Soient M un v^-module p-etale sur 5, M' un sous-<^-module de M et M" — M/M' le (^-module quotient. On dit que M' est un sous-objet strict de M ou que M" est un quotient strict de M si M' et M" sont tons deux p-etales. Ceci revient done a demander que M" soit sans p'-torsion. Remarquons que, avec les notations de la proposition 1.2.7, si M est un <;^-module p-etale sur 5, les Mt sont des sous-objets stricts de M, chaque
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JEAN-MARC FONTAINE
Mi^i etant un sous-objet strict de M,. La proposition 1.2.7 implique done que tout <^-module p-etale pent se fabriquer par un nombre fini d'extensions successives de (^-modules p- etales element aires. 1.4. 1.4.1.
Les foncteurs j * et j ^ On note j : S —^ Oe I'inclusion. Si M est un <^-module sur S, f{M)
= Oe^sM
= Oe ^s^^^ {S(p) 0 5 M)
et le lemme de Nakayama implique que, si M est de type fini et sans p'-torsion, alors M est p-etale si et seulement si j*{M) est etale. Comme Os est plat sur S, le foncteur j * est exact; sa restriction a la categorie des ^-modules sur S qui sont p-etales, est un foncteur exact et fidele, que nous notons encore
Nous allons voir que ce foncteur admet un adjoint a droite. Pour cela, pour tout <^-module ^f sur Of, notons Fs{Af) Tensemble des sous-5modules de type fini de Af, stables par (p. 1.4.2. T h e o r e m e . (i) Soit Af un (p-module etale sur Os- Alors, si N € Fs{Af)f N est p-etale sur S et rapplication naturelle Os 0 5 N —^ J\f est injective. En outre j.{Af) = UNeFs(Ar)N est encore dans Fs{Af). (ii) Pour tout objet J\f de^M^ , j*j*Af —>• J\f est un monomorphisme; c^est un isomorphisme si M est de p-torsion. (iii) Si Af est un objet de^M^ , sans p-torsion, alors j^^Af est libre sur S, de rang inferieur ou egal au rang de Af sur Os et j^j^cAf —^ Af est un isomorphisme si et seulement si rangs {j*Af) = range^ (Af). (iv) Les foncteurs j * et j ^ sont adjoints (i.e., si M est un objet de ^M^ et Af un objet de ^MQ on a Eom {M,j,Af)=
Eom(rM,J^)).
Remarquons que, si A^'i, N2 E Fs{Af), alors Ni + ^^2 aussi. En particulier, j*(A/') est un <^-module sur S. On pent done au moins considerer j * comme un foncteur additif
et il est clair que ce foncteur est exact a gauche.
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
283
1.4.3. L e m m e . Si J\f est de p-torsion, alors Fs{J\f) contient un (pmodule No sur S qui engendre Af en tant que Os-module. Preuve. Choisissons des elements e i , e 2 , . . . ,6^ E A/' et des entiers n i , n 2 , . . . ,nrf > 1 tels que Ann (e^) = p^^ Oe et Af = ®i<j
et le fait que Af est etale implique que la matrice A est inversible. Choisissons q E ms fl (5 — pS) tel que (pq = gP(mod p") (comme (fir ~ 7r^(mod p), on pent prendre, par exemple, q = TT^ ). Comme Os^n = Sn[l/q], il existe un entier s tel que tons les q^aj^i E SnSi je choisis r tel que (p — l)r > s, on a
et il suffit de prendre pour No le sous-5-module engendre par les q^Cj. 1.4.4. par p.
Lemme. Les assertions (i) et (ii) sont vraies lorsque J\f est tue
Preuve. Commengons par remarquer que jV est un E- espace vectoriel de dimension finie d. Tout objet de Fs{Af), comme n'importe quel souses £;-module de type fini de Af, est un C^^^-module libre de rang < c? et Tapplication Os 0 5 N -^ Af est injective. Elle identifie done Oe <S>s ^ a un sous-O^r-module de Af stable par (p, qui est done etale et N est bien p-etale. Soit No un element de Fs{Af) contenant une base (ej)i<j<^ de Af sur Os^i = E. Soit C = A^Af. Si e = ei A 62 A . . . A e^, il existe une unite r] de Si = OE et un entier m tels que (pe = ir^rje. On en deduit que, pour tout s' E N, il existe une unite ifg de OE telle que ^
( e ) = TT
V
"^ "^
"^
^ -Tlse.
Soient A^'i le sous-5-module de Af engendre par les Cj et r un entier tel que (p— l ) r > m. Monirons que, si N E Fs{Af), alors N C 7r""''+^7Vi: Quitte a remplacer N par N -^ Ni, on pent supposer que les Cj E N. Si A'' (jt. TT'^'^^Ni, on pourrait trouver a:i,a:25- - - ,Xd ^ E^ avec I'un des Xj egal a 7r~'' tels que x = '^ XjCj E N. Quitte a changer la numerotation, on pent supposer que Xd = T^~^. On aurait done y = eiAciA.. .Acd-i Ax ^ L = A^AT, qui est un sous-O^j-module libre de rang un de £, stable par (p. Mais ce n'est pas possible car y ~ 7r~^e, done, pour tout s E N,
et les (p^{y) engendrent C comme 5-module.
284
JEAN-MARC FONTAINE
Par consequent, toute suite croissante d'elements de Fs(Af) est stationnaire. Done i*(A/') est un objet de Fs{J^) et est bien p-etale. La double inclusion A^i C i*(A/') C 7r~^'^^Ni montre que j*(A/') est un sous-0£;module du £'-espace vectoriel de dimension finie Af, contenant une base de E et ayant son rang egal a la dimension de AT, et
est bien un isomorphisme. 1.4.5. L e m m e . Les assertions (i) et (ii) sont vraies lorsque Af est de p-torsion. Preuve. Pour tout A^ G Fs{Af), Tinjectivite de I'application Os <S>s N -^ M resulte de ce que c'est injectif lorsque Ton se restreint au noyau de la multiplication par p; en particulier Oe 0 s ^ s'identifie a un sous-objet de M et est done etale et N est bien p-etale. Comme S est noetherien, pour achever de prouver (i), il suffit de verifier que ji,{N) est un 5-module de type fini. On va proceder par recurrence sur le plus petit entier n tel que p^J\f = 0. La suite exacte 0 —> Ker p —> M —^ Im p —^ 0 induit une suite exacte 0 —^ i*(Ker p) -^ j*(A') -^ i*(Im p). D'apres le lemme precedent, j*(Ker p) est un 5-module de type fini; il en est de meme, par hypothese de recurrence de j*(Im p), et done aussi de Pour (ii), on remarque que, d'apres le lemme 1.4.3, I'application
est surjeetive, tandis que I'injectivite resulte de (i). 1.4.6. Lemme. Les assertions (i), (ii) et (iii) sont vraies lorsque Af est sans p-torsion. Preuve. Soient Afi = J\f/pAf^ Q la projection de M sur Mi et Ni la somme des ^(A'') pour N G Fs{M). II est clair que Ni C i*(A/i); c'est done un O^j-module libre de rang r inferieur ou egal au rang d die M sur
Oe. Soit M G Fs{M) tel que la projection de M sur A'^i est surjeetive (il est clair qu'un tel M existe). Soient (ej)i<;
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
285
Si X G M, on fabrique, de proche en proche des suites Xz E P et yi E M* telles que
done X — ^p^Xi G P' Ceci prouve que jit.{Af) = P , et les assertions (ii) et (iii) en resultent. Si A^ G Fs{Af), N C P; I'application Oe^sN ^ Oe 0 5 P est injective, done aussi son compose avec Tapplication Oe^sP -^ J^\ ceci nous permet en particulier d'identifier Os^sN kun sous-objet de J\f qui est done etale, d'apres la proposition Al.1.6 et iV est bien p-etale. 1.4.7. Terminons la demonstration du tkeoreme: Les assertions (i) et (ii) ont deja ete verifiees lorsque, ou bien Af est de p-torsion, ou bien A/* est libre. Le cas general s'en deduit en considerant la suite exacte
L'assertion (iii) vient d'etre prouvee. Enfin, Tassertion (iv) est evidente. 1.4.8.
Si N est un objet de ^ M J ^ , on dit que A^ est normal si
est un isomorphisme. Pour tout objet A^ de ^Aft , j*j*{N) malise de N.
est normal; on Tappelle le nor-
1.4.9. On dit qu'un 9?-module etale Af sur Os est de S-hauteur finie si j*j*Af -^ Af est un isomorphisme. II revient au meme de dire qu'il existe un objet N de ^AfJ tel que j*iV est isomorphe a Af. Tout y?-module etale sur Os n'est pas de 5-hauteur finie, mais le theoreme 1.4.2 montre que tons ceux qui sont de p-torsion le sont. La sous-categorie pleine ^AfJ (5) de ^MQ formee des <^-modules de 5-hauteur finie est I'image essentielle du foncteur j * : ^Mi —^^M^ . 1.5. La q-hauteur 1.5.1. On note N I'ensemble qui est Tunion disjointe de N et de I'ensemble des r"^, pour r G N. On munit N d'une relation d'ordre total en convenant que, pour tout r G N, r < r"^ < r + 1. 1.5.2. Soit q E S — pS. Si M est un <^-module p -etale non nul sur 5, on note hq{M) et on appelle q-hauteur de M Telement de NU{H-OO} ainsi defini: si M/^{M<j) n'est annule par aucune puissance de g, hq(M) = -f-oo; sinon, soit r G N le plus petit entier tel que q^ annule M/^{Ma)\ - si M n'a pas de quotient strict (cf. n° 1.3.6) non nul M' tel que V?(MO C q'M', alors hq{M) = r;
286
JEAN-MARC FONTAINE
- sin on hq{M) = r"^. 1.5.3. Si u est une unite de 5 , on a huq{M) = hq{M), pour t o u t ^-module p-etale non nul M sur S. Soit M un ^-module p-etale non nul sur 5 . Si M est de p-torsion, il est de g-hauteur finie. Dans le cas general, on pent trouver des ideaux premiers {c{i)i 1, et une application S'-lineaire
M/^(M,)->ei<,<,5/(q,r*' a noyau et conoyau de longueur finie. Alors M est de ^-hauteur finie si et seulement si q appartient a chacun des q^. 1.5.4. On note $ A f J ( g ) la sous-categorie pleine de ^Af J formee des objets de g-hauteur finie. Pour tout r G N, on note ^M^g(q) (resp. ^M5^.Qj.(g)) la sous-categorie pleine de ^M'g formee des objets de qh a u t e u r < r (resp. et de torsion). D'apres la proposition 1.3.5, pour t o u t r G N, tout sous-objet strict et tout quotient strict d'un
La restriction de j * a ^M'^(q) admet un adjoint a gauche ji: un objet de ^ M ^ , et si Etq{J\f) designe I'ensemble des s o u s - 5 ^s de Af stables par (p qui sont des yp-modules p-etales de c|f-hauteur prend JKAf) =
UNeEtqi^)N.
C'est bien un 5-module de type fini car c'est un sous-module de j^{Af). Si J\f est de p-torsion, on a jl{M) = j*{Af). Si M est sans torsion, jt{M) est un S-module lihre. Pour le prouver, il suffit de verifier que, pour tout A^ E Etq{M), il existe N' G Etq(J\f) qui contient N et est un 5-module fibre. Mais N s'identifie a un s o u s - 5 module du (/?-module J\f' = Os^s ^ etale sur Oe- D'apres la proposition 1.2.4, iV' = N[l/p]f]Af' est un 5-module libre. II est clair qu'il est stable par cp et contient N. II existe un entier s tel que p^ N' C A^, done N' isomorphe a un sous-^-moduJe de N est p-etale, en particulier N^/^(N^^) est tue par un element a E S — pS. Par ailleurs, on voit que si q^ annule N/^{Na), q'^p^ annule N'/^{N'^). Done N'/^{N'^) est annule par une puissance de q. On dit que M est de q-hauieur finie si j*j*{Af) = M (ce qui est toujours le cas si M est de p-torsion). II revient au meme de dire qu'il existe un obj e t N de^M'g{q) tel que j*A/" est isomorphe a M . S'il en est ainsi, on note
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
287
hq (Af) et on appelle g-hauteur de Af la plus petite des ^-hauteurs de tels N. On note aussi ^M^ (S, q) (resp. $Af^ (5, q)) la sous-categorie pleine de<MfJ (5) formee des objets de ^-hauteur finie (resp. de g-hauteur < r). Ce sont des sous-categories stables par sous-objet et quotient. 1.5.6.
Pour tout a G 5, on note (a) Tideal engendre par a.
Proposition. Supposons q irreductible, (TT) stable par a et (TTT £ (q). Alors, pour tout objet N de^M^{q)j le conoyau de N^
Norq{N):=Jtr{N)
est annule par une puissance de w. II est meme de longueur finie, s'il existe un ideal premier non nul de S, different de (TT) et (q) contenant CTTT. P r e u v e . Soit M — ^oiq{N). diagramme commutatif
Avec des notations evidentes, on a un
0
X
Na
Ma
M'l
4"
•4'
4'
N
y M
y M"
V
L"
0
0
dont toutes les lignes et les colonnes sont exactes. Pour tout S'-module A de type fini et de torsion, notons Ass(A) Tensemble des ideaux premiers q de 5 tels qu'il existe x G A, dont I'annulateur est q. C'est aussi Tensemble des ideaux premiers qui contiennent I'annulateur d'un module elementaire "quasi-isomorphe" a A (cf. n° 1.2.2). Comme Os 0 5 M" = 0, M" est de p'-torsion, done aussi M'^. Par hypothese L' et L sont de g-torsion, done X et L" aussi. On en deduit que Ass(M'0 U {(g)} = Ass{M'J) U {{q)}. Si q' et q" sont deux ideaux premiers de hauteur 1, distincts de 5, les ideaux premiers de hauteur 1 qui divisent crq' sont distincts de ceux qui
288
JEAN-MARC FONTAINE
divisent crq''. On en deduit que, si A s s ( M ' ' ) = {(c{i)i 1 tel que cr^(\i C qi, ce qui contredit le lemme suivant: 1.5.7. L e m m e . Soit r > 1. Sous les hypotheses de la proposition qui precede (TT) et (p) sont les seuls ideaux premiers de hauteur I de S stables par a^. P r e u v e . Soit q un ideal premier de S stable par cr^, different de (TT) et p. L'anneau quotient 5 / q est une VF-algebre finie et plate, integre, munie d'un endomorphisme a^ verifiant cr^{x) = x^ (mod p). On en deduit que 5 / q s'identifie a I'anneau W =^ W(k') des vecteurs de W i t t a coefficients dans une extension finie k' de k. Mais il existe A G 5 tel que cr''(7r) = ATT, et il est clair que A n'est pas une unite. Si p designe la projection de 5 sur S'/q, on a ^(TT) ^ 0, et il existe un entier m et une unite u G W tels que ^(TT) = p'^u; done aussi une unite v de W telle que cr^(^(7r)) = v.g{7r). Mais a^(g{7r)) = ^((7^(7r)) =: ^(A)^(7r) et ^(A) n'est pas une unite. 1.6.
Relevement
des modules de q-hauteur
finie
1.6.1. T h e o r e m e . Soit q un element de S dont Fimage dans Si est une uniformisante. Soit ^ G N. Tout oh jet de ^Mg{q) est quotient d^un ohjet de^Mg{q) sans torsion. 1.6.2. Supposons d'abord que q est un element quelconque de 5 — pS. Si M est un <^-module sur 5 , sans g-torsion, on definit la q-filtration sur M(j et la q-filtration conjuguee sur M en posant, pour tout i G N, F ; M ^ = {xe
M^l^x
G q'M}
et
Cf M
= q'\^
{F'^M)
.
On a F^M, = M , , q.F'^M, C F'^-^'M, C F^M, et CfM C C ^ ^ M . Si M est un <^-module p-etale non nul, dire qu'il est de ^-hauteur finie equivaut a dire que C / M — M , pour i assez grand. Si r est le plus petit des i tel que c'est vrai, la ^-hauteur de M est r s'il n'existe pas de quotient strict non nul M " de M tel que F^M'J = M'J, et r + sinon. 1.6.3. Supposons maintenant q comme dans le theoreme et soit M un
R E P R E S E N T A T I O N S p-ADIQUES DES C O R P S LOCAUX
289
F^Ma dans JI^ et Cf M Timage de Cf M dans M . Posons aussi gr^^M^r = F ^ M ^ / F ^ + ^ M ^ et g r f M =: Cf'M/Cf_J^
(en convenant que Cljd
=:= 0).
Si r G N, alors M est dans ^M''^^ (q) si et seulement si C^M = M. 1.6.4.
L e m m e . Conservons
les hypotheses
et notations
(i) Pour tout i G N, il existe une unique application Car,-: g r ; M . telle que le
ci-dessus.
k-Iineaire
gr?M
diagramme F\Ma
^ F^M
CaTj
gr^M^ soit
y grfM
commutatif.
(ii) Si M est un objet de ^Af^ (g), on a gr* M^^ = 0, p o u r i > r et Car est un
:= eo
gr?M
isomorphisme.
[La notation Car traduit le fait que cet isomorphisme semble jouer dans ce contexte un role analogue a Tinverse de Tisomorphisme de Cartier.] P r e u v e . Montrons (i). L'unicite de Car, est evidente. revient a verifier que le noyau de I'application composee F'^M,
-^-^
F^m
Son existence
. grfM,
qui contient F^'^^M, contient aussi (qM^) fl F^M^, ce qui est clair car I'image par g"*^ de ce sous-5-module est contenu dans F^_^M. Enfin, I'application Car est surjective par construction, et I'assertion (ii) resulte immediatement de ce que dimjt Ma = dimj^ gr^M. 1.6.5. Prouvons le iheoreme: Soit r Tunique entier tel que Q G {r,r"^} et soit M un objet de ^Mg{q). (a) Supposons M annule par p et reprenons les notations du n° 1.6.3. Choisissons une base (ej)i<j<^ de M adaptee a la filtration conjuguee, i.e., telle que, si ij designe le plus petit entier i tel que Ej G CfM, alors CfM soit le sous-^-espace vectoriel de M de base les ey, avec ij < i. Pour chaque j , choisissons aussi un relevement ej de ij dans Cf. M et notons e' Tunique element de Ma tel que ^e'j = q^ tj. II resulte du lemme precedent que les images des e'j dans Ma forment une base de Ma\ par consequent.
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JEAN-MARC FONTAINE
les ej forment une base de M sur Si tandis que les e'- forment une base de M(j sur 5 i . Autrement dit, si Ton ecrit
l
la matrice des aj^i est inversible dans 5*1. Choisissons alors des relevements ctj^i des aj^i dans S et notons (ej)i<j< ei E Nfj forment une base de N^ sur 5 . On munit AT d'une structure de (^-module en posant ^ij — q^^ ej. L'application 5-lineaire de N sur M qui envoie e^ sur ej identifie M a un quotient du <^-module sans torsion N. II est clair que A'' est un objet de ^ M ^ (q). Enfin, si A^' est un sous-objet strict non nul de A^ tel que A^" = N/N' est non nul et verifie (pN" C q^ N", alors M " = N/{N' -h pN) est un quotient strict non nul de M verifiant (pM" C q^M"\ done si M est dans ^ M ^ ( g ) , alors A^ aussi. (b) Passons maintenant au cas general. Pour tout S'-module N de type fini sans p'-torsion, on definit, comme dans la proposition 1.2.7, A^j = {a? G A^ I 3 r G N tel que TT^'X
G p'A^}
.
Pour I suffisamment grand Mi est sans torsion, et il suffit de fabriquer une suite M = M ° ^ M^ ^ . . . ^ M* ^
...
de morphismes surjectifs dans la categorie # M | ( g ) telle que, pour tout i, la projection de M* sur M^ induise une application injective de Ml^^ sur Mtor dont r i m a g e est exactement (Mt)torOn procede par recurrence sur i, le cas 2 = 0 etant clair. Supposons 2 > 0 et soit A/'* le quotient de M*"^ par ( M * " ^ ) i . C'est un objet de * M | ( g ) tue par p et (a) nous permet de realiser AT* comme la reduction modulo p d'un objet U de $ M | ( ^ ) qui est lib re comme S'-module. On verifie immediatement que le produit fibre M* = M*~^ x^i U convient. 1.7.
Les modules de petit
1.7.1.
hauteur
Pour tout r G N, soit ^Af5 (TT) (resp. #Af^(7r)) la sous-categorie
pleine de ^M^g{7r) (resp. 4WI(f^ (S, TT)) formee des objets M tues par p. T h e o r e m e . Tout objet categorie
est abelienne,
de la categorie ^Af^~ la restriction
(TT) est normal
de j * a ^ M ^ ~ (TT) est
fidele et induit une equivalence entre cette categorie restriction de j * est un quasi-inverse.
et ^M^^
Cette pleinement (TT). La
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291
P r e u v e . Toutes ces assertions resultent facile me nt de la premiere. II suffit done de verifier que, si M est un objet de ^Af^~ (TT) et si A^ = j^j*M, alors rinclusion M ^^ N est une egalite. Remarquons (cela ne nous sera utile que si MlM ^ k<S>N C N o r ( A r 0 M ) ) . _ Supposons M ^ N\ posons L - N/M, M = M/^M^, N = N/^Na, notons X le noyau de M —^ N. On a un diagramme commutatif
et
0
X
1
y Na
^ La
M
y N
^ L
1 X
i
M^
i
M
y N
I
1
0 0 dont toutes les lignes et les colonnes sont exactes. Comme Si — A:[[7r]] et comme L est un 5i-module non nul de longueur finie, il existe un entier c/ > 1, des entiers r i > r2 > . . . > r^ > 1 et des elements e i , 6 2 , . . . ,ed E L tels que Ann(ej) = (^'^O et L = 0 i < i < d 5 i e f . On a alors L^ = ^i
292
JEAN-MARC FONTAINE
i^e\ = ^aije'j
-f 6,-,
ou les 6j G M et la matrice des a^j est une matrice carree inversible a coefficients dans k. Soit M = M/TTM] c'est un Ar-espace vectoriel de dimension finie que I'on pent considerer comme un ^-module sur k. Si 6j designe Timage de bi dans M, le systeme d'equations
a une solution dans k\ en effet, si Ton choisit une base de M sur /?, on obtient des equations etales (grace au fait que la matrice des aij est inversible) qui ont une solution (puisque Ton a suppose k algebriquement clos). II est immediat que Ton pent relever les xi en des xi G M de fagon que ^Xi - ^aijXj
= hi.
Autrement dit, quitte a remplacer e'- par e^- — xi, on pent supposer que les hi sont nuls. Mais alors, si on pose e'l — ire'-, on voit que le sous-S*!module M' de M engendre par les e'/ est un facteur direct de M stable par yp, done est un sous-objet strict de M. Comme
on a
1.7.2. Le theoreme precedent implique que, si M est un objet de ^M'g(q)^ admettant une filtration M = M^D M^ D "-M'
D M*+^ D • • •
telle que fl M* = 0 et chaque M^/M^'^^ est un <^-module p-etale de qhauteur < p— 1, alors M est normal. En outre la sous-categorie pleine de $Af J(g) formee de tels M est abelienne. On en deduit en particulier la proposition suivante: 1.7.3. Proposition. Soient g G 5 — pSj e{q) le plus grand entier e > 0 tel que TT^ divise rimage de q dans Si et r £ N verifiant r.e(q) < (p — 1)"*". Alors, tout objet de ^M^g{q) est normal, la categorie ^M^g{q) est abelienne, la restriction de j * a^M^g{q) est pleinement Gdele et induit
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
293
une equivalence entre cette categorie et ^MQ (S,q); la restriction de j * est un quasi-inverse. 1.8.
Representations de hauteur finie
1.8.1. Dans ce n°, on reprend les notations du §A2. On note en outre R Tanneau des entiers de E; on ecrit Fr R au lieu de E. On a done Oc^ C WCFi R). On pose RQ = R^E et on note Fr RQ = (Fr R)^E son corps des fractions. On sait ([Ax70]) que Fr RQ est le complete de la cloture radicielle de E = ^((TT)) et que RQ est I'anneau de ses entiers. On pose aussi A+ = W{R) n O^^^ C W{Fv R) et
Bj = A+[l/p] = WK.iR) n ^nr C
WK,{FT
R).
Pour tout Zp-module M et tout entier n > 1, on pose Mn — M/p'^M et Moo = lim.ind. Mn = {%/Ip) (g) M. En particulier, O^ ^ = £nr/0^ , W{R)oo = WK,iR)/W{R) et Al^ = Bj/A^ On choisit q E S — pS tel que Vimage de q dans Si est une uniformisante. 1.8.2. Pour tout <^-module A sur 5, on note Fs{h) (resp. Et^(A)) Tensemble des sous-5-modules de type fini de A, stables par (^, qui sont p-etales (resp. et de ^-hauteur finie). On pose j * A = UiV6F5(A)A^
On a J** A C j*N, avec Tegalite pour la topologie p-adique et sans i*(A[l/p]) = 0 ; A ) [ 1 / P ] . Si A' C j^A' = j^A (resp. jiA' = j*A). un endomorphisme de la structure anneaux.
et
jlK = Uue EtqiA)N.
si A est de p-torsion. Si A est separe p-torsion, jt(A[l/p]) = (jlA)[l/p] et A et si i,A (resp. j | A ) C A', alors Si A est une 5-algebre et si (p est d'anneaux, jlA et j^A sont des sous-
(a) Proprietes de A J 1.8.3.
P r o p o s i t i o n , (i) On a i*(Fr R) = E^^^ nR=
O^sip-
(ii) Pour tout entier n > I, j^Wn(FT R) = A'^^; (iii) Les anneaux jiW{FT R) et J^W{FT R) sont contenus dans A J et denses dans A^. (iv) Pour tout n > 1, A+^ s'identiGe a Wn{R) Pi O^ ^(C Wn(Fr R)). 1.8.4. L e m m e . Pour tout entier n > 1, j*Wn(Fv R) - Wn{R) fl Osnv^n Gt Papplication naturelle j*Wn-\-i{R) —^ j*Wn{R) est surjective.
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JEAN-MARC FONTAINE
P r e u v e . Soit N G Fs{Wn(PT R)). On va commencer par montrer, successivement que A'' C Wn{R)y puis que A^ C O^ ^ = Ofnr.n (ces deux assertions prouvant Tinclusion j*VF„(Fr R) C Wn{R) ^0£^j.^n) ^t qu'il existe N' G Fs{Wn+i{Fv R)) dont Timage dans PVn(Fr R) est AT, ce qui prouvera la surjectivite. Tout d'abord, A^ -f S'n-l G Fs{Wn{FT R)). Le sous W'(iJ)-module de VFn(Fr R) engendre par AT + 5n.l est un Wn(^)-module L de type fini stable par (p contenant Wn{R)] si L 7^ Wn{R)y on pourrait trouver x E L de la forme x = ( 0 , 0 , . . . ,0,c) avec c ^ i2 et le sous-W„(iJ)-module engendre par les (p^x = ( 0 , 0 , . . . ,0,c^ ) n'est pas de type fini, et on a hienN CWn{R). Si J\f = Os 0 5 A'^, rinclusion de A^ dans H^n(Fr R) se prolonge en une application O^-lineaire t commutant k (p de Af dans PVn(Fr R). Si J^ est un O^r-module de longue ur r, Hom$.M(^j W^n(Fr R)) est un groupe abelien fini d'ordre p^ [si (ej)i<j
^vec
{aj^t) G GLn {Oe),
et on voit que I'ensemble des solutions, dans ^ ^ ( F r R)^ du systeme d'equations p^j Xj = 0
et
^Xj — 2_]cLj/^£,
pour
1 < i < c?,
a exactement p^ ^J = p^ elements]. Mais p^ est aussi I'ordre de FJ(A/') = Hom^M(-A/', C^^^r.n); Tinclusion Hom4^iif(A/', Ofnr n) C B.oni^M{J^jWn{Fi R)) est done une egalite et teV*^{Af)doncNcOe^r.n^ D'apres le theoreme 1.6.1, il existe un objet M de ^AfJ(^) sans torsion tel que N s'identifie au quotient de M par un sous objet strict M'. La suite exacte courte O-^M'-^M-^AT^O induit une suite exacte courte de (^-modules etales sur Oe ^-^Oe^s
M' -^0£®s
M -^Oe^sN
-^ 0.
On a i G VJ((9£:05 A/'). L'exactitude du foncteur VJ implique que cette application se releve en un homomorphisme de Os ®s M dans Oc^ Son image est un <^-module A''' quotient de M sans p'-torsion et c'est done un objet de ^M'g^q) dont la reduction mod p""*"^ est un element de Et5(M^n+i(Fr R)) = Fs{Wn+i{FT i?)), dont Timage dans Wn(Fr R) est bien N, II ne reste plus qu'a montrer que I'inclusion j^Wn{FT R) C Wn{R) H ^^nr,n ^st uue egalite et nous allons le faire par recurrence sur n.
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295
Si n =: 1, il s'agit de verifier que R Ci Os^^^^i = O^sep C i*(Fr R). Mais O^sep = U OLJ pour L parcourant les extensions finies separables L de E et il suffit de verifier que chaque OL E: FS{FT R). Mais, c'est un 5i-module libre de rang fini, Tapplication x ® y ^-^ x^ y definit un isomorphisme de {OL)(7 sur un sous-anneau de (P~^{OL) et $ : (C7L)<7 -^ OL devient, via cet isomorphisme, le Frobenius usuel; comme il est injectif, OLGF5(Fr R). Si maintenant n > 2, on a une suite exacte courte de 5-modules 0 - ^ Fr iZ - ^ Wn(Fr R) -^ W^n-i(Fr R)-^ Q ^n-l.
(ou la premiere application est celle qui envoie x sur ( 0 , 0 , . . . , 0, x^ )— p^~^x, X etant un relevement quelconque de x dans Wn(Fr R)). On en deduit un diagramme commutatif 0 ->j*Fr R -^ j.WniFv
II 0 ^
O^sep -^Wn{R)
R)
^
j.Wn-i{Fi
i
R)
-^ 0
II n Osnr,n
—Wn-l{R)
H
Osnr,n-l
dont les lignes sont exactes; la fleche verticale du milieu est done bien une egalite. 1.8.5.
Terminons la preuve de la proposition 1.8.3.: Soit N G Pour tout n, son image dans VFn(Fr R) est contenue dans i*PVn(Fr R) = Wn(R) nO^ ; il en resulte que A^ C W(R) D O^ done FS{W{FT
R)).
que j:^l^(Fr i?), et a fortiori jtW{FT R) sont contenus dans A^. Par ailleurs, A^npW{Fv R) = pA^ et A^^ s'envoie injectivement dans Wn(R) n Oc^ . Pour achever la demonstration, il suffit de verifier que riiomomorphisme compose
JtWiFv R)^At^
W„iR) n O-^^^ = i . W„(Fr R)
est surjectif, ou encore qu'il existe M G Etg(P^(Fr R)) dont Timage dans Wn(Fr R) est N. Si Ton choisit M comme dans la demonstration du lemme ci-dessus, on voit que Ton pent trouver un homomorphisme de (pmodules t : O^ 0s ^ ^ W{FT R) relevant t. L'image M de M par t est un quotient de M sans torsion; c'est bien un element de Etg(VF(Fr R)) qui releve N. (b) Representations de p-torsion 1.8.6. On a vu (1.4.2 et 1.5.3) que la sous-categorie pleine $ M ^ ^^^ (resp. ^M'g J.QJ.) de la categorie des ^-modules etales sur Oe (resp. petales sur S) formee des objets de p-torsion est une sous-categorie de $M+^(5,g)(resp. de*M+(
296
JEAN-MARC FONTAINE
Rappelons (n° Al.2.7) que, le foncteur F J qui a un tel -^-module M sur Os associe n{M)=
Hom<,M(M,05,,,oo)
induit une anti-equivalence entre ^Af^ ^^j. et la categorie Rep^.torlG^;) des representations de GE de p-torsion, un quasi-inverse etant m{V)=
HomGe(K,0£„„„o).
Pour tout objet M de $ M J n(M)=
j.^^, on pose Hom*M5(M,4^).
II resulte de la proposition 1.8.3 que Ton a aussi n ( M ) = Hom*Af5(M,Py(i^)oo)
= Hom*Af5(M,M/(Fr ii;)oo). Pour tout objet V de Repp.tor(G£;), on pose D*s{V)=
HomG^(y,vl+^).
II est immediat que V J est un foncteur contravariant additif exact et fidele de ^Mi ^^^ dans Repp.tor(G^£j)) qui s'identifie a V^ o j * et que Z>J est en foncteur contravariant additif pleinement fidele (mais non exact) de Repp.tor(G£;) dans ^ M J ^^^, qui s'identifie a j ^ oD}. En particulier, si r G anti-equivalence entre la torsion, de g-hauteur < r representations de GE de (c) Representations
N est tel que r,e{q) < p — 1, D*g induit une categorie abelienne des <^-modules p-etales de et une sous-categorie pleine de la categorie des p-torsion, stable par sous-objet et quotient.
Qp-adiques
1.8.7. P o s o n s A ^ o = Fv^c W et SRQ = Ko ^w S = S[l/p]. Si M est un ^-module sur 5 , MKQ — Ko<^w ^ ^ ime structure naturelle de 99-module sur SKQ- Si M' et M " sont deux (^-modules sur 5 , on a H o m * M ( M ; , ^ , M ^ J - Q p ( S ) z ^ Horn*Af ( M ' , M ' ' ) • On note I S ^ M J ^ (resp. I s ^ M j ( g ) , I s $ M ^ ( ^ ) ) la sous-categorie pleine de la categorie des v^-modules sur SKQ formee des N tels qu'il existe un objet M de ^Mt (resp. ^ M J ( ^ ) , ^ M ^ ( g ) ) et un isomorphisme de M^Q sur A^. On definit la ^-hauteur d'un objet de Is<Mf J ( g ) comme la plus petite des ^-hauteurs des M tels que MKQ — N.
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297
Si M est un objet de Is^AfJ^, ^{M) = S <S>SK ^ ^st un objet de ^M^. On obtient ainsi un foncteur additif, exact et fidele de I S ^ M J dans ^M£. Le foncteur j * (resp. ^1) est un adjoint a droite de j * (resp. de sa restriction a ls^M'g{q)). 1.8.8.
Rappelons (Al.2.7) que le foncteur M ^ Ve{M) = ^x e ?nr ^£ M \ (px = x^
induit une equivalence entre RepQp(G£;) et la categorie ^ M | des Sespaces vectoriels de dimension finie munis d'un Frobenius de pente 0, le foncteur V i-^Z>£:(F) = (^nr ^Qp V)^E etant un quasi-inverse. On dit qu'une representation Qp-adique de GE est de S-hauteur finie (resp. de q-hauieur finie) siD£{V) Test, et, dans ce dernier cas, on appelle q-hauteur de V celle de D£{V). Pour tout objet M de Is^^AfJ , on pose \rs{M)=
Eom^MsiM^BJ)
et
VsiM)
=
(iTsiM))';
ce sont des objets de RepQp(GjE;). De meme, pour toute representation Qp-adique V de GE, on pose
D*siV) = Eoma^
(M,5+)
et
(lesp.lTs^^iV) = EomG^iMJtWKoi^
DsiV) = (5+ (8)Qp vf^ ,R))
et
Ds^V)
=£>s(V^*) = ITs^.iV*)) ;
ce sont des objets de Is^JWj (resp. Is^M^(q)). La proposition suivante est immediate: 1.8.9. Proposition, (i) Pour tout objet M dels^Mi, on aVg{M) = Hom*M5(M,i*PyKo(Fr R)) = Eom^Ms{M,£nr) = H o m ^ ^ {M,WKQ{R)) Eom^Mg{M, WKQ(FY R)); si M est dans Is^M^{q), on a aussi V*s{M) = Eom^MsiMJlWKoi^v R)). (ii) OnaVs=Veoj\Ds=j.oDe et Ds^q = jl oDe. (iii) Pour toute representation Qp-adique V de GE, DS{V) (resp. Ds,q{V)) est un Sx^-^^odule libre de rang fini inferieur ou egal a la dimension de V sur Qp; on a Fegalite si et seulement si V est de S-hauteur finie (resp. de q-hauteur finie). 1.8.10. Remarques. (i) On a bien sur des resultats plus precis lorsque Ton se restreint aux <^-modules ou aux representations de g-hauteur < r. Remarquons en particulier que la proposition 1.7.3 montre que, si r.e{q) < p — 1, la categorie ls^M^g{q) est abelienne et que Vs induit une
=
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JEAN-MARC FONTAINE
equivalence entre cette categorie et celle des representations Qp-adiques de ^-hauteur < r. (ii) Si Ton suppose satisfaites les conditions de la proposition 1.5.6, et si en outre, on suppose qu'il existe un ideal premier non nul q de 5, distinct de (TT) et de (g), divisant (CTTT), il resulte facilement de cette proposition que la categorie Is#MJ(g) est abelienne, la restriction de j * (resp. Vs) a cette categorie etant un foncteur pleinement fidele induisant une equivalence entre cette categorie et la sous-categorie pleine de ^M^ dont les objets sont de g'-hauteur finie (resp. la categorie des representations Qp-adiques de GE de g-hauteur finie). (d) Le diciionnaire covariant 1.8.11. Supposons maintenant Ihdeal (TT) de S stable par a et an E. {q). Si M est un S'-module sans 7r-torsion, A'g (8)5 M s'identifie a un sousAj-module de Aj[l/7r] (S>s ^' Si M est un objet de #AfJ(g), on pose Vs{M) = {x e A'^[l/7r] 0 5 M I 3 r G N avec TT^'X £ A'^ 0S M et (p^TT^x) = (cr7r)^a:}. On voit que Vs est un foncteur exact et fidele de <^M'g{q) dans Repzp(G^). 1.8.12. Proposition. Soit M un objet cfe^AfJ(g). (i) Si M est de p-torsion^ V'^{M) s'identifie a (VsiM))*
~
Fo/n2p(F5(M),Qp/Zp).
(ii) Le (p-module M[l/p] est dans Is^M'^{q) et Vs{M[l/p])=Vs{M)[l/p]. P r e u v e . Soit M = j*{M) = Os 0 5 M. Si M est de p-torsion, on voit que Vs{M) s'identifie a Ve{M) tandis que V*s{M) s'identifie a FJ(A^), dou (i). L'assertion (ii) s'en deduit par passage a la limite. 1.8.13. Si 1^ est une representation de GE de p-torsion, il existe M dans ^ M j ( g ) tel que V c:^ Vs{M). On appelle q-hauteur de V la plus petite des g-hauteurs de tels M. Le theoreme 1.6.1 implique le result at suivant: T h e o r e m e . Toute representation V de p-torsion de GE Gst isomorphe a un sous-quotient d'une representation Qp-adique V de GE q-hauteur finie. On peut choisir V de meme q-hauteur que V. B2.
Les cas e — 1
Dans ce paragraphe, on reprend les hypotheses et notations du paragraphe A3, avec K — KQ. On pose TT = TTQ, et on note E le corps i((7r)).
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299
Ceci nous permet de reprendre aussi les notations du paragraphe Bl. On pourra verifier qu'elles ne se contredisent pas. En particulier, S = W[7r} C W{R), q = Tr + p. Comme au §A3, on note Os (C W{Fr R)) le separe complete pour la topologie p-adique de S[l/q] et S le corps des fractions de Os. On pose F = TK — Ga,l{Koo/J^)- On pose aussi SK = S[l/p] = K ®w S. 2.1.
Les ip-T-modules de hauteur finie
2.1.1. Soient Ai et A2 deux anneaux commutatifs. On suppose chacun d'eux munis d'un endomorphisme note a et d'une action de F commutant a celle de a. Soit a : Ai —^ A2 un homomorphisme d'anneaux commutant a cr et a F. Si M est un (^-module sur Ai, on a defini au n^ A.1.1.6, a*{M) — A2 ®Ai ^ qui est un ^-module sur A2; si M est un (p-Tmodule, Faction de F s'etend par semi-linearite a a*{M)y ce qui fait que a* pent aussi etre considere comme un foncteur covariant additif
2.1.2. On note j : S ^ Os Finclusion et i : S = W[7r} -> W^ la reduction mod TT. Remarquons que i n'est autre que la restriction a S de Fhomomorphisme 9c : t^I^Te] —> W, qui est Funique homomorphisme de V7-algebres qui envoit [e] sur 1. On voit que 6ej done a fortiori i commute a Faction de <^ et F (ce dernier groupe agissant trivialement sur W). Comme il en est de meme de j , pour tout ^-F-module M sur 5, j*{M) (resp. i*{M)) a une structure de 9?-F-module sur Os (resp. W), Si M est un <^-F-module sur 5, on dit que Vaciion de F sur M est i-unipoiente si F opere trivialment sur i*{M), 2.1.3. Soit 5 ' = (p'^iS) C W{R). L'application 0^ se prolonge en un homomorphisme d'anneaux, commutant a GK
en envoyant [e'] sur €^^\ On voit que le noyau de la restriction de 9^ = 9 o (f~^ a 5 est Fideal engendre par q, qui est done aussi stable par F (mais pas par (p). On note MWlf J la sous-categorie pleine de la categorie des ^p-F-modules sur S dont les objets sont p-etales, de ^-hauteur finie et tels que Faction de F est i-unipotente et PMf J^^^. la sous-categorie pleine de F ^ M j formee des objets de jp-torsion. Ce sont des categories additives Z^-lineaires. On note I S F ^ M J la categorie Qp-hneaire deduite de F^AfJ "en rendant p inversible." On pent la voir aussi comme la sous-categorie pleine de la categorie des <^-F-modules sur SK formee des M tels qu'il existe un objet A'' de r$Af^ et un isomorphisme de N^ = K <Sfw ^ sur M. 2.1 A. Si M est un objet d e l ^ M J , on dit qu'un sous-<;^-F-module de M (resp. un quotient) est strict si le <^-module sous-jacent Fest (cf. n° 1.3.6).
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JEAN-MARC FONTAINE
On definit la hauieurh(M) € N d'un objet M deV^M^ (resp. IsI^MJ) comme etant la g-hauteur du v^-module p-etale de g-hauteur finie sousjacent (cf. n^ 1.5.2. et 1.8.7). On voit que, ici encore, si M est d a n s F M f J et si r > 0 est le plus petit entier tel que q^ annule M/^(Ma)j - si M n'a pas de quotient strict (dans la categorie P ^ M j ) non nul M " tel que ^M'J) C q''M", alors h = r; - sin on h = r^. Pour tout r G N, on note T^M'^g (resp. I^Af^t^j., resp IsI^M^) la sous-categorie pleine de P^AfJ (resp. r^AfJ^^j., resp. IsPMf J ) formee des objets de hauteur < r. 2.1.5. Pout tout ^-F-module A sur 5, on note Fcr(A) (cr = cristallin) I'ensemble des sous-5-modules de type fini de A, stables par <^ et F, qui sont des objets de I^AfJ et on pose
Proposition, (i) Pour tout objet M deV^M^, j*{M) est un (p-T-module etale sur Oe • I/^ foncteur
j* : r$M+ -^ r$Mg^ est exact et fidele. (ii) Pour tout (p-T-module M etale sur Os, Jl^M est un objet d e l ^ M j . (iii) Les foncteurs j * et jl^ sont adjoints. (iv) Pour tout objet Af de T^M^ , j*Jl^M -^ M est un monomorphisme. Si Af est sans torsion, j^^Af est un S-module libre de rang inferieur ou egal au rang de Af sur Os et Papplication j*j^^Af —^ Af est un isomorphisme si et seulement si ces deux rangs sont egaux. Demonstration. Ce sont des consequences immediates du theoreme 1.4.2, sauf peut-etre le fait que, si Af est sans torsion, alors j^^Af est libre sur S. Pour cela, il suffit de remarquer que, si N E Fcr{Af), il existe N' G Fcr{Af) qui contient A^ et qui est libre. On a deja vu au n*^ 1.5.5 que si Af' - Oe 0 5 N, al ors A^' = N[\lp]r\Af' s'identifie a un sous- S'-module libre de Af contenant N^ stable par (p et de g-hau teur finie. II est clair que N' est stable par F, et il suffit de verifier que Faction de F sur A^' est i-unipotente. Si s est un entier tel que p^ N' C A^, comme i*{N') est un FP^-module libre, la multiplication par p^ induit par passage au quotient, une application injective de i'{N') dans i*{N) et F opere trivialement sur z*(^')2.1.6. On dit qu'un objet Af de FMf^ est de cr-hauteur finie s'il est dans Fimage essentielle de j * et on appelle alors cv-hauteur de Af la plus petite des hauteurs des N dans r $ M j tels que j * N c^ Af. On note
R E P R E S E N T A T I O N S p-ADIQUES DES C O R P S LOCAUX T^MQ
^J. (resp. F M f ^ ^j.) la sous-categorie pleine de T^M^
301
formee
des objets de cr-hauteur finie (resp. < r ) . La categorie r ^ M j n'est pas abelienne, parce que rapplication injective
n'est pas toujours un isomorphisme. On a toutefois le resultat suivant: 2.1.1. P r o p o s i t i o n . Pour tout objet N de r$Af^~ , rapplication N —^ j ^ ^ o j*N est un isomorphisme. Pour tout r £ N verifiant r < p — 1, la categorie F ^ M ^ est abelienne, j * induit une equivalence entre cette categorie et T^MQ ^^ et la restriction de j ^ ^ a T^M^ ^^ est un quasi-inverse. Si M est dansT^M^Q ^^, les inclusions naturelles j^^{^f) C ji{^J') C 3*{M) sont bijectives. Preuve.
Cela resulte immediatement de la proposition 1.7.3.
2.1.8. La situation est nettement plus agreable lorsque Ton travail a isogenie pres. Rappelons (cf. A3.4.4) que Ton a note T^M^ la categorie des ^-F-modules sur S de dimension finie sur S dont le ^p-module sousjacent est de pente 0. II est clair que la correspondance M — i > j*{M) = £ 05^^ M definit un foncteur additif f : IsraMj ->raM^, et que j ^ ^ definit un adjoint a droite. On dit qu'un objet Mi de T^M^ est de cr-hauteur finie si I'application
est un isomorphisme et on appelle alors cr-hauieur de M. la hauteur de Jl^i-M)' On note T^M^^^ la sous-categorie pleine de IXfeM^ formee des objets de cr-hauteur finie et, pour tout r G N, IX^M^ ^^ la sous-categorie pleine de I^M^^^. formee des objets de cr-hauteur < r. Les categories obtenues sont stables par somme-directe, sous-objet, quotient. 2.1.9.
T h e o r e m e . (i) Pour tout objet N d e l s I ^ M j ,
rapplication
est un isomorphisme. (ii) Pour tout objet M de F^Af^, on a dim^ j ^ A ^ < dim^: A^, avec Vegalite si et seulement si M est de cr-hauteur finie. (iii) La categorie I s P M f J est abelienne et le
foncteur
302
JEAN-MARC FONTAINE
est pleinement fidele et induit une equivalence entre ces deux La restriction de j ^ ^ aT^Af^ ^^ est un quasi-inverse.
categories.
Preuve. Compte-tenu de ce que I'on sait deja, la seule chose a prouver est (i). Posons M = Ji^j*iN)] Tapplication TV ^^ M est injective et nous notons M" le conoyau. On a vu au n°2.1.2 que I'ideal de 5 engendre par TT est stable par a. Avec les notations du n°2.1.3, on voit que 6'(air) = ^(TT) = 2(7r) = 0, done que air appartient au noyau de 6' qui est I'ideal engendre par q. On p e n t done appliquer la proposition 1.5.6 et il existe un entier r tel que TT^ annule M". On se convainc facilement qu'il existe un caractere, d'ordre infini
tel que, pour tout g ET, ^(TT) = T]{g).7r (mod TT^) (en fait rj est la puissance (p— l)-ieme du caractere cyclotomique). Pour tout 5-module A, i*(A) = A/TTA et la suite exacte
induit une suite exacte 0 - ^ M'^{rj) --^ e{N)
-^
e{M)
- > M ' V T T M ' ' - ^ 0,
compatible avec Taction de F. Par hypothese, Faction de F est triviale sur i*{N) et i*{M). Elle doit done I'etre aussi sur M"I'KM" et se fait via ry"-^ sur M^'. Si celui-ci n'etait pas nul, on aurait Ann ( M " ) = (TT''), avec r u n entier > 1. La multiplication par TT^'^ induirait alors une application non nulle
mais e'est impossible puisque F agit sur le premier /\-espaee vectoriel via rf~^ et via ry~^ sur le second. 2.2. (a)
Representations
de GK de CT - hauteur
Representations
fime
Qp-adiques
2.2.1. Si M est un objet de I s P M f J , les Qp-espaces vectoriels Vs{M) et V*g{M) definis au n° 1.8.8 sont munis d'une action naturelle de GKDe meme, si V est une representation Qp-adique de GK^ Ds.criV)
= JTDsiV)
et
Dl,,iV)
=
j^'ITsiV)
sont des objets de IsF^AfJ. Disons qu'une representation Qp-adique V de GK est de cr-hauteur finie s'il existe M dans I s P M / ^ tel que V ~ Vs{M). Ces representations
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
303
forment une sous-categorie pleine RepQ ^cr(^^) ^^ ^^P^piGx), stable par sous-objet, quotient, somme-directe, produit tensoriel. La proposition suivante resulte de la proposition 1.8.9 et du theoreme 2.1.9: 2.2.2. Proposition, (i) Pour toute representation Qp-adique V de GKI Ds,cr{y) ^st un SK-module libre de rang fini inferieur ou egal a la dimension de V sur Q^; on a Fegalite si et seulement si V est de cr-hauteur finie. (ii) Le foncteur
est pleinement fidele et induit une equivalence entre ces deux categories, la restriction du foncteurDs^cr a R e p i CJ,{GK) ^n etant un quasi-inverse. Si V est une representation Qp-adique de GKJ de cr-hauteur finie, on appelle cv-hauteur de V la cr-hauteur de Ds^cr{V). Pour tout r G N, on note HepQ ^^J,{GK) la sous-categorie pleine de Rep Q criGx) formee des V de cr-hauteur < r. (b)
Representations Tp-adiques
2.2.3. Les resultats du n°1.8 ont une traduction evidente en termes de representations Zp-adiques de GK et de ^-F-modules. Contentons-nous de remarquer que, si M est un objet de I ^ M J (resp. et de p-torsion), le Zp-module Vs{M) (resp. V*s{M)) defini au nn.8.11 (resp. 1.8.6) a une structure naturelle d'objet de Repzp(G/<'). Disons qu'une representation Zp-adique V de GK est de cv-hauteur finie si elle est isomorphe a un sous-quotient d'une representation Qp-adique V de GK de cr-hauteur finie et appelons alors cr-hauteur de V le plus petit r G N tel que Ton puisse choisir V de hauteur r. Notons R e p J CAGK) (resp. Hepj ^J.{GK)) la sous-categorie pleine de Repzp(Gi<:) formee des V de cr-hauteur finie (resp. < r). La proposition suivante resulte facilement des n°s 1.8.10, 1.8.12 et 2.1.7: 2.2.4. Proposition. Soit r G N verifiant r < p — 1. La restriction du foncteur Vs induit une equivalence entre les categories ^TM^^ et R-ep^2p,cr(GK)-
2.3.
Le lien avec les representations cnstallines (resultats et questions)
2.3.1. Appelons (cf. [FL82], Section 1) (p-module filtre positif sur W la donnee d'un PF-module D muni - d'une part, d'une filtration decroissante {FpD)i^f^y par des sous-W^modules, verifiant F^D = D et Di^N^p^ = 0; - d'autre part, d'une famille
304
JEAN-MARC FONTAINE
^;
-.F^D^D
d'applications cr-semi-lineaires verifiant (Pp{x) = p.(p]^^{x)j pour tout x G Ces (^-modules filtres positifs sur W forment, de maniere evidente, une categoric Zp-lineaire que nous notons MFy^. Notons MFIY' la sous-categorie pleine de MF^ formee des D tels que le WP^-module sous-jacent est de type fini, les F!^D sont des facteurs directs et D — Yl^pi^p^)' C'est une categorie abelienne {op cit., prop. 1.8). Notons MF'j^ la sous-categorie pleine de MFjy formee des D tels que les FpD sont des A'-espaces vectoriels (les (p^ sont alors determines par (p =z (p^^ via 0 tel que Fp'^^D = 0]si D n'a pas de quotient non nul D' tel que F^D' = D', alors la hauteur de D est r; sinon, c'est r"^. Pour tout r G N, on note MF^ (resp. MFj^^) la sous-categorie pleine de MF^^ (resp. MFj^^) formee des D de hauteur < r. 2.3.2 Si C designe le complete de A, on dispose d'un homomorphisme continu surjectif de VF-algebres e : W{R) -^ Oc : (n)
c'est celui qui a (xo,a:i,... , x„,. ..) associe ^ p ^ x „ . Notons A^ns 1 anneau (souvent note W^^{R), cf. [Fo83a], n°3.6) qui est le separe complete pour la topologie p-adique de I'enveloppe a puissances divisees de W{R) relativement a I'ideal Ker 9 et, pour tout i G N, FiPAcris I'adherence de la i-eme puissance divisee du pd-ideal correspondant. L'anneau Acris est sans p-torsion et s'identifie a un sous-anneau de ^^^g = K ^w ^cris = Acris[l/p]- L'action de GK et celle de (p s'etendent de maniere naturelle a ylcris et a S^is- On munit alors Acns (resp. 5^is) d'une structure d'objet de MF^^ (resp. MF^), compatible avec Taction de GK^ en posant F;A^r,s
2.3.3.
= {xe
F i P A e n s | ^X G P ^ e n s } et F;B^^^
= K ^w
FiP'^cns •
Pour toute representation Q^-adique V de G^, posons
rappelons ([Fo83b], §5) que c'est un A'-espace vectoriel de dimension finie, inferieure ou egale a la dimension de V sur Qp, muni d'une struc-
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
305
ture naturelle d'objet de MF^. Nous disons que V est cristalline positive si ces deux dimensions sont egales. Notons R e p i cris(^^) ^^ ^^^^ categorie pleine de RepQ (Gi<:) formee des representations cristallines positives. Alors {loc, cit.), R e p i cris(^^) ^^^ stable par sous-objet, quotient, 0 , (g). Par restriction, D^^^^ induit une equivalence entre RepQ ^cns(^^) ^^ une sous-categorie pleine MFj^ de MFjf^, stable par sous-objet, quotient, 0 , 0 , dont les objets sont appeles les (p-modules filtres admissihles sur K. La correspondance D ^
y c r i s ( i ^ ) ={xe
F^ {Bens ^ D)
\ ifX = x}
(oil 5cris peut-etre defini comme Tanneau obtenu a partir de 5^-g en rendant TT inversible) est un quasi-inverse. Si V est dans R e p i cris(^-K')5 appelons hauteur cristalline de V la hauteur de D^^-^^{V). De meme, disons qu'une representation Z^-adique V de GK est de hauteur cristalline finie si elle est isomorphe a un sousquotient d'une representation Qp-adique cristalline positive V et appelons alors hauteur cristalline de V la plus petite des hauteurs cristallines de tels V. Pour tout r G A^, on note RepQ^ cris(^-ft:) (resp. RepJ^ ^nsC^i^)) la sous-categorie pleine de RepQ cris(^^) (^*esp. Ilepzp{GK)) formee des V de hauteur cristalline < r. 2.3.4. Rappelons enfin que Ton conjecture que tout objet de MFj^'^ est admissible et que Ton sait que c'est vrai pour tout objet de MFjf~ . Cela resulte de la proposition suivante, essentiellement contenue dans ([FL82] thm. 3.3 et 6.1): P r o p o s i t i o n . Soit r < p — I. Pour tout objet D de MF^f, Vcns{D)
^ [x e F^ (Acns ^W
D)
\ (^^(x) = x] ( - r )
(ou (—r) designe la "torsion a la Tate" usuelle) est un objet de ^ ^ P z cris(^^)- ^^ correspondance D i-^ Vcns{D) induit une equivalence entre MF{^' et Rep^2^,,,,3(Gx). 2.3.5. Soit M un objet de ^ M j ( g ) (resp. Is^Af J(g)). Alors D = z*(M) = M / T T M a une structure naturelle d'objet de MF^ (resp. MF^): pour tout i G N, on definit F^D comme etant I'image du sous-5-module F^M forme des x tels que (px £ q^M (attention: il faut prendre I'image de F^M et non I'image du sous-S'-module F^M^ de M^ defini au n° 1.6.2); si X G FpD est I'image de x G F^M^ on definit
306
JEAN-MARC FONTAINE
II est clair que Fon peut considerer i* comme un foncteur additif exact, de^M^{q) (resp. Is^Mj(g)) dans AfF^ (resp. M F j ) . 2.3.6. En composant avec le foncteur "oubli de Taction de F" on obtient une recette fonctorielle pour cissocier un (^-module filtre sur W (resp. sur K) a tout objet de PMfJ (resp. I S P ^ M J ) . Dans la deuxieme partie de ce travail, nous prouverons le result at suivant: Theoreme. (i) Pour tout objet M de IsF^AfJ, r ( M ) est un objet de MF^^'^ et la, hauteur de i*{M) est egale a celle de M. (ii) Le foncteur r : Isr$M+ - . MF^^'+ est pleinement fidele et induit une equivalence entre I S T ^ M J et une souscategorie pleine de MF^' stable par sous-objet et quotient. (iii) Les foncteurs Vs^Veoj* sont naturellement
et
l^cnsor:IsreAf+-.RepQ^(GK)
equivalents.
En particulier, toute representation Qp-adique de cr-hauteur finie est de hauteur cristalline finie (et la cr-hauteur de cette representation est egale a sa hauteur cristalline). II ne me parait pas tres difficile de prouver et il me semble done raisonable de conjeciurer que la reciproque est vraie, autrement dit que le foncteur 2* : Isr$M+ ^ MF^^'-^ est essentiellement surjectif. Pour prouver simultanement cette conjecture et celle qui affirme que tout objet de MFj^ est admissible, il suffirait de verifier que tout objet de MFj^'^ est dans Timage essentielle de f*. 2.3.7. Comme on le verra dans la deuxieme partie de ce travail, on dispose de resultats partiels dans cette direction (qui donnent en outre une autre demonstration des resultats de [FL82]): Proposition, (i) Pour tout objet D deMFff~ tel que D ~ i*{M).
il existe M dans IsV^M^
(ii) Pour tout objet M deT^M^f^, i*{M) est un objet de MF{^~^, dont la hauteur est egale a celle de M; en outre i* induit une equivalence entre T^M^g~^ et MF{y^^. Les foncteurs Vs^Veoj"^ sont naturellement
et
Fcrisoi* : m M + - . R e p 2 ^ ( G x )
equivalents.
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX
307
Principales notations 0 0.1: RepQp(G), Repip(G), Repp.tor(G), RepFp(G). 0.2: 0:F, W„iA), WiA), [a], ip, a. A 1.1: M„, ^MA,O =^MA 1.2 E''^, ^("), E^'P, 2.0: Jb, W, Ko, K. 2.3: N^, T^, Col. 2A:6^[x,u). 3.0:
= * M , A^[p], A„, *Af^'.
GE, S,
Snr, Snr, De, Ve, IT,, V},
iM°.
K,GL,C.
3.1: R, X,
EQ, SO-
3.2: WA{R),
T „ SO, SO, ^O-
3.3:r^M^,r*Af^', 5, 3.4: r * M ^ .
H,T.
B 1.3: *M+p. 1.4:i*,i.,*M+^(5). 1.5: N, *M+(g), *JlfJ( ^Mi^{S,q),
*A/^^(5,g).
1.6: F^M„, C / M , Car. 1.7: e(
2 1: isTOAf, r*jif+, r$M+^„„ isr*M+, i - , r$M+^ ,„ r$M+„. 2.2: R e p + p „ ( G K ) , R ^ p + t o r , crCCic),-D*S,cr-
2.3: ylcris, ^cris.oc, S+i^, MF{^-^, JlfF^+, MF^^'+, Repii3(G*), Rep;.t,,(Gx). BIBLIOGRAPHIE [Ab89] [Ax70] [BK89] [Bu88]
Abraskin, V. A., Ramification in etale cohomology, manuscrit, 1989. Ax, J., Zeros of polynomials over local fields. The Galois action, 3. Algebra 15 (1970), 417-428. Bloch S., and Kate, K., L-functions and Tamagawa numbers of motives, preprint, 1989. Seminaire sur les periodes p-adiques, IHES, Bures-sur-Yvette, 1988, en preparation.
308 [Co79] [C082] [Co85] [Co87] [De72] [DM82]
[Fa88a] [Fa88b] [Fo77] [Fo83a]
[Fo83b] [FL82] [FM87] [FW79]
[Gr70] [Ka89] [La78] [Sa72] [Sen80] [Se58] [Se68]
JEAN-MARC FONTAINE Coleman, R., Division values in local fields, Inv. Math 53 (1979), 91-116. Coleman, R., Dilogarithms, regulators and p-adic Abelian integrals, Inv. Math 69 (1982), 171-208. Coleman, R., Torsion points on curves and p-adic Abelian integrals, Ann. of Math. 121 (1985), 111-168. Coleman, R., letter to B. Mazur, March 1987. Demazure, M., Lectures on p-divisible groups, Lecture Notes in Math. 302, Springer, Berlin, 1972. Deligne, P. and Milne, J. S., Tannakian categories, in Hodge Cycles, Motives and Shimura Varieties, Lecture Notes in Math. 900, Springer, Berlin, 1982, 101-228. Faltings, G., Crystalline cohomology and p-adic Galois representations, preprint 1988. Faltings, G., F-isocrystals on open varieties, results and conjectures, preprint 1988. Fontaine, J.-M., Croupes p- dwisibles sur les corps locaux, Asterisque, 47-58. Soc. Math, de France, Paris, 1977 Fontaine, J.-M., Cohomologie de de Rham, cohomologie cnstalline et representations p-adiques, in Algebraic Ceometry Tokyo-Kyoto, Lecture Notes in Math. 1016, Springer, Berlin, 1983, 86-108. Fontaine, J.-M., Representations p-adiques, Proc. Int. Congress Math., (1983), Warsaw, 475-486. Fontaine, J.-M. and LafFaille, G., Construction de representations p-adiques, Ann. Scient. E.N.S., 4° serie, 15 (1982), 547- 608. Fontaine, J.-M. and Messing, W., p-adic periods and p-adic etale cohomology, Contemporary Math. 67 (1987), 179-207. Fontaine, J.-M. and Wintenberger, J.-P., Le ^'corps des normes" de certaines extensions algebriques de corps locaux, C.R.A.S. 288 (1979), 367-370. Grothendieck A., Croupes de Barsotti-Tate et cristaux, Actes Congres Int. Math. Nice 1970, t.l, Gauthiers-Villars, Paris, 1971. Kato, K., The explicit reciprocity law and the cohomology of FontaineMessing, Bull. Soc, Math. France, a paraitre. Lang, S., Cyclotomic Fields. Springer, Berlin, 1978. Saavedra Rivano, N., Categories Tannakiennes, Lecture Notes in Maths 265. Springer, Berlin, 1972. Sen, S., Continuous cohomology and p-adic Calois representations, Inv. Math. 62 (1980), 89-116. Serre, J.-P., Classe des corps cyclotomiques, Seminaire Bourbaki, exp. 174, (1958), Benjamin, New York, 1966. Serre, J.-P., Corps locaux,, 2°ed., Hermann, Paris, 1968.
REPRESENTATIONS p-ADIQUES DES CORPS LOCAUX [Wi83]
[Wit37]
309
Wintenberger, J.-P., Le corps des normes de ceriaines extensions infinies des corps locaux; applications^ Ann. Sci. E.N.S. 16 (1983), 59-89. Witt, E., Zyklische Korper und Algehren der Charakteristik p vom Grad p"" ^ Struktur diskret bewerteter perfekter Korper mit vollkommenem Restklassenkorper der Characteristik p (sic), J. reine ang. Math. 176 (1937), 126-140.
Universite de Paris-Sud Centre d'Orsay Departement de Mathematiques - Bat. 425 91405 Orsay Cedex, France
Rectified Homotopical Depth and Grothendieck Conjectures
HELMUT A. HAMM and LE DUNG TRANG In honor of Alexander Grothendieck on the occasion of his sixtieth birthday
Introduction In SGA 2 ([G]), A. Grothendieck introduced the notion of rectified homotopical (resp. homological) depth. He conjectured that it gives the level of comparison for the homotopy type (resp. the homology) between a complex algebraic variety and a hyperplane section, as stated in theorems of Lefschetz type for singular algebraic varieties. In the case of non-singular varieties, the rectified homotopical (resp. homological) depth equals the complex dimension of the variety. But in the case of local complete intersections, one can show that this is still true. In fact, using the comparison theorem of Grothendieck as formulated by Mebkhout for X>-modules in [Me], the constant sheaf C. of complex numbers on a variety which is locally a complete intersection is perverse and one can prove that the constant sheaf C. of complex numbers on the variety is perverse if and only if the rectified homological depth for the rational homology equals the complex dimension of the variety. So the rectified homological depth for the rational homology measures how far the constant sheaf C. of complex numbers on the variety is from being perverse. In this paper we give a positive answer to the conjecture of Grothendieck. Actually, we prove all the conjectures given by Grothendieck on this theme in SGA 2, except Conjecture A, which is obviously incorrect as stated, but can be easily corrected.
312
H.A. HAMM and LE D.T.
In fact, the conjectures of Grothendieck cover a greater generality than the situation of a variety and its hyperplane section because he considers the case of a variety and a subvariety such that its complement is Stein. The proofs of the conjectures of Grothendieck become possible because of a more handy formulation of the notion of rectified homotopical depth, using Whitney stratifications, that we give here. We first prove the equivalence of our definition and the one of Grothendieck. This equivalence involves the proof of a local Lefschetz theorem for singular complex analytic spaces. We give a proof of this local Lefschetz theorem by using the local Lefschetz theorem we have already proved in the non-singular case (see [H-L3]). We already used Grothendieck's notion of rectified homotopical depth to get a Lefschetz type theorem for quasi-projective varieties (see [H-L2]). Following our previous work [H-L3], [H-Ll] and [H-L2], it becomes obvious that there are Lefschetz type theorems for open varieties, replacing the hyperplane section by a good neighbourhood of the hyperplane section as has been done by Deligne (cf [Dl]). So we give several theorems of Lefschetz type in the non-proper case and especially a strong local version (Theorem 4.2.1 or its corollary 4.2.2). Some of the theorems proved here fit in the theory of M. Goresky and R. MacPherson and could have been proved by using their stratified Morse theory. In this case the local rectified homotopical depth would appear through the connectivity of the complex links of the strata of a Whitney stratification. Theorem 4.1.2 gives a precise relation between the rectified homotopical depth and the connectivity of these local complex links. The idea of Grothendieck can be extended to define the rectified homotopical depth of complex analytic maps. This will lead to theorems of Lefschetz type for analytic maps. For instance, one can obtain another proof of a conjecture of Deligne, proved by Goresky-MacPherson in [GMl] (Part II, Chapter 1, 1.1), and extend its statement to maps between singular spaces. We shall deal with maps in a forthcoming paper. We wish to thank the referee and Z. Mebkhout for drawing our attention to the link between the cosupport condition for perverse sheaves and the rectified homological depth and B. Teissier for fruitful discussions and his concern about this work. Part of this work was done under the sponsorship of the program PROCOPE of cultural cooperation between France and the Federal Republic of Germany and supported by the NSF grant DMS-8803478.
1. Rectified h o m o t o p i c a l d e p t h . Let X be a reduced complex analytic space. Consider a Whitney
HOMOTOPICAL DEPTH
313
stratification iS of X . Let Xi be the union of strata of dimension < i. We have: 1.1 Definition. We say that the rectified homotopical depth rhds(X) of X is > n if J for any i and any point x in Xi — X,_i, there exists a fundamental system (Ua) of neighbourhoods ofx in X such that^ for any a J the pair {Ua, Ua — Xi) is (n — 1 — i)'Connected. Of course, the rectified homotopical depth rhds{X) maximum of the integers n such that rhds{X) > n.
of X ,£ 0 is the
Remarks. 1.1.1 First notice that, if V is any good neighbourhood of x in X relatively to Xi in the sense of D. Prill ([P]), there is a fundamental system of neighbourhoods Ua of x such that the pairs (Ua, Ua — Xi) are (n — 1 — i)connected, if and only if the pair (V, V — Xi) is (n — 1 — i)-connected. So in practice to calculate the rectified homotopical depth it will be often better to consider a good neighbourhood of a? in X relatively to Xi instead of a fundamental system of neighbourhoods Ua as above. For instance, assuming X locally embedded in some C in a neighbourhood of the point x, for any e > 0 small enough, the intersection of X by the closed ball Be{x) of C ^ centered at x with radius € is known to be a good neighbourhood of x in X relatively to Xi. 1.1.2 Moreover we can consider a normal slice Af of Xi at ar in a suflftciently small neighbourhood of x in X (see [G-Ml] Part I §1.4 definition). Because the stratification «S is a Whitney stratification, for any point x in X,- — Xj_i there is a good neighbourhood F of a: in X relatively to Xi such that (Vj V — Xi) is homeomorphic to the product of the pair (A/',JV — {x}) by an open neighbourhood t/ of a: in X,-, and moreover the homotopy type of such pairs {M^M — {x}) is the same for all the points of a connected component of Xi — Xi_i. 1.1.3 In the same way we say that the rectified homological depth of X is > n if, for any i and any point x in X, — Xt_i, there exists a fundamental system {Ua) of neighbourhoods of x in X such that, for any a, we have that Hk{Ua, Ua - Xi\Z) = 0, for any fc < n - i. We shall denote rHds{X, Z) the rectified homological depth of X. For the rational homology we define the rectified homological depth for the rational homology rHds{X,Q) in the same way. Notice that the vanishing of the Hk{Ua,Ua — Xi] Z), for any k < n — i
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implies the vanishing of the local cohomology groups Hx nc/ i^alQ), any k < n — i.
for
Example 1.2. Let X 7^ 0 be smooth and of pure dimension n. Then we have that rhds{X) = n for any Whitney stratification «S of X . The condition rhds{X) ney stratification S :
> n is independent of the choice of the Whit-
Lemma 1.3. Consider another Whitney stratification T of X and let Yi be the union of all strata of dimension < i. Then rhds{X) equals rhdj;{X). Proof: It is enough to prove the lemma when T is finer than 5 , because otherwise we can consider a Whitney stratification which is finer than both. Let 5 be a (sc. connected) stratum of S of dimension i. At any point X of 5, for any good neighbourhood F of x in X relatively to X,-, the pairs {V^V — Xi) have the same homotopy type, because the stratification S satisfies the Whitney condition. Furthermore there is a stratum T of T of dimension i which is open and dense in 5. So/\i rhdr{X) >^ n^ ai the point X oiT C S^ for any good neighbourhood F of a: in X relatively to X,-, the pair {V^V — Xi) is n — 1 — f-connected. This implies that at any point X of 5, for any good neighbourhood V^ of a:; in X relatively to Xi, the pair (V, V — Xi) is n — 1 — i-connected, which means that rhds{X) > n. Conversely, let y he a point of 1} — Yj-i; there is an i such that y is in Xi — Xi__i. Then we have i > j . Let M and A/" denote the normal slices of Yj and Xj at y respectively. Because T is a Whitney stratification finer than S , the space A4 is homeomorphic to the product of AT by a ball of real dimension 2i — 2j. Thus the pair (Af^Af — {x}) being (n — 1 — i)connected, the pair (A^, M — {x}) is (n — 1 — i + (2i — 2j))-connected and n - 1 - i + (22 - 2j) > n - 1 - j . 1.3.1 Because the definition of rectified homotopical depth does not depend on the chosen Whitney stratification, we shall denote it by rhd(X). 1.3.2 We have a similar result for the rectified homological depth that we shall denote rHd{X,Z). 1.3.3 In practice we can start with the canonical Whitney stratification of X (cf [T] VI §3; [L-Tl] 6.1; see [L-T2] (1.3)) to calculate its rectified homotopical (resp. homological) depth. So the definition 1.1 coincides with the one of (2.1.5) in [H-L2].
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315
In fact our definition coincides with the ones of A. Grothendieck in [G] (Expose XIII p.27 definition 2): T h e o r e m 1.4. Let X be a reduced complex analytic space and n a positive integer. The following conditions are equivalent: a) rhd(X) > n; b) If Y is a locally closed complex analytic subspace of X of dimension i, for any point y in Y there is a fundamental system of neighbourhoods [/« ofy in X such that the pair {Ua,Ua — Y) is (n — 1 — i)-connected; c) IfY is a locally closed irreducible complex analytic subspace of dimension i, there is an open dense analytic subset YQ ofY such that J for any point y in YQJ there is a fundamental system of neighbourhoods Ua of y in X such that the pair (Ua, Ua — Y) is (n — 1 — i)-connected. 1.5 Remark. Using the existence of good neighbourhoods of a? in X relatively to any analytic subset in the sense of Prill, the condition b) above is equivalent to the definition given by Grothendieck in [G] (Expose XIII p.27 definition 2). The condition c) corresponds to the condition of Grothendieck following the preceding one in loc.cii. We shall see that the equivalence between our definition and the ones of Grothendieck is a consequence of a local theorem of Lefschetz type. Proof of Theorem 1.4: The implication "b) implies c)" is obvious. The implication "c) implies a)" is obtained by chosing Y as any stratum of a Whitney stratification of X. Let us prove that a) implies b). The question being local we can assume that X is embedded in C , the subspace Y is closed in X and that y = 0. We shall prove that, under the assumption rhd{X) > n, for any e > 0 sufficiently small, the pair {B, H X.B.CiX -Y) is (n - I - i)-connected, with B^ = B,{0) and i = dimY. This gives b), because for any e > 0 sufficiently small, the spaces B^nX form a fundamental system of good neighbourhoods of x in X relatively to Y. Let L be a general linear subspace of codimension f, so, in particular, LDY nB^ = {0}. Suppose that under our assumptions the pairs
(5, n X - y, {Benx-Y)n
L)
and
(Benx-{0},(5,nx-{0})nL)
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are (n - 1 - i)-connected. We have (B^nX -Y)nL = {BeHX - {0}) H L and, by choosing a Whitney stratification of X where {0} is a stratum, the assumption rhd{X) > n implies that
{B,nx,B,nx-{0}) is (n — l)-connected. Now B^H X is contractible (cf [B-V]) so
Benx-{0} is (n — 2)-connected; by this we mean that any continuous map S^ -^BeCiX-
{0},-1
admits a continuous extension over the unit ball B "'"^. Since the pair
{BeHX- {0}, (B, nx - {0}) n L) is supposed (n — 1 — i)-connected, the space
(Be n X - {0}) n L = (5e n X - y) n L is (n—2—i)-connected. This implies that B^C^X—Y is (n —2—i)-connected, because the pair (5e n X - y, (5, n X - y) n L) is also supposed (n — 1 — f)-connected. So
(Be n X, Be n X - y) is (n — 1 — i)-connected. Now the pairs
(Be n X - y, (Be n X - y) n L) and
(Be n X - {0}, (Be n X - {0}) n L) are in fact (n — 1 — i)-connected because of the following local theorem of Lefschetz type. Theorem 1.6. Let X be a reduced complex analytic space embedded in C . Consider a closed complex analytic subspace Z of X. Assume that rhd{X — Z) > n, then for any point z in Z, for any general affine space L through z of codimension i with i > dimzZ, and for any e > 0 smaii enoughj the pair {Be{z)nX — Z,Be{z)nXnL — Z) is (n—l — iyconnected. Proof: The local conic structure theorem for analytic spaces (cf [B-V]) shows that B^{z) OX — Z and B^(z) d X Ci L — Z are homeomorphic to
HOMOTOPICAL DEPTH
317
the product of Se{z) CiX - Z and Se{z) nX r\L- Z respectively by (0,1]. Moreover, as we have assumed L to be general of codimension i, we have
Seiz) nxnL-z
= Se{z) nxnL.
Thus we have to prove that the pair
{Se{z)nx-z,Se(z)nxnL) is (n — 1 — 2)-connected. Now let ^fi = . . . = ^r^- = 0 be i linear equations which define L in C ^ , we denote tp{x) = ^ \9j{^)\^- We can define Va{L) := {x G C^\tp{x) < a} when a > 0. The restriction of tp to Se{z) n X being real analytic and proper, for a > 0 small enough the space Se{z) H X H Va{L) retracts onto Se{z) HX Ci L. On the other hand Se{z) n X n Va{L) is contained in Se{z) HX -Z. Our theorem is therefore a consequence of the following lemma: Lemma 1.7. There is eo > 0 such that for any e, CQ > e > 0, there is a^ such that for any a, a^ > a > 0, the space Se{z) H X — Z has the homotopy type of a space obtained from S'e(z) OX Ci Va{L) by adding cells of dimension > n — i. Proof of lemma 1.7: When X — Z is non singular this theorem was proven by us in [H-L3] (Theorem 1.1.1) (see also [H4] Theorem 2). We are going to use this previous result to prove our lemma. First we stratify the space X by a Whitney stratification adapted to Z (i.e. Z is a union of strata). We call Zk the union of Z and the strata of X of dimension < k. We shall actually prove that the space
Se{z) n [{X - Zk) u {{x -z)n Va{L))] has the homotopy type of a space obtained from Se{z) n [{X - z,+i) u {{X -z)n
V^{L))]
by adding cells of dimension > n — i. In fact, as ^^(z) fl ^ fl L = 0 when e > 0 is small enough, with a > 0 small enough, we have the equality
Se{z) n{x-z)n
Va{L) = Se{z) n X n VaiL).
Then our lemma is obtained by comparing the spaces for the cases k = 0 and k = dirUzX. Let us fix an integer Ar, dirUzX — I > k > 0. Consider equations / i = ... = /y. = 0 of Zk^i and hi = ... = kg = 0 of Zk in a neighbourhood C/ of z in C ^ . Denote (f>{x) = J^ Ifji^W and x(^) = E l ^ i W P . we define T^{Zk+i) := {x e X nU\{x) < /3}
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H.A. HAMM and LE D.T.
and
T^{Zk)-{xexnu\xix)<j}, We denote dT^iZk^i)
:^{xeXn
U\ix) = /3}
and
dT^{Zk):={xexnu\x{x) = j} 1.7.1 There is eo > 0 such that: i) for any e, eo > 6 > 0, the spheres Se{z) are contained in U and are transverse to the strata of X; ii) the choice of CQ satisfies the conditions 1.2.2 of [H-L3] so that, for any e, eo > 6 > 0, there is ao > 0 such that for any a, ao > a > 0, the space Se{z)n {Zk-\-i — Zk) has the homotopy type of a space obtained from Se{z)n {Zk^i — Zk)r\Va{L) by adding cells of dimension > Ar + 1 — ^ (as it is stated in the Theorem 1.1.1 of [H-L3]). 1.7.2 Fix e, eo > e > 0, and a, ao > a > 0. If 7 > 0 is small enough, the space A^ := Se{z) D [{X - T^{Zk)) U {X - Z) H Va(L)] is a deformation retract of A := Se{z) H [{X - Zk) U (X - Z) fl Va{L)], To prove this statement it is enough to prove that
Se{z) n [(X - T^{Zk)) u (X - Zk) n v^iL)] - V^{L) is a deformation retract of
Se{z)n{x-z„)-VaiL). This is true because Seiz)niX-T^iZk))-Va{L)
is a deformation retract of both spaces: first notice that there is a Whitney stratification of Seiz) n (X - Zk) - Va{L) induced by the Whitney stratification of X; then, since % is real analytic, any 7 > 0 small enough is not a critical value of the restrictions of x to the strata of
s,iz)n{x-Zk)-Vc{Ly, by integrating controlled vector fields, we obtain the retractions. 1.7.3 Now fix such a 7.
HOMOTOPICAL DEPTH
319
Similarly, if y3 > 0 is small enough, the space Bp := Seiz) n [{X - T^{Zk+i)) U{X-Z)n
Va(L)]
is a deformation retract of B := 5e(z) n [(X - Z,+i) L){X-Z)n
VaiL)].
Moreover, with /? > 0 small enough, the space ^7./? := Se{z) n [{X ~ T^{Zk) n Tp{Zk+i)) U{X-^Z)n
VaiL)]
is a deformation retract of A. To prove this last assertion, as we already did above, it is enough to prove that S,(z) n [{X - T^iZk) n TpiZk+i)] - Vc,{L) is both a deformation retract of Seiz)niX-Zk)-Va(L)
and of
s,{z) n [{x - TyiZk) n Tp{Zk+i)) u {{x - z^) n Va{L))] - v^iL) Again the Whitney stratification of X induces a Whitney stratification of
Se{z)n(x-Zk)-v^{L), On each of these strata, on a sufficiently small neighbourhood of Zk, the differential of the restriction of x is non-zero and those of % and are not collinear with a negative ratio (compare to [HI] Lemma 2.13). This observation allows us to build up a controlled vector field which realizes the desired deformation retractions (compare to (2.2.6) of [H-L2]). 1.7.4 Now fix ^ > 0 to get the deformation retractions as above and so that the restrictions of (f> to all the strata of
dT^iZk)niB-VaiL)) induced by the ones of X have no critical values in (0,/?]. This second restriction on the choice of /3 implies that the space
s,iz) n ^p(z,+,) - m{z,) u v„(L))]
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is a topological fibration on
s,{z)n[Zk+i - {mZk)uVa{L)] (see the Appendix). 1.7.5 Now we observe that the union of the spaces Bf^ and
E := Seiz) n ^p{Z^+i) - miZk) U Vaim, where Tp(Zk^i) is the union of T^(Z;k+i) and 9T^(Zjfc_|.i), equals the space By p. According to a theorem of Blakers and Massey (homotopy excision theorem) (cf.[B-M] (Theorem I); see [G-Ml] Part II §4.3 and compare with [H2] (Lemma 2)), if we want to compare the homotopy groups of Bj^(3 and J5^, we have to compare the homotopy groups of E and E H B^. So it is enough to prove that E is obtained from E H Bp by adding cells of dimension > n — i. As we have a Whitney stratification, the space E fibers over
s := s,{z) n [Zk^i - (T^iZk) u Vaim (see the Appendix). But the fibers of E over 5 are nothing else but the normal slices of the strata of X with dimension fe+1. Let Af be one of these normal slices. Then the pair (Af, dAf) is (n — k — 2)-connected, because we have rhd{X) > n by hypothesis, and the pair (5, S H dVa{L)) is (k — i)-connected because S is a deformation retract of Se{z)n[Zk+l-ZkUVa{L)]
and we can apply the theorem 1.1.1 and the rennark 1.1.2 i) of [H-L3] (see the condition ii) in the choice of eo made above). We shall show with the following lemma that this implies that the pair of spaces
(E.EnBp) is ({n —fc— 2) + (fc — f) -f l)-connected, i.e. {n — i — l)-connected. As we deal here with triangulated spaces (see [Lo]), this homotopy result gives us the result with cells as formulated in Lemma 1.7 above (see [Sw] 6.13). It remains to prove: L e m m a 1.8. Let {E,E') be a pair of locally trivial topological fibrations over a CW-complex S, let TT: E —^ S be the corresponding projection. Let {NjN') be a pair of fibers of these fibrations. Consider S' a subcomplex of S. If (SjS') is (/ — 1)-connected, / > 1, the space N is contractible and
HOMOTOPICAL DEPTH Hj{N,N']Z) = 0, for j < n - I - I, then the pair {E,E' U -K'^^S')) (n — l)-connected.
321 is
In our situation we apply this lemma to the case E and S are as above, E' := Se{z) nXn{cl> = /3}- (T^iZk) U Va{Y)) and
5' := s,{z) n{Zk+i -Ty{Zk))ndVa{Y) = sndVa{Y), Proof of Lemma 1.8: We may assume that S is obtained from 5 ' by adding cells of dimension > /, because of 6.13 [Sw]. Therefore it is enough to consider the case ( 5 , 5 ' ) = {D^,S^^^) with j > /; then E and E^ are trivial fibrations, i.e. E = D^ x N and E' = D^ x A'''. As E is contractible, we have to show that E' U 7r"'^(5') is (n — 2)-connected. Using Kiinneth formula and Hurewicz isomorphism it is enough to show that E^ \Jir~^{S^) is simply connected if n > 3, i.e. the product {D^,S^~^) x {E,E') is 2connected in this case. This is obvious if j > 2; if j = 1, we observe that (TV, iV') is 1-connected (because TV' is ^i^ 0 and 0-connected). We can compare this argument with the one already used in [H3] (p.552). This ends the proof of Lemma 1.8 and Theorem 1.4. We have a theorem as Theorem 1.4 for the rectified homological depth: T h e o r e m 1.9. Let X be a reduced complex analytic space and n a positive integer. The following conditions axe equivalent: a) rHd{X, Z) > n; b) If Y is a locally closed complex analytic subspace of X of dimension i, for any point y in Y there is a fundamental system of neighbourhoods Ua of y in X such that the homology groups Hj{Uoc, Ua - Y] Z) are 0 for k < n - i; c) IfY is a locally closed irreducible complex analytic subspace of dimension i, there is an open dense analytic subset YQ ofY such that J for any point y in YQ, there is a fundamental system of neighbourhoods Ua ofy in X such that the the homology groups Hj{Ua, Ua - Y; Z) are 0 for k < n - i. The proof of this theorem is similar to the one of Theorem 1.4. Instead of using the theorem of Blakers and Massey as we did above we only need to use the classical excision theorem for singular homology. The Lemma 1.8 is replaced by a similar lemma for homology groups proved by an argument of Mayer-Vie tor is type or the Leray spectral sequence for fiber spaces.
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A consequence of the preceding theorem is the following relation between perversity and rectified homological depth: Corollary 1.10. Let X be a reduced complex analytic space of dimension rij then the rectified homological depth for the rational homology rHd(Xj Q) equals n if and only if the constant sheaf of complex numbers on X is perverse. Proof: Since the constant sheaf of complex numbers on X obviously satisfies the support condition for perversity, it remains to prove that it satisfies the cosupport condition if and only if the rectified homological depth for the rational homology rHd{X^ Q) equals the dimension of the variety X. If X is embedded in a non-singular complex analytic space M, the cosupport condition (see [L-M] (4.6.1) (ii)) means that the k-ih. cohomology of the Verdier dual l^Jiomo{Q.x[dirnX — dimM\,£jvf) ^^Q~x\divfiX — dimM\ on M has its support in codimension > fc in M. As the cosupport condition is a local condition, for simplicity, we can assume that X is embedded in a non-singular complex analytic space M, Following the presentation of [D2] (§1, condition (BN)) or [B-B-D] (condition (c") of the Introduction p.9) for algebraic varieties, we can express this cosupport condition in terms of local cohomology (in a way it does not depend on the local embedding). If Y is a closed subspace of X, the biduality theorem of Verdier for constructible complexes ([V2] Corollaire 2.6.2) implies that we have an isomorphism between the restriction of the Verdier dual (se [V3]) Dx{Qx) to Y and the Verdier dual of the restriction of the local cohomology complex ILTY{Q.X) (see [G-M2] §1.13 Identity (5), compare to [V2] Corollaire 2.6.4). The complex of sheaves Dx{C_x) has cohomology sheaves which are locally constant on each stratum of a Whitney stratification of X, For any Y which is the closure of a Whitney stratum of X of dimension f, the support condition for DxiS.x) means that the A;-th cohomology of f^xiSix) vanishes at the general point of y if Ar > n — i. Using the remark of 1.1.3 and the Theorem 1.9, we have that rHd{X, Q) = n, if and only if, for any Y which is the closure of a Whitney stratum of X of dimension z, the stalk of the k-ih cohomology of R r y ( C j ^ ) at any general point of Y is zero, for any fc < n — f, i.e. if and only if, for any fc > n — i, the stalk of the Ar-th cohomology of the complex Dx{Q.x) restricted to Y, which is the Verdier dual of Rry(^jj^) on Y, is zero at any general point of Y (compare to [V2] Corollaire 2.6.5), i.e. if and only if the support condition for Dx(Cj^) is satisfied.
2. Lefschetz T h e o r e m s and rectified homotopical depth. In [H-L2] we have already proven a quasi-projective Lefschetz theorem.
HOMOTOPICAL DEPTH
323
namely (theorem (2.1.4)): Theorem 2.1. Let XQ be a quasi-projective variety. Suppose that the rectified homotopical depth rhd{Xo) > n. Let / : XQ —> P be an algebraic morphism with finite fibers. Let L be a projective subspace of P of codimension c and {XQ : ... : X^) be homogeneous coordinates of P such that XQ
— ... = Xc-l
= 0
are equations of L and let VR{L) be the neighbourhood of L defined by: N
Vfl(L) := {(^0 : ... : x^) G P^ I Y.\xj\' > C
c-1
R^^lxkl'} 0
then for all R > 0 except a finite number the pair (Xo,Xo fl is (n — c)-connected.
/~^(VR(L)))
Naturally there is a local version of this Lefschetz type theorem which generalizes the Theorem 1.1.1 of [H-L3] and the Lemma 1.7 we have proved above to show that our definition of rectified homotopical depth coincides with the one given by Grothendieck. Let X be a reduced complex analytic space embedded in C . Consider a complex analytic closed subspace Z of X and a complex analytic subspace y of C defined by the holomorphic equations gi z= .,. = g^ = 0. Let iP{x) := J2\gj\^ and Va{Y) := {x e C^ \ '(pix) < a}. Consider a point z of ZnY and denote Se{z) the real sphere of C ^ centered at z with radius 6. We have: Theorem 2.2 (Weak local Lefschetz Theorem). Suppose that the rectiEed homotopical depth rhd{X — Z UY) is > n. There exists €Q > 0, such thatj for any e, eo > e > 0, for all a > 0 except a finite number^ the space Se{z) H {X — Z) has the homotopy type of a space obtained from Se{z) n {X — Z) n Va{Y) by adding cells of dimension > n — i. Proof: We shall proceed as in the proof of lemma 1.7 with Y instead of L. First stratify X with a Whitney stratification adapted to the subspace Z, i.e. Z is a. union of strata. Denote Zk the union of Z and the strata of X of dimension < k. Our theorem will be an imimediate consequence of the following lemma: Lemma 2.3. Suppose that rhd(X - Z UY) > n. There exists CQ > 0, such that for any e, eo > c > 0^ for all a > 0 except a finite number, the space Se{z) H [{X - Zk) U {{X - Z) H Va{Y))] has the homotopy type of a
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space obtained from Se{z) H [{X - Zk+i) U ({X - Z) 0 Va{Y))] by adding cells of dimension >n — i. Theorem 2.2 is obtained from this lemma by comparing the cases k = 0 , . . . ,fc = dirrizX — 1. Proof of Lemma 2.3: Consider equations / i = . . . = /^ = 0 of Zk-\-i and hi = ... = hs = 0 oi Zk in a neighbourhood U oi z in C . A s above denote <j){x) =: ^ |/i(^)P and x(^) = J2 I^i(^)p5 ^^ define
Tp{Zk+i) := {x e Xnu \ {x) < p} and T^{Zk):={xeXnU\x{x)<7}. As we have done in the proof of Lemma 1.7, we choose eo as in 1.7.1. Now fix 6, 60 > € > 0. The Whitney stratification of X induces a Whitney stratification on Se{z) nX because of the condition i) of 1.7.1. The restrictions of ip to the strata of Se{z)r\X are real analytic, so they have only a finite number of critical values. We choose an a > 0 which is not one of these critical values. As in 1.7.2 we find that, for j small enough, the space A^ := Seiz) n [{X - T^(Zfc)) U ((X -Z)n
V^{Y))]
is a deformation retract of A := Seiz) n [{X - Zk) U {{X -Z)n
V^{Y))].
Fix such a 7. As in 1.7.3, since /? > 0 is small enough, the space Bp := Se{z) n [{X - T0{Zk+i)) U {{X -Z)n
VciY))]
is a deformation retract of B := S,iz) n [(X - Z^+i) U ((X -Z)n
K.(Y))]
and moreover with /? > 0 small enough the space B^^p := Se{z) n [{X - T^{Zk) n Tp{Zk+,)) U ((X -Z)n Vc^iY))] is a deformation retract of A. We want to compare the homotopy groups of the spaces A and B. In view of the homotopy equivalences we have just considered, it is enough
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325
to compaxe the homotopy groups of the spaces B^^f^ and B^. We observe that -B7,/? equals the union of B^ and E:=Ser\
(Ti3{Zk^i) - T^{Zk) U Va{Y)).
As in 1.7.5 we observe that £* is a locally trivial topological fibration over the space
S :=
Sen[Z,^i-mZk)UVa{Y)]
with fiber the normal shce Af of Zjb+i in X (here see the Appendix). But the hypothesis rhd{X — Z UY) > n imphes that the pair (AT, dJ\f) is {n — k — 2)-connected. Because of the Theorem 1.1.1 and the Remark 1.1.2 of [H-L3], we know that the pair ( 5 , 5 fl dVa{Y)) is {k — 2)-connected, so that as in 1.7.5 the pair (B^^^.Bp) is ((n — Ar — 2) + (Ar — 2*) + l)-connected by using the theorem of Blakers and Massey and lemma 1.8. This proves Lemma 2.3. R e m a r k 2.4. In the proof of Lemma 2.3 we only use the fact that the pair (AT, SAT), defined by a normal slice of a stratum of dimension fc + 1, is {n — k — 2)-connected. We shall express this fact by saying that X — YUZ has rectified homotopical depth > n along Zk+i — Zk- This motivates the following definition: Definition 2.5. Let X be a reduced complex analytic space and Z be a complex analytic subspace. We say that the rectified homotopical depth of X along Z is> n if, for any locally closed complex analytic subspace Y of Z, for any point zofYj there is a fundamental system of neighbourhoods Ua of z in X such that the pairs {Ua^Ua — Y) are (n — iz — 1)-connected, with iz = dirrizY. We shall denote rhd{X^Z) the rectified homotopical depth of X along Z. Of course we have a similar definition of the notation rHd{X, Z, Z) for the rectified homological depth of X along Z. A corollary of Lemma 2.3 and its proof is then the following: T h e o r e m 2.6. Let X,Y,Z be as in the theorem 2.2. Consider complex analytic subspaces Z',Z" of X such that Z C Z' C Z". Assume now that the rectified homotopical depth rhd{X -Y UZ,Z" - Z') of X - Y U Z is > n along Z" — Z'. There is eo > 0 such that for any e, eo > e > 0, there exists a i , . . . ,a,n such that for any a ^ { a i , . . . , 0:^}, the space Se{z) n ((X - Z') U {{X -Z)n Va(Y))) has the homotopy type of a space
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obtained from Se{z) D ((X - Z") U ((X - Z) D Voc{Y))) by adding cells of dimension > n — i. For the proof let us choose a Whitney stratification of X adapted Z, Z\ Z" and Y, Let Z^ be the union of Z' U Y and of all the strata Z" — ( Z ' u y ) of dimension < h. Then we have the statement of Lemma and obtain Theorem 2.6 by comparing the cases Ar = 0 , . . . , A: = divfizZ" Theorem 2.6 implies:
to of 2.3 — \.
T h e o r e m 2.7. Let X be a complex analytic space embedded in C . Let Z' C Z" be closed complex analytic subspaces of X and z £ X. Assume that the rectified homotopical depth rhd{X, Z" - Z') of X along Z" - Z' is > 72; then J for any e > 0 small enough, the space Sc{z) f] (X — Z') has the homotopy type of a space obtained from Se(z) n{X — Z") by adding cells of dimension > n — dimz{Z"). Proof: We may assume z ^ Z'. Let L be an affine subspace of C through z of codimension dirUzZ" and such that z is an isolated point of L f l Z " . So Se{z) n{XZ") contains Se{z) H X fl L. The Theorem 2.6 (with Z = Z') shows that Se{z) r\{X — Z') has the homotopy type of a space obtained from the space
Se{z) n {{X - z^') u {{X - z') n Va{L))) = Se{z) n(x-
z")
by adding cells of dimension > n — dirUzZ". This shows Theorem 2.7. Remark 2.8. In the Theorem 2.7 we can replace the sphere Sc{z) by the ball Be{z). It is clear if z G Z'. If z ^ Z', and z G Z", it is enough to consider the case Z" = {z}, because we might first compare the spaces Be{z) n(X - Z") and Be{z) f\{X - {z}) by the beginning of the remark. Then the latter case is an immediate consequence of the hypothesis on the rhd(X) by considering a Whitney stratification of X where {z} is a stratum and using Lemma 1.3. When Y is "in general position" we can replace the spaces Va{Y) by Y when a is small enough. Namely we have a consequence of Theorem 2.2: T h e o r e m 2.9. Let X and Y be complex analytic spaces embedded in C ^ , and suppose that Y is defined by i equations. Let Z be a closed complex analytic subspace of X and z £ Z DY. Assume that rhd{X — YUZ)>n and there is a Whitney stratification S of X adapted to Z and a Whitney stratification of Y the strata of which are transverse to the strata of Z except maybe at z. Then for any e > 0 small enough the space Se(z) D (X — Z) has the homotopy type of a space obtained from Se{z)n(X — Z)nY by adding cells of dimension > n — i.
HOMOTOPICAL DEPTH
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This theorem is an immediate consequence of a lemma similar to Lemma 1.4.1.1 of [H-L3] and of Theorem 2.2: Lemma 2.10. Let X and Y be complex analytic spaces embedded in C , let Z be a closed complex analytic subset of X. Assume that there is a Whitney stratification of X adapted to Z and a Whitney stratification of Y, such that the strata ofZ and the ones ofY intersect transversally except possibly at the point z ofX. Then, for any e small enoughj there is ao? such that for any a, ao > a > 0, the space X HY D Se{z) — Z is a deformation retract of X H Va(Y) H Se{z) - Z. If Z (resp.y) is defined by a non negative real analytic function for which the stratification of X (resp.Y) satistisfies the Thom condition the proof of this lemma is the same as the one of Lemma L4.1.1 in [H-L3]. The only difference is that we are on a set stratified by a Whitney stratification so that we have to build vector fields on each stratum and patch the construction into an integrable rugged vector field in the sense of Verdier (see [VI] p.308) to realize the desired retraction. In general, see the appendix. Remark 2.11. i) As z E Z, we may replace S^ by the ball B^ in the statement of the Theorem 2.9. Obviously we may only suppose z £Y. ii) IfYnZ = {z}j the transversality hypothesis is trivially true. This shows that Theorem 1.6 is a corollary of Theorem 2.9. Finally we can state a strong local Lefschetz theorem on a singular space: T h e o r e m 2.12. Let X be a complex analytic space embedded in C . Consider a closed complex analytic subspace Z of X and a point z of X and suppose that rhd{X — Z) > n. There is an open dense Zariski subset Q in the projective space of linear hyperplanes of C ^ , such that, for any H E Q, and any holomorphic function g such that g{z) = 0 and Ker dgz = H, there is eo > 0 such that for any e, 6o > e > 0, and any ^ / 0 smaii enoughj the pair
{B,{z) n (X - z), B,{z) n{x-z)n y^, with Yt := {g = t}, is (n — l)-connected. Proof: We fix a Whitney stratification of X adapted to Z. As in the proof of Lemma 2.3, we denote Zk the union of Z and the strata of X of dimension < k. Denote B^ := Be{z) and Se its boundary. Our theorem will be an immediate consequence of the following lemma:
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H.A. HAMM and LE D.T.
L e m m a 2.13. Suppose that rhd(X — Z) > n. There exists an open dense Zariski subset Q in the projective space of Unear hyperplanes of C ^ , such that, for any H £ Q, and any holomorphic function g such that g{z) = 0 and Ker dgz = H, there is eo > 0 such that for any e, CQ > e > 0, there is rj > 0 such that, with Yt := {g = i} and 0 < \t\
(B, n [(X ~ Zk) u {{X -z)n Ytl B , n [{x - Zk+i) u ((x -z)n Vt]) is (n — l)-connected. Theorem 2.12 is obtained from this lemma by comparing the case k = 0 and k = dirUzX — 1. P r o o f of Lemmia 2.13: We begin as in the proof of Lemma 2.3. Consider equations / i = . . . = /^ = 0 of ^AT+I and hi = ,..
= kg = 0 of Zk
in a neighbourhood {/ of z in C . As above denote (j){x) = Yl |/i(3?)P,
and
Ty{Z,):={x€XnU\x{x) 0 such that for any c, 0 < e < Ci, the restriction of g to the strata of the stratification on Zk+i fl Be induced by S has maximal rank. We choose H E Cl and a complex analytic function g such that the kernel of dgz is H and g{z) = 0. As we have done in the proof of lemma L7, we choose ei > eo so that: i) for any e, 6o > e > 0, the spheres S^ are contained in U and transverse to the strata of X and of S] ii) the choice of €o gives the conclusions ILL4 of [H-L3] so that, for any 6, eo > 6 > 0, there is r/i > 0 such that for any t £ C, T]I > \i\ > 0, the pair of spaces
{B, n (Zfc+i ~ Zk), B, n (Zfc+1 - Zk) n Vt) is fc-connected.
HOMOTOPICAL DEPTH
329
Now fix e, eo > e > 0. The Whitney stratification of X induces a Whitney stratification on 5^ fl X because of the condition i) above. The restrictions of y to the strata of B^ flX are real analytic, so they have only a finite number of critical values. Let r] be smaller than r}i and so that for any t, r; > |t| > 0, Yj is transverse to the strata oi B^OX. Similarly to 1.7.2 we find that, for 7 small enough, the space
c^ := B, n [{X - mzk)) u ((X -z)n y,)] is a deformation retract of
c := 5e n [(x - Zk) u {{X -z)n y,)]. Fix such a 7. As in 1.7.3, for /? > 0 is small enough, the space Dp := B, n [{X - TpiZk+i)) U {{X -Z)n
Yt)]
is a deformation retract of D:=B,n [{X - Zk+i) U ((X -Z)n and moreover with /3 > 0 small enough the space
Yt)]
£>^,^ := B,(z) n [(X - T^iZk) n TpiZk+i)) u ((X -Z)n y,)] is a deformation retract of C. We want to compare the homotopy groups of the spaces C and D. In view of the homotopy equivalences we have just considered, it is enough to compare the homotopy groups of the spaces Dp and Dy^p. In fact we shall estimate the connectivity of the pair of spaces {Dy^p^Dp). We observe that DJP equals the union of Dp and
E:=B,n[Tp{Zk^,)-mZk)l As we noticed in 1.7.5, E is 3. locally trivial topological fibration over the space with fiber the normal slice Af of Z^+i in X (see the Appendix). But the hypothesis rhd{X — ZUY) > n implies that the pair {J\f,dJ\f) is {n — k — 2)connected. Because of the Theorem II.1.4 of [H-L3], we know that the pair (SjSnYt) is Ar-connected, so that, repeating the argument in 1.7.5, the pair (Dy^pjDp) is ((n — k — 2) -\- k-\- l)-connected by using the theorem of Blakers and Massey and Lemma 1.8. This proves Lemma 2.13.
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H.A. HAMM and LE D.T.
3. Some conjectures of A. Grothendieck. In [G] (Expose XIII p.27-30), A. Grothendieck proposed several conjectures in relation with the rectified homotopical depth and theorems of Lefschetz type. In this chapter we shall prove several of these conjectures. 3.1 Conjecture A. Let us recall the formulation of Conjecture A: Conjecture A ("local Hurewicz Theorem"). Let X be a topological space, Y a locally closed subset, eventually satisfying "smoothness" conditions such as local triangulability of the pair {X,Y), let n be an integer > 3. In order to have profhtpY{X) > n, it is necessary and sufficient that one has H!Y{?LX)
= 0
for
i
(then one says that X has cohomological depth > n along Y), and that the local fundamental groups Trl{X,x)=
lim7ri([/-f/ny) xeu
vanish (then one says that X is pure along Y). First we specify what is apparently meant by local triangulability and homotopical depth {prof htp) along a subspace: 3.1.1 Definition. Let X be a topological space and Y a locally closed subspace of X. We say that X is locally triangulable along Y if for any point of X ofY there is a fundamental system of neighbourhoods Ua of x in X such that the pair {Ua^Uaf^Y) is triangulable. Let us assume for the rest of (3.1) that {X,Y) satisfies this condition. Then for any point a: of y there is a good neighbourhood of x in X with regard to Y in the sense of Prill [P]. Therefore we may simplify Grothendieck's original definition of homotopical depth in the following way: 3.1.2 Definition. The space X has homotopical depth > n along Y if, for any x EY, there is a fundamental system Ua of neighbourhoods of x in X such that the pairs {Ua,Ua — Y) are (n — l)-connected. Note that this condition is equivalent to the following one:
HOMOTOPICAL DEPTH
331
For any good neighbourhood U oi x in X with regard to Y the pair of spaces (t/, U — Y) is (n— l)-connected. On the basis of these definitions, as stated, Conjecture A is incorrect; this is related to the fact that the classical Hurewicz isomorphism theorem refers to singular homology, not cohomology. 3.1.3 Example: Let X = {z G C" : zl + z^ + .^ + z^ = 0}, n > 4, Y = {0}. Then X is a good neighbourhood of {0} in X with respect to {0}, because X is weighted homogeneous. According to a result of E. Brieskorn ([B], [H5] Satz 1.1, 1.2), the space X - {0} is (n — 3)-connected and has the same homotopy type as a rational but not integral homology sphere; so i J „ _ i ( X , X - {0}; Z) = Hn-2{X ~ {0};Z) ^ 0, which imphes that the homotopical depth of X along {0} is not > n. On the other hand ^V(Zx)o = H\X,X-
{0};Z) = H'-\X
- {0};Z) = 0,
SO S y ( ^ x ) = 0, i < n. Since X — {0} is simply connected we have that 7ro(X, 0) = 0 (in the sense of Grothendieck). An obvious modification of Conjecture A is to require furthermore that the local cohomology has no torsion in dimension n. It naturally leads to the theorem: Theorem 3.1.4 ("local Hurewicz Theorem"). Let X be a topological space, Y a locally closed subspace such that X is locally triangulable along Y. Let nbe a non-negative integer. The following conditions are equivalent: a) The space X has homotopical depth > n along Y; b) for any point x EY, there is a fundamental system of neighbourhoods Uoc of X in X such that: H^{Ua,Ua-Y;Z) = 0, 7 r i ( t / a - y , y ) = 0,
u
if
n>3
Proof: Because X is locally triangulable, we can choose the neighbourhood Ua contractible and it is enough to consider such a good neighbourhood UaNow we apply the classical Hurewicz Theorem (cf [Sw] (Theorem 10.25)) to the space Ua — Y. The existence of good neighbourhoods makes the theorem very elementary. 3.2 C o n j e c t u r e B . We recall the statement of Conjecture B:
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H.A. HAMM and LE D.T.
Conjecture B ("purity"). Let E be an analytic space, X an analytic subset of E. One supposes that E is non-singular and of dimension N at x, and that X can be described by p analytic equations in the neighbourhood of any point. Then the rectified homotopical depth of X is > N — p. Actually this conjecture may hold even if E is singular, as it is stated below: Theorem 3.2.1. Let X be a reduced complex analytic space, rhd{X) > n, and let Y be a complex analytic subspace locally defined on X by i complex analytic equations. Then rhd(Y) > n — i. Proof: As the condition rhd{Y) > n — i is local, we can assume that Y is closed in X, defined by i global equations on X, and that there is a finite Whitney stratification of X adapted to Y. Let 5 be a Ar-dimensional stratum of Y. Let y E S and consider a normal slice JV of 5 in X at the point y. Locally at y this normal slice is defined by k equations. So locally at y the space Af DY is defined hy i -\- k equations. For simplicity assume moreover that locally at y the space X is embedded in C . Denote B^ := Be{y) and Se its boundary. We want to prove that, for e > 0 small enough, the pair (BeCiY, BeOY — S) is (n —i —^ —l)-connected. Because of Remark 1.1.2, it is equivalent to prove that B^nYnAf—iy} is {n — i—k—2)connected. This latter fact is consequence of the Theorem 2.9. So, with the notations of the Theorem 2.9, replace Z by {y} and Y by jV H Y, for e > 0 small enough, the pair
(SenX-{y},SenYnX-{y}) is (n — i — k — l)-connected. As in Remark 2.11, we obtain that the pair
{B,nX-{y},B,nYnAf-{y}) is (n — i — k — l)-connected. As we have assumed that rhd{X) > n, the space B^ nX — {y} is (n — 2)-connected which implies that
BenYnAf-{y} is (n — i — k — 2)-connected as expected. Because of the example 1.2 the Conjecture B is an obvious corollary of our theorem. Furthermore, as Grothendieck already noticed:
HOMOTOPICAL DEPTH
333
Corollary 3.2.2. If the complex analytic space X is locally a non empty complete intersection of dimension n, we have rhd{X) = n.
3.3 C o n j e c t u r e C. Let us recall the formulation of Conjecture C: Conjecture C ("local Lefschetz T h e o r e m " ) . Let X be an analytic space, Y a closed analytic subset, and let x be a point ofY. Suppose that X — Y is Stein in some neighbourhood of x (for example that Y is defined by one equation in x), and that X — Y has rectified homotopical depth > n in some neighbourhood ofx (for example that X — Y is near x locally a complete intersection of dimension > n, cf conjecture B). Then the canonical homomorphism 7rf (Y) -^ 7rf{X) is an isomorphism for i < n — 1, an epimorphism for i = n — 1. Furthermore, Grothendieck adds the following remark: If one replaces the hypothesis that X — Y is Stein by the hypothesis (which will play the role of a topological "concavity" condition) that X — Y is a finite union of c + 1 open Stein subsets (sc. in some neighbourhood of x) the conclusion should be modified simply by replacing n by n — c. We will prove that Conjecture C holds even in its stronger version described in this remark. Furthermore we may look at a more general situation where a subspace Z o{ X is removed, just as in Theorem 2.2 (Weak local Lefschetz Theorem). Let W be an open neighbourhood of the origin in C ^ . Consider a closed complex analytic subspace X of W. Let Y and Z be closed complex analytic subspaces of X. Let ^fi,... ,^j : W —^ C be holomorphic functions such that Y = {z e yV\9i{z) = ... = gi{z) = 0} , and let ^ : ^ -^ R be defined by tP(z) = ^ \9jiz)\^. For a > 0 define Va(Y) = {z e yV\^p{z) < a}. We shall denote B^ the closed ball of C centered at 0 with radius e and Se its boundary. Then we have: Theorem 3.3.1 (Generalized weak local Lefschetz Theorem). Let us suppose that X — Y is the union ofc-{-1 open Stein subsets Ui,...,Uc-\.i and that the rectified homotopical depth rhd{X — Y U Z) is > n. Then there is an CQ > 0 such that for any e, 0 < e < CQ, there exists ao > 0 such that for any a, 0 < a < ao, the pair
(5e n X - z, 5e n X n v^iY) - z)
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H.A. HAMM and LE D.T.
is [n — c — 2)-connected. R e m a r k 3.3.2. We may take c = i — 1,
Uj =
{zeX\gj(z)^0},
j = 1 , . . . , i; in this case Theorem 3.3.1 follows from Theorem 2.2. But in some cases a choice of a smaller c is possible. Proof: We have supposed that X-Y
=
UiU...Uc^i
where each I7j, 1 < j < c + 1 is a Stein open subset. From [N] (Theorem 6), there is an integer m such that for any j , l < i < c + l , there is an analytic isomorphism hj of Uj onto a closed analytic subset of C"^. Let <j)j:X -^Hhe defined by:
^
[
0
zex-Uj
then denote <^ = 0i + . . . -f (^c+iNow let us fix a Whitney stratification of X adapted to Z. We shall assume 0 ^ Z because we consider spaces on the sphere Se- Consider the function x - ^ -^ R- defined by N
X{Z):=J2\'A' I/ = l
Now let 60 > 0 such that: (i) for any e, eo > e > 0, the sphere Se intersects transversally all the strata of X; (ii) for any stratum 5 of X and any point z of 5 — Y, and any A E R, such that x{z) ^ ^o ^^^ dz{rp\S) = Acf^(x|S'), then one has A > 0. The first condition is satisfied because the square of the distance function to the point 0 is real analytic and so is the Whitney stratification (because it is complex analytic). The second condition can be satisfied because of [HI] (Lemma 2.13). Let us fix 6, 60 > e > 0. The Whitney stratification of X induces a Whitney stratification oi X D B^. There is ao > 0 such that for any a, ao> a > 0^ the space dVa{Y) is transverse to all the strata o{ X (13^The space 5en(X —Z) is a deformation retract oi Ben{X — Z)y because of the choice of 6 as in (i) above (compare to the local conic structure [B-V]). For any a, ao > a > 0 the space
Benxn Va{Y) - z
HOMOTOPICAL DEPTH
335
is homeomorphic to B^ D {X — Z). This can be shown using a suitable controlled vector field the existence of which is guaranteed by the choice of e as in (i), (ii) above; cf. Lemma 1.3.2 of [H-L3] in a particular case. Moreover the spaces
SenxnVa{Y)-z are a fundamental system of good neighbourhoods of 5e fl X fl Y relatively to Se D Z, so that for any a > a', ao > a > a' > 0, the space
SenxnVa'{Y)-z is a deformation retract of (actually homeomorphic to)
SenxnVa{Y)-z. Now consider the neighbourhoods of X HY defined by: VpiX nY)
:= {z e X\iz) < 0}
for yS > 0. Since (f) is continuous and B^ D X and S^ f) X are compact, the spaces (V^(X fl V) fl B^) and {Vp{X nY)n S^) form a fundamental system of neighbourhoods of X HY D Be and X DY H Se in X HB^ and X nSe respectively. Notice that, even if the /? > 0 are small enough, these spaces are not necessarily good neighbourhoods, because we do not have the condition (ii) above, as we do not know that (f> is real analytic on X. Thus we fix /3 small enough such that Vp{X nY)n Be is contained in
BenxnVa,{Y)-z and moreover /? is not a critical value of the restriction of <j) to the strata of X n Be', this is possible because of Sard's theorem. Therefore we have an induced stratification of Be r\Vp{X HY). We shall actually prove that: L e m m a 3.3.3. The pair
{Be nVpixnY)- z, Se nVpixnY)- z) is (n — c — 2)-connected. First let us show how Lemma 3.3.3 implies the Theorem 3.3.1. We have the following inclusions of pairs:
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H.A. HAMM and LE D.T.
(JB, n X n Va{Y) -z,Senxn vdY) - z) c {B,r\Vp{xr\Y)-z,s,f\Vp{xp^Y)-z) c (5, n X n K.„(y) - z,5, n x n V^,{Y) - z) with a < OfQ small enough. Because the spaces S^ n X fl Va{Y) (resp. 5e n X n Va(y)) are good neighbourhoods of S c D X n y (resp. 5 ^ 0 X 0 7 ) relatively to Zr\Bc (resp. Z f\Si), the inclusion:
(Be n X n Vc(y) - z,5, n x n K.(y) - z) c(Be n X n i/„„(y) - ^, Se ' n X n K„o(y) - z) is a homotopy equivalence. So, if
(5e n V/3(x n y) - z, Se n v^{x n y) - z) is (n — c — l)-connected, for any a, ao > <^ > 0, the pair
(5, n X n Va{Y) -z,s,nxn
Va{Y) - z)
is {n — c — l)-connected. Now Theorem 3.3.1 follows because B^f) X f) Va(Y) — Z and B^f^X — Z are homeomorphic (see above) and Bef\X — Z retracts by deformation on S^r\X — Z. It remains to prove Lemma 3.3.3. Proof of Lemma 3.3.3: The proof proceeds by stratified Morse theory in the sense of Goresky and MacPherson in [G-Ml] on the space
M:=:5enV/5(xny)-z. The space M having "boundary and corners" we have to adapt this stratified Morse theory to this situation as we have done in the classical case in [H-Ll] (3.1.7). The Morse function here is a small perturbation cr of the restriction of —x on a neighbourhood of this space starting from the level —6^ until a level —e\ where ei > 0 is small enough. For the sake of simplicity we shall assume that a is defined on the whole space X. More specifically, ei is chosen in such a way that B^^ fl X is contained
in5enV/?(xny). According to the choice of e the restriction of —x to the strata in the interior of M has no critical points, so we can assume that the same is true for cr.
HOMOTOPICAL DEPTH
337
Therefore we have only to look at the critical points of the restriction of a to the strata of the boundary of M. As in (1.2.4) of [H-L2] in order to estimate the indexes of c at Morse critical points let us look at the Levi form of — X first. If p is a smooth real function in C"^, we shall denote by dg the sum ^dgjdzy^dzyz and by LxQ the Levi form of y at a? defined for any tangent vector V at a: by:
^xg{y>) - Y^{d'^g/dzkd'zj)vkvi Similarly we define the Levi form of a smooth function in C or a complex manifold. We shall denote the Levi form in the same way. Let p : C"^ —^ R be the square of the norm in C^ defined by p{x) = ^ \xj\^. With this notation <j)j\Uj = e-^^'^K Now let 5 be a stratum of X — Z and z £ S. If z E Uj we have:
d,ij\s) = -jiz)d,ipohj\s) and dz{<j>j\S) = 0 otherwise because from its definition the restriction of (j)j to the stratum is smooth and its derivatives are 0 outside of Uj. Then inside Uj, the Levi form of the restriction (i>j\S is Mj\S) = -4>jiz)L,ipohj\S)
+
cl>j(z)mpohj\S)\^
and outside of Uj it is 0. The complex linear subspace Vz of the tangent space TzS where the complex differentials a.(^l|5),...,5.(<*e+l|5) vanish has codimension < c + 1. As the Levi form of Lz{pohj \S) is positive definite if ^r G Uj, the Levi form of Lz{(i>\S) is negative definite on this complex subspace. We consider now the Levi form of cr at a point of S on the boundary of M in the interior of B^. Because ^ is a small deformation of —X on a neighbourhood of M, the Levi form of the restriction (7|5 at a point z E S is negative definite (compare to (1.2.5) of [H-L2]). Now consider a critical point z of the restriction of cr on a stratum 5*0 of dV^{X n y ) n ^ e , where B^ is the interior of B^- Because the stratification of M is induced by the one of X, the stratum 5o is the intersection of a stratum 5 of X with dVp{Xr\Y). As 2: is a critical point of the restriction of a on the stratum 5*0 there is a real number A such that dz{(T\S) = \dz{\S), Because we are concerned with Morse theory on a stratified set with boundary, the only critical points which will contribute to the change of topology
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are the critical points z where A < 0. The Hessian Hz at z of the restriction of and the real part 3?(Q) of a complex quadratic form (see [H-L2] (1.2.5)). The Levi form o{ a — \(j) is therefore negative definite on the intersection of the subspace Vz oi TzS with TZSQ. In fact, at the critical point z the space Vz is contained in TZSQ. Using an argument similar to the one of A. Andreotti and T. Frankel in [A-F], if for some vector t; of V^, ^{Q){v) > 0, we have
HziV^v) < 0. Hence, the index at the critical point z of the restriction cr|So is > dimS — c — 1. A normal slice AT at z of 5o in dVp{X DY) is a normal slice at z of 5 in X . The hypothesis on the rectified homotopical depth rhd(X) > n tells us that the pair (A/" — {{z} U Z),Fz — Z), where Fz is the complex link of 5 in X at the point z (see [G-Ml] Part II §2.2) is {n - dim S - 1)connected. In fact we use the Theorem 3.2.1 to find that the normal slice being defined by dim S equations locally at z the rectified homotopical depth of AT — {{z} U Z) is > n — dim, 5; then Theorem 2.12 gives our assertion. Therefore the local Morse data (see [G-Ml] Part I §10.3 and Part II §2.6) are (n — dim, S — l)-connected. According to the stratified Morse theory of Goresky and MacPherson (see [G-Ml] Part I §10.5), we obtain that
{Be n Vpix ny) - z,5, n Vp{x nY)-z) is (n —c —2)-connected (using [Sw] (Prop. 6.13) we can show that the cells which are locally added have dimension > n — c — 1). This proves Lemma 3.3.3. Theorem 3.3.1 has the following corollary if we assume transversality conditions between Y and Z: Corollary 3.3.4. Under the same notations and hypothesis of Theorem 3.3.1 J assume that there is a Whitney stratification ofX adapted to Z and a Whitney stratiGcation ofY, such that the strata of Z and the ones ofY intersect transversally except possibly at the point 0. Then, for any e small enough, the pair (X f) B^ — Z,Y DX H B^ — Z) is {n — c — 2)-connected. This corollary is an immediate consequence of Lemma 2.10. 3.4 Conjecture D. Let us quote Grothendieck's Conjecture D:
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Conjecture D ("Global Lefschetz Theorem"). Let X be a compact analytic space, Y an analytic subspace of X such that U = X — Y is Stein and has rectified homotopical depth > n (for instance a local complete intersection of dimension > n at any point). Then the canonical homorphism n(Y)
- 7r.(X)
is an isomorphism for i < n — 1, an epimorphism for i — n — 1. Furthermore Grothendieck adds a remark similar to the one already quoted in the statement of Conjecture C in the case X — Y is the union of several Stein open subsets. As we have done in the case of Conjecture C we are led to prove the following theorem, which extends the Theorem 1.1.1 of [H-L2] for the singular case. Theorem 3.4.1 (Generalized global Lefschetz Theorem). Let X be a compact reduced complex analytic space, Y and Z be closed complex analytic subspaces of X such that X — Y is the union of c-\- 1 open Stein subsets. Assume that the rectified homotopical depth rhd(X — YUZ) is > n. Let V{Y) be a good neighbourhood ofY in X relatively to Z. Then the pair {X — Z, V(Y) — Z) is {n — c— l)-connected. Proof: We assume that X — Y = UUj where each Uj^l < j < c + 1, is a Stein open subset. We procceed in the same way as in the proof of Theorem 3.3.1. According to [N] (Theorem 6) there is an integer m such that, for any j , 0 < j ' < c + 1, there is an analytic isomorphism hj of Uj onto a closed analytic subset of C " . Let (^j: X ^ R be defined by:
''
\
0
zex-Uj
then denote <^ = <;^i + . . . -f (^c+iNow let us fix a Whitney stratification of X adapted to Z. Consider the neighbourhoods of Y defined by:
v^(y) := {z e x|<^(z) < /?} foi /3 > 0. Since (j) is continuous and X is compact the spaces V(3{Y) form a fundamental system of neighbourhoods of Y in X . In general these spaces are not good neighbourhoods of Y relatively to Z even if/? is small enough. Fix /3 such that V/3(y) is contained in the given good neighbourhood V{Y) and /? is not a critical value of the restriction of <^ to the strata of X . This is possible because of Sard's Theorem. As in the proof of Theorem 3.3.1, we actually prove:
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L e m m a 3.4.2. The pair {X - Z, Vp{Y) - Z) is {n - c -
l)-connected.
Let us show first how this lemma imphes Theorem 3.4.L We choose another good neighbourhood V^{Y) of Y relatively to Z which is contained in Vp{Y). We have the following inclusions of pairs:
{X - z, y'(y) -z)c{x-z,
Vp{Y) -z)c{x-z,
V{Y) - z)
Because V'{y) and V{Y) are good neighbourhoods of Y in X relatively to Z, the inclusion
(X - Z, V\Y) -Z)C{X-Z,
V{Y) - Z)
is a homotopy equivalence, hence if Lemma 3.4.2 is true, Theorem 3.4.1 is true. Proof of Lemma 3.4.2: We procceed as in the proof of Lemma 3.3.3. We use stratified Morse theory on the space X — Z. The Morse function r will be a small perturbation of the function \S at z and the real part ^(Q) of a complex quadratic form Q on the tangent space Tz{S). As above the Levi form of (l)\S is negative definite on the complex linear subspace of Vz of the tangent space T^S where the complex differentials dz{(j>i\S),... ^dz{
0 = dzWS) = 2mz{ct>i\s) + . . . + 2mz{c+i\s) hence there is a linear dependence between the complex differentials
Again the argument of A. Andreotti and T. Frankel gives that the index of the Hessian of (^15 at z is > dimS — c. We are only interested in the critical points of (f) on the strata of X which lie in the closure of X — V/3. This closure being compact, we do as in [M2] Lemma 22.4, and we approximate <^ by a function r which is a Morse function in the sense of Goresky and MacPherson in [G-Ml] Part I §2.1. If the approximation is good enough, the index of the Hessian at the critical point of the restriction T\S of r on a stratum 5 is > dimS — c. So we can estimate the connectivity of the local Morse data of the restriction to X — Z of T at z. Let JV be a normal slice at 2: of 5 in X. The homotopy normal
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Morse data is {M — {z} U Z,Fz — Z) and, as above, it is (n — dimS — 1)connected because of the hypothesis rhd{X — Y U ^ ) > n, Theorem 3.2.1 and Theorem 2.12. The local Morse data of the restriction of r to X — Z at z is therefore {n — c — l)-connected. The stratified Morse theory of Goresky and MacPherson (see [G-Ml] Part I §10.4) gives that the pair {X — Z, V^{X nY) — Z) is {n — c — l)-connected as stated in the Lemma 3.4.2. Remark 3.4.3. If y is defined by c + 1 global holomorphic equations, the complement X — Y is covered by c + 1 Stein open subsets, so we can apply our Theorem. We have the following consequences of Theorem 3.4.1: Corollary 3.4.4. Let f:X -^ X' he a. finite complex analytic map between compact complex analytic spaces and Z be closed complex analytic subspace ofX. Suppose that the rectified homotopical depth rhd{X — Z) is > n. Let Y' be a closed complex analytic subspace of X' such that X' — Y' is the union of c + 1 Stein open subsets. Then, for any good neighbourhood V{Y) ofY := f-\Y') in X relatively to Z, the pair {X - Z, V{Y) - Z) is {n — c— l)-connected. It is enough to observe that the inverse image of a Stein space by a finite complex analytic map is Stein (see [K-K] §73.1). Then we apply Theorem 3.4.1. Adding some transversality conditions in Theorem 3.4.1, we can replace the good neighbourhood V{Y) by Y using the lemma of the Appendix (compare to [H-L3] Assumption (T) of §1.4.1.). 3.5. Conjecture D ' . In [G], Grothendieck proposed a version of Conjecture D for the case dimX = 3. Let us quote his conjecture: Conjecture D ' (Global Lefschetz Theorem for the fundamental group). Let X be a compact complex analytic space, Y be a closed analytic subset such that U = X — Y is Stein. Suppose moreover that the following conditions are satisfied: (i) For any x element ofU, the local fundamental group 7r2(X, x) is zero (i.e. X is "pure in x^'), or only: the local ring Ox,x is pure, (ii) The local rings of the points of U are "connected in dimension >2". (Hi) The local rings of the points ofU are of dimension > 3.
342
H.A. HAMM and LE D.T. Under these hypothesesj for any x £ X, the
homomorphism
7ri(y,y)-^7ri(X,x) is an isomorphism (and 7r2{Y^y) -^ 7r2(X,:ir) is an epimorphism). We want to apply the results of Theorem 3.4.1. It is enough to show that the hypotheses (i), (ii), (iii) correspond to the hypotheses of Theorem 3.4.1 in the case dimX = 3. T h e o r e m 3.5.1. Let X be a complex analytic space. Suppose that: (i) For any point x of X there is a good neighbourhood Vx ofx in X such that 7ri{Vx — {x}, y) = 0, for any y EV^ — {x}. (ii) For any point x of X and any locally closed complex analytic subspace Y of dimension < 1 which contains x, there is a good neighbourhood Vx of x in X relatively to Y, such that Vx —Y is connected, (iii) For any point x of X, dirUxX > 3. Then rhd{X) > 3. 3.5.2 R e m a r k s : According to the definition given by Grothendieck in [G] (Expose XIII p.26), the point (i) of our theorem corresponds to the point (i) of Conjecture D' as far as the local fundamental group is concerned. The point (ii) of our theorem corresponds to the point (ii) of Conjecture D', because of the definition in Theorem 2.1 bj^ in Expose XIII p.6 of [G]. So the proof of Theorem 3.5.1 and Corollary 3.4.5 imply Conjecture D'. Proof of Theorem 3.5.1: Choose a Whitney stratification of X and denote by Xi the union of strata of dimension < i. Let x be a point in Xi — Xi^i. Consider a good neighbourhood 1/ of x in X relatively to Xi. By definition we have to prove that the pair (F, V — Xi) is (2 —2)-connected. If 2 = 0, it is sufficient to prove that the pair (F, V^—{a?}) is 2-connected: it is 0-connected because dimX > 1 (see condition (iii)); it is 1-connected because of conditions (iii) and (ii); it is even 2-connected since we have also condition (i). If 2 = 1, then {V^V — Xi) is 1-connected because of (ii) and (iii). If i = 2, the pair of spaces (F, V — X2) is 0-connected because of (iii). R e m a r k 3.5.3. We are not sure whether Conjecture D' still holds if we only assume in (i) that the local ring Ox,x is pure.
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4. Further Results. 4.1. Other characterizations of Rectified Homotopical Depth. We have the following characterization of the rectified homotopical depth where it is only necessary to consider local homotopy groups: T h e o r e m 4 . 1 . 1 . Let X a complex analytic space. The following conditions are equivalent: a) rhd{X) > n; b) for any locally closed complex analytic subspace Y ofX, any point z £ Yj where Y is defined by k holomorphic equations, there is a fundamental system of neighbourhoods U of z inY j such that the pair of spaces {JA^U — {zY) is {n — k— l)-connected; c) for any point xofXj any local embedding at x of X into C and any k, dirUxX > k > 0, there is a non-empty open subset W of the Grassmannian of linear suspaces of codimension k inC , such that the following holds: for any L £ W and any holomorphic functions flfi,... ,5^jb whose differentials at z define L, there is a fundamental system of neighbourhoods U of z in Y, where Y is defined by gi^,.. ^g^^ such that the pair of spaces {U,U — {x}) is (n — k — lyconnected. Proof: The implication a) =^ b) is easily obtained using Theorem 3.2.1 above which gives that the space Y has rectified homotopical depth > n—k. The implication b) =^ c) is obvious. Now let us prove c) => a). Let us choose a Whitney stratification of X. Let 5 be a stratum of X and x E S. We can embed X locally at x into C . So any affine space transversal to 5 at a? in C defines a normal slice Af of S in X at X (see [G-Ml] Part I §1.4). A normal slice of codimension dimS being defined by dimS analytic equations which are non singular at X, condition c) tells us that the pair {Af^Af— {x}) is {n — k — l)-connected and Remark 1.1.2 above implies that, for any good neighbourhood V of x in X, the pair (F, V — S) is (n — k — l)-connected. We have another characterization of the rectified homotopical depth in terms of the connectivity of the complex links of M. Goresky and R. MacPherson (see [G-Ml] Part II §2.2). Before stating the theorem, let us introduce some notations and definitions. If X is a complex analytic space that we stratify with a Whitney stratification, we define the complex link of the stratum S in the following way: Consider a point x of S and a local embedding at a? of X into C^. Denote d = dimS and consider a general linear projection p of C ^ onto C^''"^. For
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e > 0 small enough, a general fiber of p over a point near to p{x) intersects Be{x) n X in the complex link of S. The fact that this complex link does not depend on the point x of 5, the general projection p, the small e and the general fiber of p is proved in [G-Ml] §2.3 (see [L-T2] §3). In this geometrical setting the normal slice is the inverse image by p restricted to Be{x) n X of a general line through p{x). Therefore we can view the complex link £ as a subspace of the normal slice Af. The homotopy type of the pair (AT, C) is the same as the homotopy type of the normal Morse data of Goresky and MacPherson (see [G-Ml] §2.5). Theorem 4.1.2. Let X be a complex analytic space, endowed with a Whitney stratification. The following conditions are equivalent: a) rhd{X) > n; b) for any stratum S the pair (AT, C) of the normal slice of S in X and the complex link of S in X is (n — dimS — l)-connected and therefore also the normal Morse data of S. Proof: We show first the implication a) => h): Theorem 3.2.1 gives that the rectified homotopy depth of the normal slice A/* is > n — dimS^ because the normal slice is obviously defined by dimS holomorphic equations in X. Theorem 2.12 just gives the desired result. The implication b) => a) is proved by induction on dimX. If dimX = 0, there is nothing to be proved. So suppose that dimX > 1, and that the theorem is true for any complex analytic space of dimension < dimX. Let Xi be the union of strata of X of dimension < i. If AT is the normal slice of Xi at X in X , we have to prove that {Af,Af— {x}) is (n — i — l)-connected. If i > 1^ rhd{J\f) > n — i by the induction hypothesis applied to A/* instead of X {M has an induced stratification where all normal discs and complex links coincide with those of X along AT). The pair {Af^C) of the normal slice and complex link of Xi at x is (n — i — l)-connected by hypothesis. So Theorem 2.12 gives that the pair (A/'— {x}, C) is (n — z — l)-connected. But M being contractible this means that {J\f,Af— {x}) is (n —i— l)-connected. If i = 0, the preceding result for i > 1 shows that rhd{X — XQ) > n and Theorem 2.12 gives that, after embedding locally X into C , for any e small enough, the pair {B^{x) D X — {x},C) is (n — l)-connected. By hypothesis the pair {Be{x) f) X,C) is (n — l)-connected. As Be{x) H X is contractible, finally {Be{x) 0 X^B^{x) H X — {x}) is {n — l)-connected. 4.2. General Strong local Lefschetz Theorem. We have a more general version of Theorem 2.12 which can be applied to non generic situations.
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T h e o r e m 4.2.1. Let X be a complex analytic space embedded in C . Consider a closed complex analytic subspace Z of X and a point z of Z and suppose that the rectified homological depth rHd(X — Z, Z) is > n. Let g:U -^ G be a holomorphic function defined on an open neighbourhood U of z and g{z) = 0. There is eo > 0 such that for any e, €o > e > 0, there is ao such that for any a, ao > a > 0, and any ^ E C, |^| = a the pair of spaces
(B^iz) n (X - z), ((5e(^) n 9-\D„)) u (5,(z) n g-'m n{x- z)) is {n — l)-connected, where Da is the closed disc of C of radius a centered at 0. Proof: To simplify the notations we shall assume z = 0 and we denote by B^ and Se the spaces 5e(0) and S'e(O). Call ip the real analytic function on U defined by \g\^. First notice that, as the point 0 is in Z, there is Ci such that for any e, ei > 6 > 0, the space Bcn{X — Z) is homeomorphic to the product of the interval (0,1] with ^e fl (X — Z) because of the local conic structure of singularities (see [B-V] (Lemma 3.2)). Now consider the function x - ^ —^ R- defined by N U= l
Let 6i > eo > 0 such that: (i) for any e, eo > e > 0, the sphere S^ is contained in U and intersects all the strata of X transversally; (ii) for any stratum S oi X and any point z of 5 — ^"^(0), and any A G R, such that x{^) ^ ^o ^^^ dz{xp\S) = Ac/^(x|«S'), one has A > 0. Fix e, eo > e > 0. There is ao > 0 such that for any a, ao > a > Oj and for any stratum S of X H Se the function g restricted to S has no critical values in the punctured disc D*. Now ^x a, ao > a > 0. It is known (see for instance [G-Ml] Part II §2.A) that the union
{S,ng-\Da))U{B,ng-\dDa)) is homeomorphic to the sphere Se by a stratified homeomorphism such that the space SeC\(X — Z) is homeomorphic to
{Se n g-\Da)) u {B, n g-\dDa) n{x~ z)) Following [L2] (Theorem 1.1) (see [L-T2] (Theoreme 2.3.1)) we know that the mappings induced by g go: B, n g-\dDa)
^{X
- Z)—^
OD^
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H.A. HAMM and LE D.T.
and
gi: s, n g-\dDa) n (X - z) —> dD^, are topological fibrations. By excision (Blakers-Massey Theorem) it is enough to show that the pair
(5, n g-\dD,) n (X - z),{s, n g-^dD^)) u {B, n g-\t) n (x - z))) is {n — l)-connected for t, |<| = a. By excising B^ fl g~^{t) r\{X — Z), using the trivial fibration over dDa — {t}^ we observe that we have to prove that the product {F,dF) x (7,97), where F := B, fl g-^{t) H {X - Z), dF := Se n g'^{t) H {X - Z), I := [0,1] and dl := {0,1}, is (n - 1)connected. We apply stratified Morse theory on F , using a small perturbation cr of the restriction of the function —%. Let x be a critical point of cr on a stratum of 7^ of dimension i. It is enough to prove that the product of the local Morse data (M, M') of cr at x by the pair (7, dl) is (n— l)-connected. First the local Morse data has the homotopy type of the product of the normal Morse data {Af,C) and a relative cell {B^ ,dB^) with j > i. The hypothesis tells us that the homology groups H^{M, £; Z) vanish for 0 < fe < n — 2 — i. Now using Lemma 1.8 for the trivial fibration (B^ ,dB^) x (7, dl) x (Af,C) over {B^,dB^) x (1,81)^ we obviously obtain the desired result. Corollary 4.2.2. Under the assumptions of Theorem 4.2.1, assume furthermore that there is a Whitney stratification of X adapted to Z such that the restriction of g to any stratum has no critical point outside z with critical value 0 (i.e. g has an isolated critical point on X relatively to its Whitney stratification in the sense of [L3] or is generic in the sense of [H3] §2J. There is eo > 0 sucii that for any e, CQ > e > 0, there is r^ > 0 such that for any t G C, 0 < |t| < r^, the pair {Be{z)n{X-Z),Be{z)ng-'^{t)n{X-Z)) is {n — i)-connected. Proof: Under this new hypothesis there is €o eventually smaller than the one chosen in the proof of the preceding theorem, so that the spheres S^ with 6, 6o > e > 0, are transverse to the traces of the strata of X on the fiber ^ - ^ 0 ) . We fix e, eo > e > 0. Then there is a such that the restriction of the function g to Se{z) fl {X—Z) has no critical value in the disc £)«, so it induces a trivial topological fibration of Se{z)fl(X — Z)flg~^{Dcx) over Da. This obviously implies our corollary by using our theorem 4.2.1.
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APPENDIX In this appendix we shall first prove the following theorem: Theorem. Let X be a complex analytic space embedded in a smooth manifold M and S = {Si)i^i a Whitney complex analytic stratification of X. Let N be a real analytic submanifold with corners of codimension 0 in M. Suppose that Si is a stratum of S which intersects N, its boundary and corners transversally in M and that K = N Cl Si is compact. Then there is a system of neighbourhoods Ua of K in NOX which admit locally trivial continuous fibrations TTa'.Ua -^ K, the fibers being homeomorphic to normal slices of the stratum Si in X. Proof. We use and follow a geometric construction of M.H. Schwartz in [S]. The manifold Si being a submanifold of M, with an adequate Riemannian metric on M we can define a system of neighbourhoods Tp of Si in M made of geodesic rays normal to Si. Furthermore we may suppose that the Riemannian metric is chosen in such a way that the geodesic rays respect the natural stratification of N. So if /x > 0 is sufficiently small, the rays of length /i starting from K form a geodesic tube T^(K) around K in N OX (see [S] 2.2). Now the projection p of T^{K) onto I{ given by the geodesies defines restrictions to all the strata of the Whitney stratification which contain points o{ K in their closures. These restrictions have maximal rank on each strata of the interior of T^{K)^ but furthermore, if p is small enough, the restrictions on the intersection of these strata and the boundary of T^{K) have also maximal rank. This last assertion results from the existence of transversal fields by M.H. Schwartz. In fact she proves that in the geodesic tube T^{K) there is a vector field v tangent to the strata Sj which vanishes only on K and which makes an angle I3{x) with the geodesic ray at any X G T^{K) which tends uniformally to 0 when x tends to any point of K (see [S] Theoreme 2.3). Therefore the geodesic tube T^{K) is naturally stratified by a stratification which satisfies the Whitney condition. Furthermore the smooth map p restricted to all the strata being of maximal rank, the first isotopy theorem of Thom and Mather ([T], [M]) implies that T^{K) is a locally trivial topological stratified fibration on K. The fiber above x £ K \s a smooth manifold transverse in M to Si at x and it obviously intersects X in a normal slice of Si at a point x. Putting Uoc '-= To(f^{K)j 0 < a < 1, we obtain the family mentioned in
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H.A. HAMM and LE D.T.
the theorem. Remarks. a) One can start with an arbitrary Riemannian metric on M above and replace T^{K) by the intersection T^{K) of XfliV by the union of geodesic rays normal to Si of length //, since T^{K) and T^{K) can be shown to be homeomorphic (sc. for small //). b) In the case of the last assertion of 1.7.4 we consider the tube T^(Zk+i) (with /? > 0 small enough) instead of T^(A"). We define K := S, n [Zk+i - mZk)
U Va{L)]
Obviously K can be written as K = 5e fl [Zk+i — Zk] fl N where TV is a real analytic sub manifold with corners of M = C . According to a), we may start from the standard Riemannian metric on C and obtain T^(K). Let ^ be the Euclidian distance from Se fl [Zk+i — Zk] in C ^ . Within a neighbourhood of this subset, the set {x e S: there is A < 0 such that d^{^\S) = Xda;((f>\S) } is semi-analytic for every stratum 5 of ^e fl X fl A'', so that the Curve Selection Lemma implies that one can construct a homeomorphism of Sc H [T^(Zfc+i) - Ty(Zk) U Va{L)] onto f^(K) for small /3 and fi by integration of a suitable vector field. In the proof of Lemmas 2.3 and 2.13 we proceed similarly. It remains to prove Lemma 2.10. L e m m a . Let X and Y be compact real analytic spaces embedded in a compact smooth manifold M, let Z be a closed real analytic subset of X. Assume that there is a Whitney stratification of X adapted to Z and a Whitney stratification ofY, such that the strata of Z and the ones ofY intersect transversally in M. Then, if V(Y) is a good neighbourhood of Y nX in X relatively to Z, the space Y f] X — Z is a deformation retract ofV{Y)-Z. Proof. Let Zi (resp, Yi) be the union of all strata of Z (resp. Y) of dimension < 2. Let us choose a Riemannian metric on M. For each submanifold A of M, /i > 0, let T^{A) be the open geodesic ^-neighbourhood of A. Now if 1 > //o > /^i > . . . > 0 and 1 > I/Q > ^^i > • • • > 0, the space
\JiT^,{Yi - y^.i) n X - u,-T,^(z,- - z,-^i)
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has a homotopy type independent of the choice of fii, Uj: this follows from the transversality condition which implies similar transversality conditions for the boundaries of the neighbourhoods. It is easy to see that
is a deformation retract of
u,r^,(y.-Y.-i)nx-z and that Y DX — UjTj^ {Zj — Zj-i) as well as of
is a deformation retract oiY nX — Z
UiT^XYi - Yi-i) n X - UjT,^{Zj -
Zj.i).
So y n X — ^ is a deformation retract of
Therefore UiTf^XYi — Yi-i) H X, which is a good neighbourhood of Y H X in X is a good neighbourhood of Y f l X in X with respect to Z. Since good neighbourhoods give the same homotopy type we obtain the statement of the lemma. Note that we can apply this lemma to Lemma 2.10, since we obtain induced stratifications of {X H Se{z)yZ H Se{z)) and of y fl S^z) such that the strata of Z H Se{z) and of y fl Se{z) intersect transversally in Se{z). Furthermore X nVa{Y)n Se{z) is a good neighbourhood of X DY Ci Se{z) in Se{z) with respect to Z fl Se{z). Bibliography [A-F] A. Andreotti - T. Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. (2) 69 (1959), 713-717. [B-B-D] A.A. Beilinson, J. Bernstein, P.Deligne, Faisceaux pervers, Asterisque 100 (1982), 1-172. [B] E. Brieskorn, Beispiele zur Differentialtopologie von Singularitaten, Inventiones Math. 2 (1966), 1-14. [B-M] A. Blackers - W.S. Massey, Tiie homotopy groups of a triad, IL Ann. of Math. 55 (1952), 192-201. [B-V] D. Burghelea - A. Verona, Local homological properties of analytic sets, Manuscripta Math. 7 (1972), 55-66. [Dl] P. Deligne, Le groupe fondamental du complement aire d^une courbe plane n'ayant que des points doubles ordinaires est abelien (d^apres Fulton), Sem. Bourbaki 543, Lect. Notes in Math.842 (1981), 1-10.
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[D2] P. Deligne, Purete de la cohomologie de MacPherson-Goresky (d'apres un expose de O. Gabber), Preprint, IHES/M/81/8 (Fevrier 1981), 1-9. [G-Ml] M. Goresky - R. MacPherson, Stratified Morse Theory, Springer Ed., Berlin-Heidelberg-New York 1987. [G-M2] M. Goresky - R. MacPherson, Intersection Homology II, Inv.Math. 71 (1983), 77-129. [G] A. Grothendieck, Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux (SGA2), Masson & North-Holland Ed., Paris, Amsterdam, 1968. [HI] H. A. Hamm, Lokale topologische Eigenschaften komplexer Raiime, Math. Ann. 191 (1972), 235-252. [H2] H. A. Hamm, Zum Homotopietyp Steinscher Raiime, J. reine angew. Math. 338 (1983), 121-135. [H3] H. A. Hamm, Lefschetz Theorem for singular varieties, Proc. Symp. Pure Math. 40 Part 1 (1983), Providence, 547-557. [H4] H. A. Hamm, On the vanishing of local homotopy groups for isolated singularities of complex spaces, J. Reine Angew. Math. 323 (1981), 172-176. [H5] H. A. Hamm, Exotische Spharen als Umgebungsrander in speziellen komplexen Raumen, Math. Ann. 197 (1972), 44-56. [H-Ll] H. A. Hamm - Le D. T., Un theoreme de Zariski du type de Lefschetz, Ann.Ec.Norm.Sup. 6 (1973), 317-366. [H-L2] H. A. Hamm - Le D. T., Lefschetz Theorems on quasi-projective varieties, Bull. Soc. Math. France 113 (1985), 123-142. [H-L3] H. A. Hamm - Le D. T., Local Generalizations of Lefschetz-Zariski theorems, J. reine angew. Math. 389 (1988), 157-189. [K-K] B. Kaup - L. Kaup, Holomorphic functions of several variables, De Gruyter Ed., Berlin, 1983. [LI] Le D. T., Sur les cycles evanouissants de espaces analytiques, C. R. Acad. Sc. 288 (1979), 283-285. [L2] Le D. T., Some remarks on relative monodromy, in "Real and complex singularities", Sijhoff and Noordhoff, Alphen aan den Rijn (1977). [L3] Le D. T., Le concept de singularite isolee de fonction analytique. Advanced Stud, in Pure Math. 8 (1986), 215-227. [L-M] Le D. T. - Z. Mebkhout, Introduction to linear differential systems, Proc.Symp.Pure Math. 40 Part 2 (1983), Providence, 31-63. [L-Tl] Le D. T. - B. Teissier, Varietes polaires locales et classes de Chern des varietes singulieres, Ann. of Math. 114 (1981), 457-491. [L-T2] Le D. T. - B. Teissier, Cycles evanescents, sections planes et conditions de Whitneyll, Proc.Symp.Pure Math. 40 Part 2 (1983), Providence, 65-103. [Lo] Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scu. Norm. Pisa (1965), 449-474.
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[Ma] J. Mather, Notes on Topological Stability, Mimeographed Notes, Harvard University, July 1970. [Me] Z. Mebkhout, Local cohomology of analytic spaces, Pub. Res. Inst. Math. Sc. 12, (1977), 247-256. [Ml] J. Milnor, Singular Points of complex Hypersurfaces, Ann. of Math. Stud. 61, Princeton, 1968. [M2] J. Milnor, Morse Theory, Ann. of Math. Stud. 51, Princeton, 1963. [N] R. Narashiman, Imbedding of holomorphically complete complex spaces, Amer. J. of Math. 82 (1960), 917-934. [P] D. Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967), 375-386. [S] M. H. Schwartz, Champs radiaux et preradiaux, Publications de TUER Mathematiques pures et appliquees, IRMA, Lille I, 3, 1986. [Sw] R. Switzer, Algebraic Topology - Homotopy and Homology, Springer Ed., Berlin-Heidelberg-New York. [T] B. Teissier, Multiplicites polaiies, sections planes et conditions de Whitney, Lect.Notes 964 (1983), Springer - Verlag, Berlin - Heidelberg - New York. [Th] R. Thom, Ensembles et morphismes stratifies, BuU.Amer.Math.Soc. 75 (1969), 240-284. [VI] J.L. Verdier, Stratifications de Whitney et theoremes de Bertini-Sard, Inv.Math. 36 (1976), 295-312. [V2] J.L. Verdier, Clause d'homologie associee a un cycle, Sem. Ec. Norm. Sup., Asterisque 36-37 (1976), 101-151. [V3] J.L. Verdier, Dualite dans la cohomologie des espaces localement compacts, Seminaire Bourbaki 300, Publications de r i l l P and Benjamin Pub., 1965-1966. [W] H. Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496-549. Helmut A. Hamm Mathematisches Institut Universitat Miinster Miinster, D-4400, FRG
Le DUng Trang Department of Mathematics Northeastern University Boston, MA 02115, USA
Automorphisms of Pure Sphere Braid Groups and Galois Representations YASUTAKA IHARA Dedicated to A. Grothendieck on his 60th birthday
Introduction Let Pn be the pure braid group of the 2-sphere, with n strings (n > 3), and Pn be its pro-^ completion {£ : a fixed prime number). We shall study what we call the special automorphism groups of Pn and P^ and apply it to Galois representations of the type proposed in Grothendieck [7]. The group P„ is equipped with the special conjugacy classes ^ij{= ^ji) defined from the (simple positive) intertwining of the f-th and the j-ih string {I 4) obtained by forgetting the n-th string, and its pro-^ completion fn - Pn -^ Pn-i- Then their kernels are stable under all special automorphisms (see §1.2). So, fn,fn induce the homomorphisms between the outer special automorphism groups : in : Out*Pn
-^
Out*Pn-U
In : Out*Pn
->Out*P„_i.
The key observation proved in this article is : The Injectivity Theorems.
im'^n «^^ ^oth injective for n > 5.
In the discrete case, what this leads to is : The Vanishing Theorem.
Out*P„ = {1}
(n > 3).
As S. Morita pointed out to me, this is also a direct consequence of a theorem announced in Ivanov [9] Th.l2 on automorphisms of mapping class groups of surfaces with boundaries. But we shall set this injectivity of in as our first main goal, because (i) Ivanov's method seems to be
354
YASUTAKAIHARA
geometric and different in nature from ours, and (ii) the injectivity of in (the pio-i case), which has apphcation to Galois representations, can be established by minor modifications in our arguments for the discrete case (the second goal). In the pro-^ case, Out*P„ is an infinite pTo-£ group for each n > 4, and the injectivity of in has the following implication. Consider the Galois representation ^^^ : Gal(Q/Q) - . Out P„ arising from the Galois action on the etale pro-^ coverings of the configuration space i^o,„P' = {(^1, • • • , 2n) € ( p i ) " ;
Zi ^ zj for i ^ j}.
Then as a corollary of the injectivity of i„ (n > 5), we obtain : T h e Galois Kernel T h e o r e m . have the same kernel.
The representations (pQ (n > 4)
The Galois extension Q/Q corresponding to the kernel of <^Q is nothing but the smallest common field of definition for all etale pro-f coverings of FO,4PV^GL2^P'-{0,1,OO}.
It is known [8] that Q is an infinite pro-£ extension of Q(/i^oo) unramified outside £. The above theorem implies that every etale pro-^ covering of FQn^^/PGL2j for n > 4, has a canonical Q-structure. If we consider each Out*P„ (n > 4) as embedded in Out*P4 via i^ o - - - o in, then their intersection contains a subgroup isomorphic to Gal {Q/Q{fi^^))] Gal(l]/Q(/i,^)) C f]Ont*Pn
C Out*P4.
n>4
We do not know at present whether the group in the middle is closer to that on the left or that on the right. For further comments related to <^Q , see §1.4. The injectivity of in,in{f^ > 5) is a reflection of some property of the braid monodromy. Roughly speaking, the (outer) action of P^-i on Ker/„ (:^ Fy^_2 5the free group of rank n — 2) is far from being the trivial action. This makes it "difficult" for a special automorphism of Pn to act trivially on the quotient Pn-i without acting innerly on Ker/^. (For some more sketch, see §1.2, and for details, see §§4, 5.) Actually, we shall work more with the special derivations of the associated graded Lie algebra Vn =
BRAID GROUPS AND GALOIS REPRESENTATIONS
355
gr Pn, in order to simplify these arguments. Thus, after precise statements of the main results (§1) and some preliminaries on P„(§2), our first task is to determine the structure of T^n over Z explicitly (§3). The structure of Vn'S'Q was determined by Kohno as a special case of his work [11], but we shall proceed directly using explicit generators. In §4, we shall prove a Lie version of the injectivity of i„, assuming two key lemmas. These lemmas will then be proved in §5. In §6, we reduce the injectivity of 2„ to its Lie version. In §7, we indicate the "new" ingredients for the injectivity of the pro-^ homomorphism in. Some analogous results on pure braid groups of punctured Riemann surfaces will be given in a joint paper with M. Kaneko (in preparation). The author is very grateful to M. Asada, J. Birman, P. Deligne, T. Kohno, S. Morita and Takayuki Oda for helpful communications. Conventions. In what follows, when topological groups are treated, subgroups will always mean closed subgroups, and the commutator {A,B) of two subgroups A, B means the topological closure of the algebraic commutator. All homomorphisms (resp. automorphisms) are continuous (resp. bicontinuous), and the short exact sequences 1-^ A-^ B-> C - ^ 1 are strict^ i.e., the topology of ^ (resp. C) is the one obtained by restriction (resp. taking the quotient) of that oi B. §1 The main results. 1.1. Let 'P]-, be the complex projective line, n be an integer with n > 3, and consider the configuration space (1.1.1)
Yn = Fo,nP^ = {(^1, ...,^n) e ( P ^ ) " ; Zi ^ zj for i ^ j}.
Its fundamental group Pn = TTI (y„, b) is the pure sphere braid group with n strings ("sphere", because P ^ « 5^ topologically). As a base point b = (6i, ...,6„) G Ym we shall choose such a point that 6i, ...,6„ lie on the unit circle S^ C P^(C) in the anti-clockwise order (Fig 1). bo
(Fl£ 1)
This is an "intrinsic base point" in the sense that if 6' = (fci,...,&^) is another such base point then there is a canonical identification of 7ri(yn, 6')
356
YASUTAKA IHARA
with 7ri(y„,6) via an isotopy on S^. This identification is well-defined, because the element of P„ induced from the rotation of 5^ belongs to the center C2 of P„ (cf §2.1). For each i,j {I < i^j < n, i ^ j), call Xij = x\^^ the element of (1.1.2)
7ri(P^ - {61, ..,6,-, ..,6„}, 60 C Pn
represented by the loop described in Fig 2. Also, for convention, put xa — 1(1 < i < n). Then Xij — ^ji (as element of Pn), and as is well-known, Pn is generated by all the x^j 's (§2). We shall consider the following subgroups of the automorphism group Aut P„ and the outer automorphism group Out P„, respectively. (1.1.3) Aut*Pn -{(re Aut Pn\ (TXij ~ Xij (1 < ij < n)}, (1.1.4) Out*P„ = Aut*Pn/Int Pn. Here, ^ denotes conjugacy in P„, and Int P„ denotes the inner automorphism group of Pn. 1.2. The following theorem is a direct consequence of Theorem 12 of Ivanov announced in [9], via a comparison theorem of Birman [3] (Theorem 4.2). The Vanishing Theorem.
Out*Pn = {1}
(n > 3).
The main ingredient for our alternative proof is the following Injectivity Theorem (i). For each n > 4, consider the natural fibration Yn —> Yn-i defined by the projection (-2^1,..., 2^n) ^-^ ('^^i ? ...j-^n-i). This induces a surjective homomorphism fn '- Pn ^^ Pn-i- Its kernel is generated by ^ni> ...J ^n,n-ij ^Lud hence is stable under any a G Aut*Pn. Therefore, fn induces a homomorphism (1.2.1)
in : O u t * P n - > O u t * P n _ l .
The Injectivity Theorem (i). If^yb, in is injective. Actually we shall prove this via its Lie version, but here, we explain the idea in terms of groups. First, Nn = Ker/n is generated by Xnj{l < j < n — 1) which satisfy a single relation Xn n—1 * * ''^ni = 1. (Thus, Nn ^ Pn-2.) The centralizer H = Cp^{xn,n-i) satisfies NnH = Pn, Nn H H = < Xn,n-i > ; hence if cr G Aut*Pn acts trivially on P n - i , it acts almost trivially on H. Suppose a acts trivially on H. Then for each j(l < j < n — 2),cr(xnj) must commute with every such element 7 of Pn that centralizes both Xnj and Xn^n-i- (Note that 7 G H;hence a{j) = 7.) From
BRAID GROUPS AND GALOIS REPRESENTATIONS
357
this we obtain (by explicit calculation of centralizers in P„) that a{xnj) G < Xnj,Xn^n-^i >• This, Combined with the relation (T{xn^n-i)' * '<^{^ni) = 1, leads to : a must be inner by some power of Xn^n-i- This last step is done by some (delicate) free Lie calculus. Remark 1.2.2. The outer automorphism group of the full braid group of E"^ is determined in [5]. As for automorphism groups of surface mapping class groups of genus > 2, see also [13]. Our method is quite different from those used in [5] [9] [13]. 1.3. Now fix a prime number ^, and denote by ^ the pro-^ completion in the category of groups. Denote the image of Xij under the canonical homomorphism Pn —^ Pn also by Xij, call fn '- Pn -^ Pn-i (^ ^ 4) the pro-^ completion of / „ , and define Aut*Pn, Out*P„ analogously (i.e., simply replace Pn by Pn in (1.1.3), (1.1.4)). In this pro--^ case, Out*Pn, for each n > 4, is an infinite pxo-£ group containing the image of a "large Galois representation" (described below). Our main result is : The Injectivity Theorem (ii). phism
/ / n > 5, the canonical homomor-
in : O u t * Pn - > O u t * P n - l
induced from fn is injective. An implication of this to Galois representations will be explained below. 1.4. Let n > 4 , use the symbol Yn now for the Q-scheme Po.nP^, and consider the exact sequence of algebraic fundamental groups : (1.4.1)
1 - irf^(y„ ® Q) - irf^Yn)
- G Q - 1,
where G Q = Gal(Q/Q). This induces a homomorphism of GQ into the outer automorphism group of TT^ ^(y„ 0 Q); and hence also into the outer automorphism group of the pro-^ completion Pn = TT^^^" (Yn 0 Q) of Pn :
(1.4.2)
^ 5 ^ G Q - . Out Pn.
It is easy to see that (fQ{o-)xij ^ xf>^^ {a G G Q , 1 < i , j < n), where X ' GQ -^ Zf denotes the ^-cyclotomic character describing the GQ—action on fi^oo . Therefore, ^Q'^ induces a representation (1-4-3)
<0^,.)^^Q(..o.)—Out*Pn
358
YASUTAKA IHARA
of GQ(/i^^) = Gal(Q/Q(^^oo ))• It is easy to see that ^Q^^ ^ ) (n = 4,5,...) are compatible with the i„. Since Ker (p^ = Ker (p)^}
N, the Injectivity
Theorem (ii) gives : The Galois Kernel Theorem.
Ker^Q^ = Ker (^Q
(n > 4).
Since Y4/PGL2 c^P^ - {0,1, 00} and (1.4.4)
P4 ^ (Z/2) X 7ri(P^ - {0,1,00}),
the representation <^Q is essentially the same as the Galois representation in Trf''"^(Pi - {0,1,00}) studied in [1],[4],[8] etc. The Galois extension n of Q corresponding to K e r ^ g is an infinite non-abehan pro-^ extension of QifJ'ioo) unramified outside £ which is generated by higher circular £-units [1]. The above corollary says that we do not obtain an extension bigger than fi by considering (pQ for higher n. (This may be regarded as an "affirmative evidence" for the question raised in [1]: "Is Cl the maximal pro-^ extension of Q{fi^^) unramified outside £ ?") On the other hand, as the Grothendieck program [7] may suggest, the consideration of (p)^^ for higher n might give more information on the image of v^Q in Out P4. Indeed, Imy?Q.^ ^ must be contained in the image of Out*P„ in Out*P4 for all n > 4 (and in fact, in that of (Out*Pn)*^", the 5^-symmetric part). But the author does not know at present whether the common image of (Out*Pn)*^" {n > 4) in (Out*P4)^^ is actually smaller than (Out*P4)'^^ §2 Preliminaries on the group P„. 2.1. Recall that Yn = Fo.nPc and P„ = 7ri(y„,6) (§1.1). We shall recall some basic known facts concerning P„ necessary for our purpose. Proposition 2.1.1.
Yn ^homeo i^3,n-3Pc >< PGL2{C)
{u > 3),
where i^3,„-3Pc = ^0,„-3(Pc - {0, 1, ^ } ) (cf. [3] §1.2 for the notation Fm,n)Corollary 2.1.2. Z/2.
P„ ~ 7ri(y„/PGL2(C)) XC2
(n > 3), where C2 =
BRAID GROUPS AND GALOIS REPRESENTATIONS
359
In particular, P3 = 7ri{PGL2{C)) 2^ C2 and P4 = ^ i ( P c " {O^l.oo}) X C2 — F2 X C2J where, in general, Fn denotes the free group of rank n. We shall also need the following basic Proposition 2.1.3 [6]. If n > 4, the fibration Yn —> Yn-i defined by (ziy...^Zn) -^ (2^1, ...,Zn-i) has PQ — {61, ...,frn-i} fl5 the fiber above (61,.., 6„_i), and induces a short homotopy exact sequence (2.1.4)
l—,Nn—^Pn^
Pn-1 — 1,
where Nn = 7ri(P^ - {61, ...,6„_i},6n) c^ Fn-2. Proof
7r2(yn-i(C)) = 1
([6] §11).
Q.E.D.
Let Xij{l < i,j < n) be the elements of P„ defined in §1. Then by the definition of x^j^s and Prop 2.1.3, it is clear that Ker/„ is generated by the Xnj^s (1 < i < ^ — 1), and is in fact freely generated by any n — 2 of them, because Xn,n-i - •' Xni = 1. As fn{Xij) = a^lj" for ij < n (§1), we obtain by induction on n that Pn is generated by all the Xij^s. Corollary 2.1.5. (i) P„ is generated by Xij (1 < iJ < n) ; (ii) Nn is freely generated by any n — 2 of Xnj (1 < J < ^ — !)• 2.2. The order of multiplication in Pn will be chosen as follows. The class [/3 o a] of the composite /? o a of two loops a, /? (a first, /? next) is, by definition, [/?] [o^]. For 71,72 G Pny we define the commutator (71,72) = 7i727rS2"^Proposition 2.2.1. The generators Xij (1 < i^j < n) of Pn satisfy the following relations. (In (R3) ^ (R5), i^j^k^i are mutually distinct.) (Rl)n
Xii = 1,
Xij = Xji
(R2)n (R3)„
Xin"'Xii = l (xki.Xij) = 1,
(1 < i, j < n),
(1<«<^), if {ij}n{k,e} = <j) and bk.bi lie
on the same connected component of S^ — {bi^bj). (R4)n 'if {ijj}ri{kji}
(xki^xij) =
{{xr^\xr/)^xij),
— (f) and bi^bj^b^^bi lie on S^ in the order as shown by (Fig
3). (R5)n
(xki^Xik) = {^iki^u)^
order as in (Fig 4).
2/ bi,bj,bk lie on S^ in the
360 Proof
YASUTAKA IHARA This can be checked directly by chasing the braid monodromy.
Remark 2.2.2 Actually (Rl) ~ (R5) together gives the system of defining relations for P„. This is also known and can be proved exactly in the same manner as in the proof of Prop 3.2.1 (its Lie version). But this will not be needed for our purpose (in this form). 2.3. By Cor 2.1.5 (ii), the subgroup < Xni, ...,^n,n-i > of P„ generated by Xni, ...jXn^n-i is normal in P^ and is free of rank n — 2. By changing indices, we see that, for each i ( l < i < n), Nn =< xn,-- - , Xin > is also normal and free of rank n — 2. Proposition 2.3.1. Let 1 < i, j < n, i ^ j^ and C{xij) centralizer of Xij in Pn- Then Pn = Nn 'C{xij).
be the
Proof If suffices to show that every P„-conjugate of Xij is an M'^conjugate of Xij. Since P„ is generated by the Xki^s and TV^ is normal in Pn, it suffices to show that XkiXij xj^/ is 7V„ -conjugate to x^j for all k,i. But this is obvious by the relations (R3) ^ (R5). Q.E.D. §3 The structure of Vn — gv Pn< 3.1. In general, for any topological group G, the graded Lie algebra gr G of G is defined as grG=
e m>l
g r " ^ G - 0 G(m)/G(m + 1), m>l
where {G(m)}m>i is the lower central series of G, (G(l) — G, G{m-\-1) = {G,G{m)) (m > 1)), and gr^G = G{m)/G{m -\- 1) is considered as an additive group. If a G G{m), b G G{n) {m,n > 1) represent a G gr^G,6 G gr"G, respectively, then (a, 6) = aba~^b~^ G G(m-h n), and its projection on gr'^'^^G depends only on a,b which is denoted by [a, 6]. By linearity [a, 6] is defined for all a, 6 G gr G, which gives gr G a structure of graded Lie algebra. As a Lie algebra, it is generated by gr^G. In particular, if G is a free (resp. free pro-^) group of rank r on a^i, • • • ,Xr, then gr G is a free Lie algebra of rank r over Z( resp. Zi) generated by the classes X i , - - - ,Xr G gr^G of iCi, • • • ,x^. Note that any homomorphism G ^ H oi topological groups induces a homomorphism gr G ^^ gr iJ of their graded Lie algebras. It is easy to see that the functor G ^ gr G is right exact . We shall need the following
BRAID GROUPS AND GALOIS REPRESENTATIONS L e m m a 3.1.1.
361
Let
(3.1.2)
i—^N—^G—^H
—. 1
be an exact sequence of topological groups such that (i) [G^N) — {N,N), and (ii) gr N has trivial center. Then the natural homomorphism gr N -^ gT G is infective ; hence (3.1.2) induces an exact sequence (3.1.3)
0 —^ gr TV —> gr G —> gi H — . 0.
Remark 3.1.4 A similar lemma is given in [12]. Proof By P. Hall (cf.e.g.[14] Th 5.2), if A, B, C are three normal subgroups of any group G, then (3.1.5)
{A (B, C)) C (B {C, A))iC (A, B)).
We shall apply this to prove {A),n.n
{G{m),N{n))cN{m+n)
(m,n > 1). Note that {A)i^i is nothing but the assumption (i). Let us prove (A)m = 'XA)m,n for all n > 1", by induction on m > 1. First, for m = 1, apply (3.1.5) for A,B,C = G,7V,A/'(n) , respectively, to prove {A)i^n by induction on n > 1. This is straightforward. Then apply (3.1.5) for A,BjC = 7V(n),G,G(m), to prove {A)m by induction on m. This is again straightforward. Thus, {A)m,n is established. Now we shall check {B)m
NnG{m)
= N{m)
by induction on m > 1. For m = 1, this is obvious. Suppose {B)m is vahd, and take any n e N D G{m + 1). Then n e N H G{m) ~ N{m). As n G G{m + l ) , n acts trivially on N/N{m + 2) (by (A)m-\-i,i) ; hence the class n G gv^N of n centralizes gr^ A'' ; hence also gr A''. By assumption (ii) this implies n = 0, i.e., n G N{m-{-l). Therefore, NnG{m-\-l) C iV(m-hl). The other inclusion is obvious. This settles ( 5 ) ^ . This implies that the natural homomorphism gv N —^ gr G is injective. Corollary 3.1.6. quence
The exact sequence (2.1.4) induces an exact se-
O^gvNn—^gvPn-^
gr Pn-l
" ^
0.
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YASUTAKA IHARA
We shall write Vn = grP„, Af^"^ = grM*^ and Afn = Af^""^ = Ker {Vn —^Vn-i)' Sometimes the subscript n will be dropped. 3.2. Now we shall determine the structure oiVn = gr Pn- Call Xij{l < h J < ^) the image of Xij in gr^P„. Note that An is a free Lie algebra over Z generated by (any n — 2 among) Xij^s {I < j < n^j ^ i). Proposition 3.2.1. The Lie algebra Vn = grPn (^ > 3) is the quotieni of the free Lie algebra over Z generated by Xij (1 < i^j < n) modulo the following relations. [Linear relations]
[LI]
Xa = 0
(1 ^ ^ ^ ^)5
[L2] Xij=Xji [L3] S!?^iXi^=0 [Quadratic relations] [Ql]
(l<2,i
[Xij^Xki] = 0, if{ij}
H {fc,^} = (j).
The same statement for (grP„) 0 Q was given by Kohno as a special case of his work [11]. We shall give a direct proof which works over Z. Remark 3.2.2
The elements Xij also satisfy
[Xij , Xij -f Xjk -f- Xjki] = 0
(i,i, Ar : distinct),
but this is a consequence of (LI) ~ (L3),(Q1). (Replace Xjfc + Xki by -T,i^ijXki and use (Ql).) Proof First, by Prop 2.2.1, the X^j's generate Vn and satisfy (LI) ^ (L3), (Ql). Call Vn the quotient of the free Lie algebra over Z generated by the symbols Xij^s modulo these relations. Let Vn —^ Vn be the projection. We shall show by induction on n > 3 that this is bijective. For n = 3, both are Z/2. For n > 4, we have a commutative diagram 0
> Nn
(3.2.3)
> Vn surjective
0
^ Afn
^ Vn
> Vn-1
>0
isomorphism by I induction a s s u m p t i o n
^ Vn-1
^ 0,
where Vn -^ Vn-i is defined by X^i —^ 0 (1 < z < n). The kernel Afn of the projection Vn —^Vn-i is an ideal generated by X„i, • • • ,X„ n-i- But
BRAID GROUPS AND GALOIS REPRESENTATIONS
363
it is, in fact, generated by X„j's as a Lie subalgebra over Z. This is clear because of (Ql) and [Xkj , Xnj]
= [—T,i:^j Xkl,
Xnj]
= —[Xkn , ^ n j ]
for k ^ j,n. Since we have [L3], Afn is generated by Xnjil < i < ?^ — 2). But Afn is free of rank n — 2 on these Xnj's ; hence Afn —^ J^n must be bijective ; hence Vn —^VnQ.E.D. 3.3. In general, for a Lie algebra £, its subalgebra £', and elements Of,/?, ••• G £, we shall denote by < a,/?, ••• > the Lie subalgebra of C generated by a,/?, • • •, and by Cc{C) the ceniralizer of C in C. We shall need the following Lie-analogue of Prop 2.3.L Proposition 3.3.1. Lei i,j (1 < i, j < n) be two distinct indices, and write C{) = C'p^{ ) for the centralizer in Vn- Then (i)
C{Xij) =< Xij,Xu
("•)
Vn = ^f^'^ +
(in)
TVi') n C{Xij) = ZXij
Proof (3.3.2)
ik,£e{l,---,n}\
{i,j})
>,
ciXij), C gv'Tn •
We may assume i — n, j = n— 1. Put C =< X „ , „ _ i , X « ilC
C(Xn,n-x).
By [L2][L3], Vn is generated by Xij for i,j
Vn-i,
Vn=^f+C'.
Since Af is free on any n — 2 of X„j(l < j < n — 1) we have (3.3.4)
^ n C ( X „ , „ _ i ) = ZX„,„_i c
C.
But from (3.3.2) ~ (3.3.4) we obtain C = C(X„_„_i). In fact, take any c G C(X„_„_i) and decompose as c = n + c' (n £ Af,c' ^ C). Then n = c-c' e C(X„,„_i) nAT C C". Therefore, c G C . Therefore, we obtain (i) ~ (iii). ' Q.E.D.
364
YASUTAKA IHARA
§4 S p e c i a l d e r i v a t i o n s o f P„ ; A Lie v e r s i o n o f t h e I n j e c t i v i t y Theorem. 4.1. A derivation D :Vn —^Vn of the Lie algebra Vn = gr Pn (^ > 4) will be called a special derivation of degree m (m > 1), if for each i, j (1 < hj ^ ^) there exists some Ttj G g r ^ Pn such t h a t (4.1.1)
D{X,,)^[T,j,
Xij].
An inner derivation Int(T) ( T G gr^Vn) is a special derivation of degree m. T h e least degree m, for which there exists a special derivation of "Pn of degree m which is not mner^ is m = 3. An example of such a derivation is given by (4.1.1), with n
(4.1.2)
Ti^ = ^ [ ^ z f c , [Xi^. Xi^]]
(1 < i , i < n ) .
fc=i
Now since TVi'^ (1 < i < n)
is an ideal of Vn generated by Xn^ - • •
each special derivation D maps Afn a derivation D ofVn-i
= Vn/J^n
into itself. In particular, D
\ which is again a special derivation of
degree m such t h a t D ( x f ~^^) = [Tij, x f "^^] (1 < ij Tij is the image of Tij in
,Xin,
induces
< n - 1), where
Vn-i-
4 . 2 . T h e main aim of §§4 '^ 5 is to prove the following T h e I n j e c t i v i t y T h e o r e m (Lie v e r s i o n ) . Lei n > 5, and D be a special derivation of Vn of degree > 1. If the induced derivation D of Vn-i is inner, then D itself is inner. In this section, we shall reduce this proof to two key lemmas. proof of these lemmas will be given in §5. 4.3 K e y lemmas. Recall §3.3 for the notation (a,/?, •••) algebra) and, Cc{C') (centralizer). Lemma 4.3.1. distinct. Then (*)
Let
n > 4, and i,j,k
C^(fc) (Cv{{Xki,
Xkj)))
(1 < i , j , Ar < n) be
= {Xki,
The (sub-
mutually
Xkj),
where A/'W == A/'i^^ V = VnL e m m a 4 . 3 . 2 . Let C be a free Lie algebra over Z on r -\- 1 variables X o j X i , • • • , X r (^ > 2), and put Cj = ( X Q , XJ) (1 < j < r). Suppose that ^j ^ A' (^ ^ i — '') ^^^ S E C are homogeneous elements of degree > 1 such that [Xi,Sl]
+•••+
[Xr,Sr]
Then S = Si = • • • = Sr = 0.
= [Xo+X,
+ --- +
Xr,S].
BRAID GROUPS AND GALOIS REPRESENTATIONS
365
4.4 Proof of the Injectivity Theorem (Lie version) assuming lemma 4.3.1, 4.3.2. Let n > 5, and Z) be a special derivation oiVn of degree m > 1 such that the induced derivation D oi Vn-i is inner. Our goal is to show that D itself is inner. Obviously, we may assume that D = 0, i.e., D{Vn) C M = ^jr\ Put DiXij) = [T,j, Xij] {Tij e gx^Pn\ 1 < hi < n). We may assume that Tnj G A^ (1 < i < n), because Vn = J^ -i- Cr{Xnj) (Prop 3.3.1 (ii)). Since we may replace D by D — Int(Tn,n-i) without affecting our assumption D = 0, we may further assume that D{Xn,n-i) ~ 0 and Tn,n-i = 0 . We claim that (4.4.1)
D{Cv{Xn,n-l))
= 0.
To check this, take any homogeneous element a G C'p{Xn,n-i)' Then [a, Xn^n-i] = 0- Apply D on this equality, noting D{Xn,n^i) = 0 ^nd D{Vn) C A/", to conclude D{a) G Cj^{Xn,n-i)- But Cj^{Xn,n-i) = Z • Xn,n-i C gr^Pn (Prop 3.3.1), and D{a) has degree = m -{- dega > 1. Therefore, D{a) = 0. This proves (4.4.1). We now claim that (4.4.2)
D(X„i) 6 Cj^{Cr{Xr,i, X „ , „ _ i » .
To check this we may assume i < n — 1, as D{Xn,n-i) = 0. Take any a € Cv{Xni, Xn.n-i). Then [X„i, a] = 0, and by (4.4.1) we have D{a) = 0. Therefore, [D(A'„i),a] = 0. Therefore, D{Xni) centralizes C'p{Xni^ Xn,n-i)' As D{Xni) G AT, (4.4.2) follows. By lemma 4.3.1, applied for k — n, j := n — 1, (4.4.2) implies that (4.4.3)
D{Xni) G {Xni, Xn,n-l)
{I < i < n - 2).
But since A' is free on any n — 2 of Xnj ( l < i < ^ — 1), the multidegree decomposition of A/' shows that (4.4.3) imphes (4.4.4)
Tni G {Xni^ Xn,n-i)
Now since Xni +
(1 < ^ < n - 2).
h Xn^n-i = 0 and D(X„,„_i) = 0, we have n-2
J2D{Xnj) = 0, or equivalently, (4.4.5)
[Tn2, Xn2] +
H [T„^n-25 ^n,n-2]
— [^nl, A^n2 "h
V Xn,n-l\,
366
YASUTAKA IHARA
with Tnj G {X„j, Xn,n-i) (^ < J < n —2). Therefore, when m > 1, lemma 4.3.2 applied to the case r = n — 3 (> 2) and XQ = X„^„_i gives (4.4.6)
T„j=0
(l<J
When m = 1, by (4.4.4) we may assume Tni — ^i-^n,n—i (1 ^ ^ ^ ^ — ^)) with Qi E Z. But then (4.4.5) imposes that aiXni + ••• + an-2-^n,n-2 commutes with Xn^-i- Therefore, ai = ••• = ct„_2, which we call a. Then Tni = '- = T„n-2 = «-^n,n-i- Therefore, if we replace D by D — alnt{Xn,n-i), we also obtain (4.4.6). Since Tn,n-i = 0 (our earlier normalization), (4.4.6) gives (4.4.7)
D{Af) = 0.
Combining this with (4.4.1) and using Prop 3.3.1 (ii), we obtain Z) = 0, as desired. Thus, the Injectivity Theorem (Lie version) is reduced to lemma 4.3.1, 4.3.2. §5 Proofs of lemma 4.3.1, 4.3.2. 5.1. In order to prove lemma 4.3.1, we shall first prove the following elementary Lemma 5.1.1. Lei C be a free Lie algebra over Z generated by X i , • • • ,Xr,Xr+i (r > 1), and D be a derivation of C of degree 1 defined by DiXi) = ---^D(Xr) = 0, D(Xr+l) = [Xr, -X'r + l]. Then the kernel of D ts the subalgebra of C generated by Xi,--,Xr. Proof Extend D to a derivation of the universal enveloping algebra A = ®m>oAm of -C, the free associative algebra over Z generated by Xi, • • • , X , + i . Thus, L»(a ± /?) = D{a) ± D{l3), D{a0) = a£>(/?) + D{a)l3 (a,/? G .4), and DiXi) = ••• = D{Xr) = 0, £>(X,+i) = X,X,+i - X , + i X , . For each monomial ^ = Xi^ • • Xi^ G A, define its (5-degree by (5(0 = 0 = Max{jy;
.--if i^9^r-f 1, allz/, i^ = r + 1)
• • -otherwise.
This measures the position of the leftmost Xr^i in <^, from the right. We shall show that if D{a) = 0 for a E Am^ then a does not contain Xr-\.i. This is sufficient, as the embedding C "^-^ A respects the multidegree decomposition. Suppose that a contained Xr+i, and express it as a Z - hnear combination of monomials of degree m. For each i/ (0 < u < m), call a^
BRAID GROUPS AND GALOIS REPRESENTATIONS
367
the partial linear combination over the monomials of (5-degree jy. Call /i the smallest i/ with a^ / 0. We may assume // > 0. Put a' = a — a^. Then it is clear that (i) each monomial in D(a') has 6-degree > /i, (ii) each monomial in D{a^) has S -degree > /i, and (iii) the contribution of D(a^) to the (5-degree // is obtained from a^ by replacing Xi^ (= ^ r + i ) by XrXr+i for each monomial Xi^ • • -Xi^^ contained in a^. Therefore, the //-part of D{a) cannot vanish. Therefore, D(a) 7^ 0, a contradiction. Therefore, a cannot contain Xr+iQ.E.D. 5.2 Lemma 5.2.1.
/ / i^j^k (1 < i^j^k < n) are distinct, then
Cj^(jc) {Xij) - {Xki ;
I
^ij,k).
Proof We may assume i = l^ j = 2, k = n. Consider the derivation D of Af — Afn of degree 1 obtained by restriction of Ad(Xi2) to the ideal A/'. Choose Xni (2 < i < n — 1) as a free basis for Af. Then D{Xm) = 0
(i>3),
D(Xn2) = [X12, X,2] = - [ X i „ , Xn2] = [W, X„2], where W = ^ Xni- Since we may choose {Xn2y'' * ^Xn,n-2^ ^ } also as a free basis of A/', and since D{W) — 0, lemma 5.2.1 is reduced to lemma 5.1.1. Q.E.D. 5.3 Proof of lemma 4.3.1. First it is clear that the RHS is contained in the LHS of (*). To prove the other inclusion, we may assume i = 1^ j = 2^ k — n. Since C-pUXni, Xn2)) contains all Xki with 3 < A:, / < n — l,"its centralizer in AT = AT^") is contained in (5.3.1)
f]
C^{Xu)-
3
But by lemma 5.2.1, Cj\j'{Xki) = {Xni; (5.3.2)
C^{Cvi{Xnu
Xn2))) C
i ^ kj,n). f]
{Xni;
Hence (take k — ^) i^SJ.n).
4
By looking at the multidegrees in Xni (« 7^ 3,n), we see easily that the intersection on the RHS of (5.3.2) is {Xni, Xn2)- Therefore, the LHS of (5.3.2) is contained in {Xni, A^n2)Q.E.D. 5.4. We shall proceed to the proof of lemma 4.3.2. First, we need the following probably well-known
368
YASUTAKA IHARA
Lemma 5.4.1. Let C be a free Lie algebra on X i , - - ,Xr (r > 2) over a domain R of characteristic 0, and embed C into the free associative algebra A on Xi,- - - , X^. If a £ C is homogeneous with degree > 1 and, as an element of A, divisible by Xi from the left (i.e., a = XiP with (3 E A), then a = 0. Proof Define an i?-linear map a : A -^ C hy
Take any homogeneous element a E .4 of degree m > 0. Then a belongs to C if and only if o'(a) = ma (Dynkin-Specht-Wever criterion; cf. [10]V §4, or [14] §5.9). Now let a E £ be homogeneous of degree m > 1 and a = ^iP (/^ G w4). Since a ^ C^ cr(a) = ma. Moreover, since a = XifS, o-(a) = [Xu cr(/?)]. Therefore, mXiP =:ma = a{a) = Xia{P) - cr(/?)Xi ; hence cr(/?) E XiA. Note that cr(/?) is of degree m — 1. If m = 2, then a{l3) = cXi [c E R)] hence ma = [Xi, cr(/?)] = 0; hence a = 0. If m > 2, we obtain by induction on m (which enables us to assume cr(/?) = 0) that a = 0. Q.E.D. 5.5 Lemma 5.5.1. Let C be a free Lie algebra on three independent variables X , y , Z over a domain R of characteristic 0, and Cx,Y be the free Lie subalgebra of C generated by X^Y, Let S E Cx,Y be homogeneous with degree > 1. / / [Z, S] E [¥,£], then S = 0. Proof Embed C into the free associative algebra A over R on X,Y,Z. Then (5.5.2)
ZS-SZ = YT-TY
with some T £ C. Decompose each a E A uniquely as a == ao • 1 + ax -X -{• ay • Y -{• az ' Z, with ao E R^ ax^ « y , az E A. Take the Z-component of each side of (5.5.2) (noting 5 E >Cx,y) to obtain —S = YTz. But then S E Y A ; hence 5 = 0 by lemma 5.4.1. Q.E.D. 5.6 Proof of lemma 4.3.2. (5.6.1)
i: = e a £ ^ ^ ^
Decompose C as a = (ao,ai,... ,a,),
where C^^^ consists of homogeneous elements of multidegree a = (ao, ••• ,ar) in {Xo,-" , ^ r ) . Now let S E C, Sj E Cj (1 < j < r) be homogeneous with degree > 1 satisfying (5.6.2)
[Xi, 5i] + ... + [Xr, 5,] - [Xo + Xi + . . . + X,, 5].
B R A I D G R O U P S AND GALOIS R E P R E S E N T A T I O N S
369
Suppose S ^ Q, and decompose 5 as 5 ^ E a 5'^''^ with S'(^) G C^^'l Let a = (ao, • • • , a r ) be a multidegree such t h a t , among those a with S^^^ ^ 0, a has the maximal value of GQ. Then such an a must be ;?wre, i.e., there exists only one j {I < j < r) with aj > 1, and moreover, aj > 1 for this j . In fact, [XQ, S^^^] is ^/le (ao + 1, a i , • • • , ay.)-component of the RHS of (5.6.2), but from the LHS of (5.6.2) appears only pure multidegrees. T h a t aj > 1 is obvious {if aj = 1, Sj cannot exist). Now choose any index i with 1 < i < r and i ^ j . (This is possible because r > 2.) We shall show t h a t (5.6.3)
[Xi, 5(<^)] € [Xo, C].
To show this, note first t h a t [X^, 5^^^] is of degree ao in X Q , 1 in X^, aj > 1 in Xj, and degree 0 in all other variables. Such an element cannot come from the LHS of (5.6.2), and hence it must be cancelled out by some linear combination of other terms [Xk^S^^^], /3 = (^O)--* ^^r) on the RHS. If k :i 0 then 6o = ao ; hence /? must also be pure. In this case ana [Xk, S^^^] can have the same multidegree only if ^ = i and /3 = a, which is an excluded case. Therefore, k = 0, and hence [Xi, 5^^^] G [XQ, £ ] . Now apply lemma 5.5.1 for X = Xj^Y = XQ and Z = Xi, to conclude gW — 0, a contradiction. Therefore, 5 = 0. Therefore, the multidegree decomposition of each side of (5.6.2) gives Si = • • • = Sr = 0. Q.E.D. §6 P r o o f o f t h e V a n i s h i n g T h e o r e m . 6.1.
T h e main ingredient of the Vanishing Theorem is the following
T h e Injectivity T h e o r e m (i). ^/ ^ > 5, ike canonical phism in : Out*P„ - > O u t * P n - i is
homomor-
injecttve.
First, we shall show how this leads to the Vanishing Theorem. As P3 :^ C2,Out*P3 = (1). T h u s , we have only to know Out*P4 = (1). B u t P4 c^ C2 X P2, where P2 = Ker(P4 - ^ P3). Thus, each element a of Aut*P4 respects this decomposition, inducing an automorphism of P2- Clearly, such an automorphism of P2 acts trivially on the abehanization Pf*. But by a classical result of J.Nielsen [15], such an automorphism of P2 must be inner. Therefore, Out*P4 = (1). Therefore, the Vanishing Theorem is a consequence of the Injectivity Theorem (i). 6.2. We shall reduce the Injectivity Theorem (i) to its Lie version (§4.2). In this step we need the following
370 L e m m a 6.2.1.
YASUTAKA IHARA Consider the projective P„=lim(P„/P„(m)).
limit
m
Then, (i) the canonical homomorphism Pn —>• Pn is mjective; (ii) if i £ Pn is such that lnt{t) : 7 -^ tjt~^ stabilizes Pn and induces on P„ an automorphism belonging to A u t * P „ , then t G PnProof (i) This is equivalent to C\^Pn{'^) = ( I ) , which we shall prove by induction on n > 3. When n = 3, P^ — C2 and ^3(2) = (1). Let n > 4. T h e n the projection fn '- Pn -^ Pn-i niaps P n ( ^ ) onto Pn-i{m). Therefore, by the induction assumption, p | ^ P „ ( m ) C Nn — ^^^fn — Fn-2- Moreover, by lemma 3.1.1, we have (6.2.2)
Pn{m)nNn=Nn{m).
Therefore, f]^ P n ( ^ ) C Hm ^ n ( ^ ) — (1) group being well-known).
(^^^ ^^^ equality for the free
Incidentally, (6.2.2) shows t h a t Nn = limNn/Nn{m)
is injectively
map-
m
ped into P „ . (ii) This proof will be by induction on n > 3, using the exact sequence
l^Nn—Pn—Pn-l-^l^ T h e induction assumption implies t h a t we may assume t G Nn- T h e n Int({) induces an automorphism of Nn stabilizing Nn =^ NnCiPn- (The last equality is obvious, as P n - i —^ P n - i is injective.) Since this a u t o m o r p h i s m is assumed to belong to Aut*Pn, and since each P„-conjugate of Xni (1 < 2 < n — 1) is an Nn -conjugate of Xm (Prop. 2.3.1), there exist ti G Nn (1 < i < n — 1) such t h a t ixnd'^ — tiXnitJ^- T h e n tj^i centralizes Xni in NnBut since gr Nn — gr Nn is free and generated by the classes of Xni^s, the centralizer of'Xni G gr^A^n (the class of Xni) in gr7V„ is Z'Xni- Therefore, (i) any element of 7Vyj(2) which commutes with Xni must be 1, and (ii) any element of Nn{l) which commutes with Xm must be an integral power of Xni (« priori, modulo 7Vn(2), but by (i), exactly so). Therefore, t~^i must be an integral power of Xni and hence belong to Nn- Therefore, t G Nn. Q.E.D. 6.3 P r o o f of t h e I i i j e c t i v i t y T h e o r e m (i). Let n > 5, and a G A u t * P „ . Assume t h a t a induces an inner automorphism of P „ _ i = Pn/NnOur goal is to prove t h a t a itself is inner. We may assume cr ^ 1. As a G Aut*Pn, a acts trivially on Pn/PnC^)- By lemiTia 6.2.l(i) there exists a maximal integer m > 1 such t h a t a acts trivially on Pn/Pn{fT^-i- ! ) • Our first (and the main) goal here is t o apply the Injectivity Theorem (Lie version) (§4) to prove the following.
BRAID GROUPS AND GALOIS REPRESENTATIONS
371
[Claim 6.3.1] There exists some tm G i^n(^) such that the action of a on Pn/Pn{f^ + 2) IS the same as that of Int(t^) on Pn/Pn{fn 4- 2). Proof of [6.3.1] As cr G Aut*P„, there exists, for each i,j (1 < hj ^ ^) ^^ element tij G Pn with o-{xij) = tijXijt~-^. By Prop 2.3.1, such tij can be chosen from the free group Nn = (ar,i, • • • ^Xin) — i^n-2- As a acts trivially modP„(m + 1), {tij, Xij) G Pn(m-|-l)nM*^ = M*^(mH-l) (lemma 3.1.1). But since Nn is free, this implies that tij can be chosen from Nn (m). In particular, we may assume tij G Pn{m). Now (7 induces a special derivation of gr Pn of degree m, as follows. As a acts trivially on Pn/Pn{'^ -h 1) and hence also on Pn{d)/Pn{m + c/) for all d > 1, cr induces a mapping Pn{d)
3jy-^
(T{J)J-^
G Pn{d +
m)
for each d > 1. It is easy to see that this induces a well-defined Z-hnear mapping D^ : gr^P„ —» gr^"'"'"P„, and that D = {D^) is a special derivation of gr Pn of degree m with D{Xij) = [Tij, Xij], where Tij is the class of tij in gr^Pn. Since a induces an inner automorphism of P n - i , D induces an inner derivation of grPn_i. Therefore, by Theorem IB (§5), D must itself be inner. This implies that there exists some T G gr'"Pn such that D = Int(T). But this implies [6.3.1]. Now, by a repeated use of [6.3.1] we can find a sequence of elements of Pn such that tW = t^'+^'^ mod Pn(m + i) and ( I n t ^ W ) - ^ acts trivially mod Pn{m -f i + 1). Let i = lim^(*) in Pn. Then a = Int({). By lemma 6.2.1 (ii), t must belong to Pn. Therefore, a must be an inner automorphism. This settles the proof of the Injectivity Theorem (i) and hence also that of the Vanishing Theorem. §7 Pro-^ analogues ; proof of the Injectivity Theorem (ii). 7.1. Fix a prime number £. denote by G the pro-£ completion Theorem (ii) taking the following (Step 1) Show that the exact sequence (7.1.1)
In §7, for a discrete group G, we shall of G. We can now prove the Injectivity steps. sequence (2.1.4) gives rise to an exact
l—^Nn-^Pn—
Pn-1 — 1
of pro-^ groups (n > 4). (Step 2) Use Step 1 to show by induction on n > 3 that gr Pn ::^ gr Pn 0
372
YASUTAKA IHARA
(Step 3) Check t h a t the Injectivity Theorem (Lie version) remains valid under the base change 0 2 Z^(Step 4) Deduce the Injectivity Theorem (ii) using Step 3. Among these, Steps 2 ^ 4 are easy. Step 2 can be carried out in a way parallel to the proof of P r o p 3.2.1. Step 3 is obvious from the proof of the Injectivity Theorem (Lie version). Step 4 is even easier t h a n its discrete counterpart, because P„ is complete (use of lemma 6.2.1 unnecessary). T h u s , to prove the Injectivity Theorem (ii), the only new point required is Step 1. This is obviously contained in the following L e m m a 7.1.2. (M.P.Anderson) Letl^N-^G-^H-^1 be an exact sequence of discrete groups such that (i) (G, N) = {N, N)^ and (ii) A^ is a free group of finite rank r > 1. Then the induced sequence of pro-£ completions 1 —^ N — ^ G — y H —^ 1 IS exact. Proof By assumption (i), the image of G in Aut A'' is contained in a pio-i group (P.Hall;see [2] Th.6). By (ii), N has trivial center. Therefore, the assertion follows from [2] Proposition 3. Q.E.D. T h u s , the Injectivity Theorem (ii) is also proved.
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properties
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3] J.Birman, Braids, links, and mapping class groups, Ann. of M a t h . Studies 8 2 (1975), Princeton Univ. Press. 4] P.Deligne, Letters to A.Grothendieck (Nov. 19, 1982, and an earlier one undated(?)), to S.Bloch (Feb. 2, March U, 1984). 5] J.Dyer and E.Grossman, The automorphism groups, Amer. J. Math. 1 0 3 (1981), 1151-1169.
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braid
6] E.Fadell and J.Van Buskirk, The braid groups of E^ and 5 ^ , Duke M a t h . J. 2 9 (1962), 243-257. 7] A.Grothendieck, Esquisse
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8] Y.Ihara, Pro finite braid groups, Galois representations multiplications, Ann. of Math. 1 2 3 (1986), 43-106.
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[9] N.V.Ivanov, Algebraic properties of mapping class groups of surfaces, "Geometric and Algebraic Topology " 18 (1986), 15-35, Banach Center Publ. [10] N.Jacobson, Lie algebras, Interscience (1962). [11] T.Kohno, Serie de Potncare-Koszul associee aux groupes de tresses pures, Invent. Math. 82 (1985), 57-75. [12] T.Kohno and T.Oda, The lower central series of the pure braid group of an algebraic curve, Adv. Studies in Pure Math. 12 (1987), 201-220. [13] J.McCarthy, Automorphisms of surface mapping class groups. A recent theorem of N.Ivanov, Invent. Math. 84 (1986), 49-71. [14] W.Magnus, A.Karrass and D.Solitar, Combinatorial group theory, Interscience (1966). [15] J.Nielsen, Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1918), 385-397.
Yasutaka Ihara Department of Mathematics Faculty of Science University of Tokyo Tokyo 113, Japan
Ordinarite des intersections completes generales LUC ILLUSIE* A Alexandre Groihendieck pour son soixantieme anniversaire
0. Introduction Soient k un corps de caracteristique p > 0, r un entier > 0, et a = ( a i , . . . , a m ) une suite d'entiers > 1. Notons S le fc-schema parametrant les intersections completes lisses de dimension r et multidegre a dans P]^"^"^, et X —» 5 la famille universelle. Nous prouvons le result at suivant : T h e o r e m e 0.1. // existe un ouvert non vide U de S iel que, pour tout s dans U^ Xg soit ordinaire. Que Xs soit ordinaire signifie que, si s est une extension parfaite de 5, le polygone de Newton defini par les pentes de Frobenius agissant sur le groupe de cohomologie cristalline H^{Xs/W){W = W{k{s)) coincide avec le polygone de Hodge defini par les nombres de Hodge /i*'''~* ==
dim,(,)i/'-'(X„n*^^/,). La terminologie "ordinaire en dimension d" a ete introduite par Mazur [25, §3] par analogie avec celle, classique, pour les varietes abeliennes. Dans cet article, nous dirons "ordinaire" pour "ordinaire en toute dimension" ; dans le cas d'une intersection complete de dimension r, il revient au meme de dire "ordinaire" ou "ordinaire en dimension r". Dans {loc. cit.), Mazur conjecture (0.1) pour les hypersurfaces. Rappelons que Ton dispose de theoremes d'ordinarite generique analogues a (0.1) pour : (i) les courbes de genre g (Miller [27], Koblitz [24]), (ii) les varietes abeliennes polarisees de dimension g (Mumford [29], Norman-Oort [30]) (iii) les surfaces K3 (Ogus [32]). Tons ces resultats ont, en un sens, leur source dans (a) le theoreme de Grothendieck ([14], [22], [4]) selon lequel dans une famille propre et lisse
* Unite associee au CNRS URA D0752.
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M —^ T sur Ar, le polygene de Newton Nwi^ de Mt en degre d croit par specialisation de t (6) Finegalite de Katz, prouvee par Mazur [26] et Ogus [31], selon laquelle Nwtd est au-dessus du polygene de Hodge Hdgd defini par les nombres de Hodge h}'^~'^{Mt) ; nous renvoyons a I'article de Mazur [25] pour un historique de ces questions. Les premiers pas en direction de (0.1) sont efFectues par Katz (non publie), qui montre que, pour m = l , s i r - f l ^ j ^ O mod a i , et s est assez general, alors les polygenes de Newton et de Hodge de Xg ont meme premiere pente, et par Koblitz [24], qui prouve qu'il existe un ouvert non vide V de S tel que, pour tout s dans F , Xg ait une matrice de Hasse-Witt inversible, i.e. Frobenius agisse bijectivement sur H^{Xs^O) , generalisant ainsi un resultat de Miller [28]. La demonstration de Koblitz repose sur I'observation que si, dans une famille propre et plate d'intersections completes (eventuellement singulieres), Hasse-Witt est inversible en un point, alors Hasse-Witt est inversible aux points voisins ; il suffit done d'exhiber un exemple d'intersection complete de multidegre a de Hasse-Witt inversible, ce qui est facile : prendre une intersection d'hypersurfaces reunions finies d'hyperplans en position generale. S'inspirant de cette methode, Deligne esquisse dans [8] une demonstra-tion de (0.1) par voie ^-adique. L'argumient de semi-continuite de Koblitz est remplace par un resultat de specialisation extrait de "Weil 11" [52, 1.10.7] : si C est une courbe sur un corps fini, E un ensemble fini de points fermes de C, L un Q^-faisceau lisse sur C — E tel que, pour presque tout point ferme s de C — E, le polygene de Newton de L en 5 soit au-dessus d'un polygene donne A^, alors, pour 5 G E, le polygene de Newton de L en 5 est encore au-dessus de N (il s'agit de polygenes de Newton relatifs au choix d'un isomerphisme entre Q^ et Q^, et leur definition aux points de E utilise le theereme de menodromie locale de Grothendieck, voir {loc. cii.) pour les details). Raisennant par recurrence sur la dimension et le multidegre, Deligne est amene a considerer un pinceau d'intersections completes de multidegre a , de fibre generale lisse, et de fibre speciale reunion de deux intersections completes (de multidegres inferieurs) lisses et ordinaires, se coupant transversalement suivant une intersection complete ordinaire. L'idee est de montrer que la fibre generique geemetrique est ordinaire. Teutefois, la reduction envisagee n'est pas semi-stable, et le calcul de cycles evanescents que requiert la determination du polygene de Newton est inacheve. Nous reprenons ici la meme idee geemetrique, mais en remplagant 1'argument £-adique par un argument de cohomolegie de de R h a m , independant des resultats de "Weil 11". Celui-ci repose sur la caracterisation de Bloch-Kato des varietes ordinaires par la nullite de la cohomolegie des bords du complexe de de R h a m [3] (cf. aussi [19]). Pour faire marcher I'argument, on a besein cependant d'une extension de cette caracterisation a une situation de mauvaise reduction. Or, dans le cas semi-stable, le
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travail de Hyodo [16] fournit ringredient m a n q u a n t . II s'agit du critere suivant (cf. 1.10) : (0.2). Soient T un trait de caracteristique p, de point ferme s et point generique ry, et Z un T-schema propre et semi-stable (cf. 1.7), de fibre speciale Y = Xs un diviseur a croisements normaux somme de diviseurs Di lisses sur s. Alors, si toutes les intersections Di^ D- • -HDij. sont ordinaires, la fibre generale Xj^ est ordinaire. Un resultat essentiellement equivalent a (0.2) est enonce sans demonstration dans [16]. Nous le prouvons en appendice. Une modification par eclatement du pinceau considere par Deligne conduit a une situation semistable, a laquelle (0.2) s'applique, comme on le verifie aisement. Quelques mots sur le plan. Le n ° l rassemble les definitions et resultats sur les varietes ordinaires necessaires a la demonstration de (0.1), donnee au n°2. Nous prouvons en fait un resultat plus fort (2.2). L'ingredient cle est 1.10. Au n°3, nous completons Targument de Deligne, donnant ainsi une seconde demonstration de (0.1), independante de (0.2). L'appendice est consacre a la demonstration de 1.10, sous une forme un peu plus generale. Les devissages, a Paide de residus de Poincare convenables, du complexe "logarithmique relatif" a;" de Hyodo sont caiques sur ceux faits par Steenbrink dans [34]. J e tiens a remercier pour d'utiles discussions A. Beauville, H. Clemens, T. Ekedahl, O. Gabber, O. Hyodo*, M. Raynaud, et tout particulierement P. Deligne, dont la lettre [8] est a Torigine de ce travail, et a qui je suis redevable de plusieurs ameliorations de presentation ; je lui dois n o t a m m e n t la formulation 2.2 du resultat principal. Sommaire 1. Varietes ordinaires : rappels et complements. 2. Le theoreme principal. 3. La methode f-adique [8]. Appendice : Le critere d'ordinarite de Hyodo. 1. Schemas a singularites croisements normaux reduits. 2. Residus de Poincare et critere de Hyodo. Bibliographie N o t a t i o n s et conventions Dans toute la suite, p designe un nombre premier fixe.
* Osamu Hyodo s'est, helas, suicide le 22 avril 1989.
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Si L = (• • • -^ U-^U^^ -^ ' • •) est un complexe, nous posons Z^L — ZU := Ker {d : U -^ L^+^) et B'L = BL' = dL'-K Si S est un schema de caracteristique p, nous notons souvent par un "prime" les objets deduits par changement de base par le Frobenius absolu Fs de S. Par exemple si X (resp. M) est un 5-schema (resp. un Osmodule), X' =zX xs {S,Fs),M' = F^M. Si X ^ y est un morphisme de schemas, nous abregeons parfois (quand il n'y a pas de confusion a craindre) la notation 0>x/Y du complexe de de Rham de X sur Y en fix o^ ^ • 1. Varietes ordinaires : rappels et complements. DefinUion 1.1. Soient S un schema de caracteristique p et f : X —^ S un morphisme propre et hsse. On dit que X est ordinaire sur S si WUBQ.\is
=0
pour tout i et tout j (5fi^/^ design ant le sous-faisceau abelien dQ.^-^,^ de ^ x / 5 ' ^^ ^^^ R"^ f* etant calcules pour la topologie de Zariski). Soient f : X' —^ S deduit de / par le changement de base par le Frobenius absolu F5 de 5, et F : X —> X' le Frobenius relatif. Alors F^BQ^-^/s est le Ox'-module dF^fi*^/^, et Rf^BQ^'^is ^^t Tobjet Rf^F^BQ^^s de D{S). Rappelons que F^BQ^^^.g est localement libre de type fini sur X ' , de formation compatible a tout changement de base [17, 0 2.2.8]. Pour S — Spec k^k wn corps parfait de caracteristiquep, on retrouve la definition usuelle ([3, 7.2], [19, IV 4.13]). Dans ce cas, et plus generalement s'il n'y a pas de confusion a redouter, nous dirons "ordinaire" au lieu de "ordinaire sur 5". Proposition 1.2. Soient S un schema de caracteristique p et f : X -^ S un morphisme propre et hsse, (a) Soit g :Y -^T deduit de f par un changement de base T ^>- S. Si X est ordinaire sur 5, Y est ordinaire sur T. (b) L^ensemble des points s de S tels que la fibre Xs soit ordinaire (sur Spec k{s)) est un ouvert U de S, appele ouvert d'ordinarite de X sur S. Les conditions suivantes sont equivalentes :
(i) s e U; L
(ii) Rf^BQ^^j^ig^o^k{s) L
(iii) Rf^BQi^xig^OsOs,s
— 0 pour tout i ; — 0 pour tout i.
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Soit g' :Y' ^^T deduit de g par le changement de base FT- Alors g' se deduit de / ' par le changement de base T -^ S. Comme / ' est lisse et que F^BQ^^,^ est localement libre de type fini sur X\ de formation compatible a tout changement de base, on a done, par Kiinneth (cf. par exemple (SGA 6 III 3.7)) (*)
RUBQ}xis^OsOT^Rg.BQ.^IT,
et (a) en resulte. Prouvons (b). La condition (iii) definit un ouvert de 5 ; il suffit done de prouver I'equivalence de (i), (ii) et (iii). L'isomorphisme (*), apphque a Spec k{s) —> 5, entraine I'equivalence de (i) et (ii). II est trivial que (iii) implique (ii), il reste done a prouver que (ii) implique (iii). On pent supposer S local, d'anneau A^ et de corps residuel k. Alors Rf^BQ^^^.^ est un complexe parfait Ei G D(A) (SGA 6 III 4.8). On pent supposer E^ borne, a composantes libres de type fini. Des lors, Ei est acyclique si et seulement si Ei 0 ^ k est acyclique, ce qui prouve (ii) =^ (iii) et acheve la demonstration. 1.3. Rappelons d'autre part ([3,7.3], [19, IV 4.13]) que si 5 = Spec k, k un corps parfait de caracteristique p, X est ordinaire si et seulement si F : W{X, W^') -> W{X,
WQ})
est bijectif pour tout i et tout j . De plus {loc. cit.)^ quand la cohomologie cristalline H^'i^X/W) est sans torsion, il revient au meme de dire que le polygone de Newton Nwid defini par les pentes de Frobenius agissant sur H^{X/W) coincide, pour tout d, avec le polygone de Hodge Hdgd defini par les nombres de Hodge /i*'^~* — dim if ^~*(X, O*) (i.e., ayant pour pente i avec la multiplicite /i*'^~*) (en general, que H''{X/W) ait ou non de la torsion, on sait, d'apres Ogus [31], que Nwtd est au-dessus de Hdgd pour tout d). Ceci s'applique en particulier au cas ou X est une intersection complete lisse, de dimension r et multidegre a — ( a i , . . . , am) dans P]^"^^. II existe en effet une intersection complete lisse relative X/W^ de dimension n et multidegre a relevant X. On a alors i7*(X/H^)-^i7p^(X/VF), done (SGA 7 XI 1.5) H\XIW) est sans torsion. l\ resulte de plus de {loc. cit.) que, pour d ^ r^ H^{X/W) — 0 pour d impair, et, pour d pair = 23,H\XIW) est libre sur W de base rf (si d < r),rf /\ai,"-am) (si d > r), ou ry € H'^{X/W) est la classe de Chern ci(Ox(l)) [2]. Par suite, Nwid = Hdgd si d ^ r (segment de pente d/2 et longueur horizontale 1 si d est pair, reduit a 0 sinon). Done X est ordinaire si et seulement si Nwtr = Hdgr.
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P r o p o s i t i o n 1.4. Soieni k un corps parfait de caracteristique p^Y un k-schema propre et hsse, E un Oy-Tnodule localemeni hbre de type fini, X — V{E) le fibre projectif associe. Alors Y est ordinaire si et seulement si X est ordinaire (sur k). D'apres [12, I 4.2.13], on a un isomorphisme de VF-modules, compatible a Taction de F
0
H'-''{Y,W^y'')^W{X,W9}x),
0
ou r -}- 1 est le rang de E. La conclusion resulte done de 1.3. On aurait pu aussi invoquer la decomposition analogue pour H^{X, 5fij^), qui resulte de [12, I (2.2.2), (2.2.3)], ou encore, quand H*(Y/W) est sans torsion, celles relatives a H*{X/W) et i / * ( X , Q * ) [2]. C o r o l l a i r e l,5,Soient Ei{l < i < n) des Oy-inodules localement lihres de type fini. SiY est ordinaire, il en est de mime deF(Ei)xY'•'X^Y^iEn)' Cela resulte de 1.4 par recurrence sur n : P(JE'I) Xy - - - Xy P{En) est le fibre projectif P ( F ) sur P{Ei) Xy - Xy P{En-i) ou F est I'image inverse de^n. P r o p o s i t i o n 1.6. Soient k un corps parfait de caracteristique p,X un k-schema propre et hsse, Y un sous-schema ferme hsse, X Veclate de Y dans X. Les conditions suivantes sont equivalentes : (i) X et Y sont ordinaires ; (ii) X est ordinaire. O n pent supposer Y purement de codimension d dans X. II resulte de [12, IV 1.1.9] qu'on a un isomorphisme de VK-modules, compatible a Taction de F ,
W{X,W^'x)^
0
H^"'{y.WQfy'')^W{X,WQ.'^).
Q
La conclusion en decoule grace a 1.3. On pourrait encore, ici, utiliser une decomposition analogue pour H^{X ,BQ>^-). 1.7. Soient S un trait, de point ferme s et point generique 77, et X un S'-schema. Nous dirons que X est semi-stable si, designant par S —^ S un localise strict et X = X Xs S,X est, localement pour la topologie etale, isomorphe au sous-schema ferme de Tespace affine A ^ — S[xi^..., x„] defini
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par Tequation xi - - -Xr = t {1 < r < n), on t est une unifor mis ante de S. L'espace total X est alors regulier, X est plat sur S, la fibre generique Xj^ est lisse, et la fibre speciale Xs est un diviseur a croisement normaux reduit dans X. Supposons X semi-stable. Le complement de I'ouvert de lissite u : U —^ X de X sur S est de codimension > 2. Un calcul local montre que le complexe (1.7.1)
ux/s '= w*^V/5
est a composantes localement fibres de type fini, et que u;^y^ = A*u;^/^. Pour X = S[xi^... ,Xn]/{xi • " Xr ~'^)y'^x/s ^^^ lihie de base dxi/xi, . . . , dxr/xj.^ dxr^i^ • • •) dxn^ soumis a la relation X^i<j
ujXs '•= ^x/s ^Os ^(«)-
C'est un complexe a composantes localement fibres de type fini sur Xs, qui depend de X (en depit de la notation). Observons d'autre part que (1.7.3)
u;x/s^OsHl)
= ^Xrj/v
Supposons 5 de caracteristique p, et X semi-stable, d'ouvert de lissite U. Soient X' ^ X Xs {S,Fs) et F : X -^ X' \e Frobenius relatif. L'isomorphisme de Cartier usuel C ^ : Vt\j, ig^^WF^UJ'uIs se prolonge (de maniere unique) en un isomorphisme (1.7.4)
C-':uj'x>is^'n'F.ujxis.
ou ^\iIS '-— ^x/s '^Os {^S, Fs) : on le verifie par un calcul local, voir par exemple [18,1.5] ou [10, 6.1]. II en resulte que les faisceaux Wcux/s^ ^^^xis^ ZLO^^IS sont plats sur S, et de formation compatible a tout changement de base T -^ S (i.e. WLOXJS '^OS OT~^W{^ ^OS OT), et de meme pour BUJ\ZLO^). Noter que ^ x / 5 ' ^ ^ x / 5 ' - ^ ' ^ x / s ^^^^ coherents en tant que Ox'-modules, mais ne sont pas en general plats sur X' (a la diff'erence de WW/5). Par le changement de base s -^ S, (1-7.4) donne un isomorphisme (1.7.5)
C-':u'x'^^n'F,u>x,,
qui prolonge l'isomorphisme de Cartier usuel sur le fieu lisse de Xg (et est caracterise de maniere unique par cette propriete). Un tel isomorphisme,
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avec la meme caracterisation, existe encore si Ton suppose seulement s de caracteristique p, au lieu de S [15, 1.5] et (Appendice 2.3)). Definition 1.8. (cf. [15, 1.10]). Soient S un trait, de point ferme s de caracteristique p, et X un 5-schema propre et semi-stable. On dit que X a reduction ordinaire si, avec la notation (1.7.2), on a
pour tout i et tout j . P r o p o s i t i o n 1.9. Soient S un trait de caracteristique p, et X un Sschema propre et semi-stable. Les conditions suivantes sont equivalentes: (i) X a reduction ordinaire ; (ii) H^{X,Bu^j^ig) — 0 pour tout i et tout j . En particuher, si X a reduction ordinaire, la fibre generale Xj^ est ordinaire (au sens de 1.1). Comme J5a;^/^ est plat sur S et commute au changement de base, on a, par Kiinneth (*)
RTiX,Bw'^,s)^Osk{s)^RnX„Bu;'xJ,
done (ii) implique (i). Notons de plus que RT{X,BLu)^/g) est un complexe parfait sur S^Bu)^^,^ etant coherent sur X' et plat sur S (SGA 6 III 4.8). Par suite, compte tenu de (*), (i) = > (ii) en resulte, par le meme argument que pour 1.2(b). La derniere assertion decoule de (1.7.3). P r o p o s i t i o n 1.10. Soient S un trait, de point ferme s de caracteristique p , et X un S-schema propre et semi-stable. On suppose que la fibre speciale Y — Xs est somme de diviseurs Yi C X{1 < i < r) tels que, pour tout I = (l < ii < ' • • < im ^ ^); ^intersection Y/ — Yi^ fl • • • D 1^^ soit lisse sur s et ordinaire {au sens de 1.1). Alors X a reduction ordinaire. Ce critere est enonce, sans demonstration, par Hyodo dans [15, p . 548, 1-10]. Nous le demontrerons en appendice, sous une forme legerement plus generale. Nous ne Tappliquerons, toutefois, que sous la forme 1.10 et dans le cas ou S est d'egale caracteristique p. 2. Le t h e o r e m e p r i n c i p a l O n fixe, dans tout ce numero, un corps k de caracteristique p .
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2.1. Soient N un entier > 1 et F un sous-schema ferme lisse de P — P ^ , purement de dimension n > 1. Soient ai^... ,am des entiers > 1, avec 1 < m < n. Si T est un ^-schema, par une intersection complete lisse de multidegre a = ( a i , . . . , a y „ ) de VT, on entend un sous-schema ferme W de VT, hsse sur T , purement de dimension relative n — m, de la formie W = VT O Hi n " • n Hm, ou Hi est une hypersurface de PT de degre a^ (i.e. un diviseur relatif de degre a^). Le foncteur associant a T Tensemble des suites {Hi,..., Hm), de multidegre a, telles que Vr fl ffi fl • • • fl Hm soit une intersection complete lisse est representable par un ouvert non vide S d'un espace multiprojectif P^^ x • • • x P]^"", ou ri — (^*^^) - 1, et Ton a un morphisme universel X —^ S, propre et hsse, dont la fibre en H ~ {Hi,..., Hm) :T ^ S est I'intersection complete Vr H i J i fl • • • fl Hm Soit T} le point generique de S. On dit que Xrj est intersection complete generique (de multidegre a.) de V. D'apres 1.2, il revient au meme de dire que Xfj est ordinaire (sur 77), ou que I'ouvert d'ordinarite de X sur S est non vide, ou que, si k est une extension algebriquement close de k, il existe une intersection complete lisse de multidegre a de F^ qui soit ordinaire. Quand Oi = 1 pour tout z, nous dirons section plane (de codimension m) au lieu d'intersection complete de multidegre a. Theoreme suivante : (2.2.1) Pour V de codimension Alo7^s, pour section complete
2.2.
Soit V C P comme
en 2.1.
On fait
Vhypothese
tout m tel que 1 <m < n, la section plane generique de m est ordinaire. tout multidegre a = ( a i , . . . ,ay^)(l < m < n), Vintergenerique de multidegre a de V est ordinaire.
Exemple 2.3. L'hypothese (2.2.1) est difficile a verifier. Voici quelques cas oil elle est satisfaite. 2.3.1. V ~P (1.4). Dans ce cas, 2.2 equivaut a 0.1. 2.3.2. V = \ine quadrique lisse de P. On pent en efFet supposer k algebriquement clos, et il suffit de montrer que V est ordinaire (une section plane d'une quadrique etant encore une quadrique), i.e., avec les notations de 1.3, que Nwtn{V) = Hdgn{V){n = dim V). Mais cela resulte de ce que, d'apres (SGA 7 XI et XII), H^^g{V/k) = H''{V/W{k)) = 0 si n est impair, et si n est pair, Hjf^ (V/k) (resp. H^{V/W{k)) est engendre par les classes de cohomologie des generatrices. 2.3.3. V = rhypersurface de Fermat d'equation XQ -\- — - -\- xj^ = 0 dans P (de coordonnees homogenes XQ,, .. ^x^), avec p \ d et p = 1 mod d. On sait en efFet qu'alors V est ordinaire (cf. [37, II 3.1]) ; il en est de meme des sections planes d'equation XQ = - • -Xr = O^r < N — I.
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II peut arriver toutefois que V soit ordinaire sans que Ton ait p = 1 mod d. Mais alors la section plane generique de V de codimension 1 n'est pas necessairement ordinaire. Prenons en efFet d=N = ^etp=2, Alors V est ordinaire d'apres 1.6, car isomorphe, sur une cloture algebrique de k, a I'eclate de 6 points de P^ en position generate (cf. par exemple [Y. Manin, Cubic Forms, North Holland 1974, 24.1 et 24.4]), mais toute section plane lisse de V est une courbe elliptique supersinguliere d'apres [24, p . 154, 1.7]. Pour une generalisation de cet exemple, voir [A. Beauville, Sur Its hypersurfaces doni les sections hyperplanes sont a module constant, ces volumes]. 2.4. D e m o n s t r a t i o n d e 2 . 2 . On peut supposer k algebriquement clos. On raisonne par recurrence sur dim V{N fixe). Pour dim 1/ = 1, 2.2 est trivial. Fixons n > 2, et supposons 2.2 etabli pour dim V < n {hypothese de recurrence (A)). Soit V C P de dimension n, verifiant (2.2.1). Prouvons que, pour tout a = ( a i , . . . , am) donne, il existe une intersection complete lisse, ordinaire, de multidegre a de 1^. Prouvons d'abord Tassertion suivante : 2.4.1. // existe une intersection complete lisse V C] Hi H - • - D de multidegre ( a i , . . . , am-i), verifiant (2.2.1).
Hm-i,
II suffit de montrer qu'il existe une suite d'hypersurfaces {Hi^... ^Hm-i) de multidegre ( a i , . .. , a ^ _ i ) et une suite d'hyperplans ( L ^ , . . ., L „ ) tels que W := V n Hi n •'' f] Hm-i soit lisse de dimension n — m + 1, et, pour m < 7' < n, Wr := W f) Lm PI • • • fl L^ soit lisse (de dimension n — r) et ordinaire. Soit S I'espace multiprojectif p a r a m e t r a n t les suites d'hypersurfaces {Hi,..., Hm-i, Lm,'-',Ln)de multidegre ( a i , . . . , otm-i, 1 , . . . , 1). La condition que V n Hi n •'' n Hm-i soit lisse de dimension n — m -f 1 definit un ouvert non vide [/ de E. D'autre part, pour m < r < n, la condition que 1/ n //"i n • • • n Hm-i n Lm n • • • n L^ soit lisse de dimension n — r et ordinaire definit un ouvert non vide Ur de E: d'apres I'hypothese (2.2.1), il existe en effet des hyperplans X ^ , . . . ,Lr tels que V f\ Lm D • • • fl L^ verifie encore (2.2.1), et d'apres I'hypothese de recurrence (A) appliquee a VDLm^'' '^Lr, il existe alors {Hi,... ,Hm-i) de multidegre ( a i , . . . ^dm-i) tel que y fl i J i fl • • • fl Hm-i H Lm fl • • • D Xr soit lisse de dimension n — r et ordinaire. Une suite {Hi,..., Hm-i^Lm-, • • - iLn) a p p a r t e n a n t a U n Um n '' • nUn verifie alors les conditions requises. Ayant choisi {Hi,..., Hm-i) de multidegre {ai,... ,am-i) de maniere que W = VnHiD'' 'HHm-i soit hsse (de dimension n — m - f 1) et verifie (2.2.1), il sufl[it de prouver qu'il existe Hm, de degre a^^, tel que W H Hm soit lisse (de dimension n — m) et ordinaire. En d'autres termes, quitte
ORDINARITE
385
a remplacer V par W, on est ramene a prouver la conclusion de 2.2 pour m = 1, i.e. a prouver Tassertion suivante : 2.4.2. Pour tout d > 1, il existe une hypersurface H de degre d telle que V C{ H soit lisse de dimension n — 1 et ordinaire. On va demontrer 2.4.2 par recurrence sur d. Pour d = l^la, conclusion decoule de Thypothese (2.2.1). Fixons d > 2, et supposons 2.4.2 etabli en degre < d {hypothese de recurrence (B)). Rappelons que Thypothese de recurrence (/S) est toujours en vigueur. CommenQons par prouver Tassertion suivante : 2.4.3. // existe des hypersurfaces Hoo = V{f),H' = V{9),H" = V{h), de degres respectifs d, d— 1,1 telles que les conditions suivantes soient remplies : (a) F = F n Hoo ^st lisse (de dimension n — 1) ; (b) Y' = VnH' etV = Vn H" sont lisses (de dimension n-l) ordinaires ; (c) Y'nV
= VnH'nH"
et
est lisse (de dimension n — 2) et ordinaire;
(d) F DY' nY'' = V n Hoo n H' n H" est lisse (de dimension n-Z) et ordinaire. (NB. On convient d'omettre (d) {resp. (c) et (d)) si n < 3 {resp. < 2).) Soit H I'espace multiprojectif des suites d'hypersurfaces {Hoo j H', H") de multidegre {d^d— 1,1). Chacune des conditions (a), (b), (c), (d) definit un ouvert non vide de H. Pour (a), c'est clair. Pour (b), cela resulte de rhypothese de recurrence (B). Pour (c) et (d), cela resulte de Thypothese de recurrence (A). D'ou 2.4.3. Choisissons done {Hoo, H\ H") telles que les conditions (a) a (d) soient verifiees. Considerons le pinceau H(x,,) = V{Xf - ngh),{\,ti)
eP^
d'hypersurfaces de degre d de P, et le pinceau
qu'il decoupe sur V. Soit M I'espace total de ce dernier :
Notons TT : A^ -^ P^ la projection induite par pri, de fibre V(^x,y.) ^^ (A,//). L'ouvert D de P^ au-dessus duquel TT est lisse, de dimension relative
386
LUC ILLUSIE
n — 1, est non vide, car F = 7r~^(l,0) = V Ci HQQ est lisse de dimension n — 1 d'apres (a). Pour etablir 2.4.2, il suffit de demontrer que Touvert d'ordinarite de 7r~^{D) sur D est non vide, ou, ce qui revient au meme, que la fibre generique de TT est ordinaire. On pent pour cela remplacer P^ par le localise T de P^ au point (0,1) et M par le schema M induit sur T . On a M =
VTnV{tf-gh)cPT.
ou t est une uniformisante de T. Notons s (resp. t)) le point ferme (resp. generique) de T , et TT : M - ^ T la projection. Pour achever la demonstration de 2.2, il sufSt done de prouver : 2.4.4.
La fibre generique Mj^ de TT : M -^ T est
ordinaire.
Pour cela, considerons I'eclate p : M —^ M de M le long d u sousschema K = FnY' n y
de la fibre speciale M^ =^ Y'UY^'
= V H V{gh)
de TT. Soient ei E — p~^{K) le diviseur exceptionnel. La fibre speciale M^ de (j) est done somme des diviseurs E,Y',Y" de M , ou Y' (resp. Y") est le transforme pur de Y' (resp. Y"). Nous allons etablir les assertions suivantes : 2.4.4.1.
(j) est semi-stable
;
2.4.4.2. E,Y\Y",Er\Y',EnY'\Y'nY",EnY'nY" ordinaires.
sont hsses ei
Verifionsd'abord 2.4.4.1. En dehors d e / ) - ^ ( y ' n y " ) , 0 est lisse. Soient a un point ferme de Y' DY" et Oa Tanneau local de VT en a. Si a ^ / \ , les images de g^h^t dans Oa font partie d'un systeme regulier de parametres, r i m a g e de / est une unite, et, au voisinage de a , M coincide avec M et est lisse sur le sous-schema de T[2/, z] d'equation yz — t^ done a reduction semi-stable sur T. Si a G A', les images de f^g^h^t dans Oa font partie d'un systeme regulier de parametres, et, au voisinage de a^M est lisse sur le sous-schema Z de T[x^y^z\ d'equation tx — yz. Done I'image inverse dans M d'un voisinage de a dans M est lisse sur I'eclate de Z le long du sous-schema d'equation x = y = z = t=:0. II suffit done de verifier que si C - V(ix -yz)Q.A^W^-
Spec h\t, x, y, z\
et C est I'eclate de I'origine (t — x — y—z — ^^ dans C, la projection c : C -^ A^ = Spec k\t\
ORDINARITE
387
composee de C -^ C et de la fleche induite par (t, x^y.z) \-^ t, a, reduction semi-stable en ^ =: 0. Si ^ est Teclate de Torigine dans A, C est le transforme pur de C dans A, done est reunion des ouverts affines standard Ui(l
OK{-HOO)
e OK{-H')
e
OK{-H"),
ou Hoo (resp. H',H") designe encore par abus le diviseur V{f) (resp. V{g), V{h)) de PT et OKi-Hoo) := OP^{^HOO)®OK (resp. OK{-H^) := OP^{-H')^OK.OK{-H'')
:= OP^{-H'')^OK)^
Si (x,)(0
est
un systeme de coordonnees homogenes sur PT et Ui Touvert ou Xi est inversible, les images de i,f/xf,g/xf~\h/xi dans OKIOK^HOO)JOK^H')^ OK{—H") respectivement forment une base t^ui^Vi^ Wi de ces faisceaux sur K nUi. On a une section Q de S'^{N)^ egale a tui — ViWi sur K fl Ui, et E
388
LUC ILLUSIE
est la quadrique V{Q) de P{N)
(cf. SGA 7 XII)). Observons que
et que EOY' - P{N'), EHY" = P(7V") sont deux generatrices de type oppose de E. Notons e :£'—>• / i la projection, et D ' (resp. D'') la generatrice E OY' (resp. EnY^'). Le faisceau F ' := C^OE^D') (resp. F ' ' := e^OE{D")) est localement libre de rang 2, et si g' : F - ^ P(i^') (resp. q" : E -^ P{F")) est le morphisme canonique (partout defini), le morphisme produit q^{q'^q"):E-.P{F')XKnF") est un isomorphisme (on a aussi F{N') ^^ P{F''),P{N'') - ^ P(i^O) i^^is ceci ne nous servira pas). Comme K est ordinaire, il en resulte que E est ordinaire (1.5), et I'assertion 2.4.4.2 est etablie. D'apres le critere de Hyodo 1.10, M a done reduction semi-stable ordinaire sur T. D'apres 1.9, la fibre generale Mfj de (f) est done ordinaire. Comme Mj^ = Mr^^ I'assertion 2.4.4 est done etablie, et ceci acheve la demonstration de 2.2. 3. La m e t h o d e ^-adique [8] Nous aliens prouver la conclusion de 2.2 par Pargument de Deligne [8] dans le cas oil V C P est une intersection complete (verifiant (2.2.1)). On se ramene, comme en 2.4, a prouver 2.4.2, et Ton choisit Hoo, H\ H" comme en 2.4.3. Par un argument standard, on se ramene a supposer V^HoQ^H'^H" definis sur un corps fini A^o, et Ton considere, comme plus haut, le pinceau TT : A^ —> P^ (sur ko) defini par les V(A,^) — ^^H(^x,^i)^ o^ H(^x,n) — y{^f — fJ'dh). II suffit de montrer qu'il existe un point ferme y de I'ouvert de lissite D C P^ tel que My soit ordinaire. D'apres (SGA 7 XI 1.5), la restriction a D de iJ^TT^Q^ ,pi est localement libre de type fini, de formation compatible a tout changement de base. Notons h^^ son rang. Pour tout point geometrique z —^ D localise en z, on a done h""^ = dimH\Mz,n'') = dim/f*(M,,Q^). Pour a-\-h ^ r (ou r = n — 1 est la dimension relative de TT), on a /i^* = 0 si a / 6, et h^^ — \sia — h {loc. cit.). Pour z G Z, notons Hdgi le "polygone de Hodge en degre i ", graphe de la fonction continue lineaire par morceaux valant 0 en 0 et ayant pour pente a avee la multiplicite h^'^~^. Fixons d'autre part un nombre premier ^ / p et un isomorphisme t : Q^ —-^ Qp. La restriction a D de J^^TT^Q^ est un faisceau lisse O de rang
ORDINARITE
389
^^ — Yla+b=t ^^'^' pour 2 ^ r^O cst Hul si I est impair, et canoniquement isomerphe a Qi{—i/2) si i est pair (SGA 7 XI 1.6). Si z est un point geometrique localise en un point ferme z de D , le polygone de Newton Nwti{z) de £* en 2r, rel. i, au sens de [7, 1.10.6], est le polygone de pentes les Vz{tct), pour a parcourant les valeurs propres de Fz sur O^Vz etant la valuation normalisee par Vz{Nz) = 1; en d'autres termes, c'est le polygone de Newton, rel. i, du polynome P:{t) = d e t ( l - FztXl)
= d e t ( l - F , t , iJ^'(M,-,Q,)).
En fait, Nwii{z) est independant de A, car Pl{t) est a coefficients entiers (et independant de ^), d'apres [6, 1.6], ou, plus simplement, d'apres la formule de Grothendieck ([13], (SGA 5 XV)) pour la fonction zeta :
et, pour i 7^ r, Pl{i) = 1 si f est impair, et 1 — q^''^i si i est pair, on q = Nz. D'apres [23], ou plus simplement d'apres la formule de Berthelot pour la fonction zeta [1], on a aussi Pi{i) = &ei{l - Ft,
H\MzlW)),
F designant le Frobenius de Mz sur A:(2:)et H^{MzlW) la cohomologie cristalline de Mz par rapport kW = W{k{z)). Pour i 7^ r, on a Nwii{z) — Hdg^, et pour z = r, on a, d'apres Mazur [26], I'inegalite de Katz (3.1)
Nwtr{z)
>
Hdgr.
II resulte d'autre part du theoreme de specialisation de Grothendieck [14], [22], [4,2.7] qu'il existe un ouvert non vide Do C D tel que, Nwtr{z) soit constant pour z dans DQ, de valeur Nwtr, et au-dessus de Nwtr pour z dans D — DQ. Soient (T,5,77) comme en 2.4.4 et fj un point geometrique de P^ localise en 77. Le polygone de Newton Nwti(s) de £* en 5, rel. i, est encore defini [7,1.10.2] : c'est le polygone de Newton, rel. t, du polynome PiiFs^t)
- det (1 - Fst.C)
= det (1 -
Fst,H\MffMi)).
ou Fs est un relevement de Fg dans W{fj/r]) ; ce polygone ne depend pas du choix de Fg. Pour i / r, on a encore Pg{Fs,t) = 1 si i est impair et 1 — g*/^/ si i est pair, oh q = Ns, de sorte que Nwti{s) = Hdgi. On ignore en revanche si Pg(Fs^i) est a coefficients entiers, independant de £. D'apres Deligne [7, 1.10.7], on a (3.2)
Nwtr{s)
>
Nwtr.
390
LUC ILLUSIE
Compte tenu de (3.1) il suffit done de prouver que Ton a (3.3)
Nwtr{s) = Hdgr
(on saura alors que Mz est ordinaire pour tout z dans DQ), Pour A E Q, notons 6f(A) (resp. Hi{\)) la longueur horizontale du segment de pente A de Nwti{s) (resp. Hdgi), et posons
6(A) = X^(-1)'6.(A), h{\) = Y,{-^rHi{\). Comnae Nwti{s) = Hdgi pour f ^ r, (3.3) equivaut a (3.4)
6(A) = h{X) pour tout A. Considerons la suite spectrale des cycles evanescents
(3.5)
E'J = H'iMs,R^rp{^e))
-> F'+^(M,-,Q,)
pour le morphisme (j) : M —^ T envisage dans la preuve de 2.4.4. EUe est compatible a Taction de Fg, et Ton a par suite (3.6)
6(A) = ^ ( - 1 ) ' + > 6 . , ( A ) ,
ou bij (A) est la dimension de la partie de pente A de E2 (longueur horizontale du segment de pente A du polygone de Newton, rel i, de Fs operant sur £'2^). Comme (j) est semi-stable, I'inertie opere trivialement sur W'ip{Qi) [33, 2.25], de sorte que R^'tp{Qi) est donne par les formules de (SGA 7 I 3.3) :
(3.7)
R'^PiQi) =
(Q^(-I)E
0
QI{-1)Y'
e Q^(-l)y'')/(Q^ diagonal),
avec les notations de la preuve de 2.4.4. Pour g > 1, notons Yq la somme disjointe des intersections ^ a ^ des composantes de la fibre speciale Y = Ms, et Sq :Yq -^Y la projection :
Yi = EUy'Uy''X
= iEnY')U{EnY'')U{y'nr'),Y3
et Yq = ^ pour ^ > 3. On a une suite exacte 0 -^ {QI)Y -^ suQi -^ e2^Qt -^ e3*Qe -^ 0
= EnY'nY^^
ORDINARITE
391
(resolution de Cech pour le recouvrement ferme Si), d'ou Ton deduit, grace a (3.7), une suite exacte 0 -^ i^V(Q^) - e2.Qi{-l)
-> e3*Q£(-l) -^ 0
et un isomorphisme i?'V'(Qf)-^e3.Q<(-2). On a par suite 6o(A) = biY,){X) - 6(y2)(A) + 6(Y3)(A),
h{X) = (3.8)
b{Y2){X-l)-biY3){\-l),
62(A) = b{Y3){X - 2),
ou Ton a pose (avec la notation de (3.6)) et
biY,){X) = Y,i-l)%iY,)iX), bi{Yq){X) designant la dimension de la partie de pente A de iJ*(y^,Q^). Finalement, (3.6) se recrit (3.9)
6(A) = 60(A)-61(A) + 62(A),
avec les 6f(A) donnes par (3.8). Par analogie, notons hi(Yq){\) la dimension de H^~'^(Yq^Q^) (on convient que hi{Yq){X) = 0 si A n'est pas entier), et posons
hiY,){X) = J2(-'^Yhi{Y,)(X),
(3.10)
/10(A) = h{Y^){X) - h{Y2)iX) + h{Y3)iX) h,iX) = h{Y2){X-l)-h{Ys){X-l) /»2(A) = h{Y3){X - 2).
D'apres le choix du pinceau w et I'etude geometrique faite dans la preuve de 2.4.4, Yg est ordinaire pour tout q. On a done hi{Yg){X) = 68(y^)(A) pour tout i, tout q et tout A, done h{Yq){X) = 6(yj)(A) pour tout q et tout A, done, d'apres (3.8) et (3.10), 6,(A) = hi{X) pour tout i. Compte tenu de (3.9), la formule (3.4) a demontrer se reerit done (3.11)
/i(A) = /io(A)-fti(A) + /«2(A),
392
LUC ILLUSIE
avec les /ij(A) donnees par (3.10). Rappelons que, par definition, h{X) = Pour verifier (3.11), on pent relever le pinceau TT - et reclatement en caracteristique zero, ce qui ramene a remplacer le corps de baise ATQ par le corps des complexes. On pent alors faire appel a la theorie de Steenbrink ([34], complete par [36]), qui fournit un complexe de Hodge mixte cohomologique A'l sur Y ayant Rip{Z) comme complexe de Z-modules sous-jacent. Le complexe bifiltre A'c est le complexe simple associe a un certain bicomplexe A'\ et Ton a une suite spectrale de filtration par le poids (3.12)
Ei^ = W+iiY,grZAc) => H\M,-) = H%Y,Ac)
qui degenere en £"2, et dont la differentielle est strictement compatible a la filtration F. Le complexe gr^A'c est le complexe simple associe au bicomplexe, de difFerentielle d" = 0, somme des complexes disposes dans le tableau suivant : ^ 3 * ^ Y3
£\*^Yi
^2*^Y2
^3*^y3
(i.e., SI^Q'Y^ 0e2*^'y2[Oj~l]®^2*^y2[-l)O]0 * • Oi^^fc^^' ^^^ ^^ sous-complexe correspondant aux complexes situes sur la diagonale j — i = k (i.e. gr^2^'c = £3*^'y3[-2],...,yrS^Ac = SI^'Y, 0 £3*^'y3[-2],...). Prenant le gradue de (3.12) pour la filtration F (qui est donnee par la filtration par le premier degre de A"), et tenant compte de ce que dimgr^H^Y, A'c) — h^>'-^{Mfj) [35, (2.11)], on obtient h{X) = hiYi){X) - h{Y2){X) + h{Ys){X) - h{Y2){X - l)-f +h(Y3){X-l)
+
h{Ys){X-2),
ce qui est precisement la formule (3.11). La demonstration est terminee. On pourrait en fait, si on le souhaitait, se passer de la theorie de Steenbrink. En effet, les hi{j) s'expriment simplement en termes des nombres de Hodge des intersections completes Y'^Y"^F et de leurs intersections, de sorte que (3.11) apparait comme une relation "universelle" entre les nombres de Hodge de certaines intersections completes. Deligne (Lettre a L. Illusie, juillet 79) a verifie cette relation a I'aide des series generatrices de Hirzebruch (SGA 7 XI 2).
ORDINARITE
393
Remarques 3.13. (a) On peut verifier la coincidence des polygenes de Newton et de Hodge des Yg directement a partir de la structure connue de la cohomologie ^-adique et de la cohomologie de Hodge des fibres projectifs et des eclates, cf. (SGA 5 VII) : la demonstration precedente ne fait done pas appel a I'equivalence, rappelee en 1.3, entre cette coincidence et la definition 1.1. (b) Le fait que V soit une intersection complete dans P intervient de fagon essentielle dans la reduction a un calcul virtuel et dans Targument de relevement. Cette methode ne permet done pas, semble-t-il, de prouver 2.2 dans le cas general.
Appendice : le critere d'ordinarite de Hyodo 1. Schemas a singulantes croisements normaux reduiis. Definition 1.1. Soient k un corps de caracteristique p, et Y un Ar-schema de type fini. On dit que Y est a singularites croisements normaux reduits (ou de type cnr) si Y est, localement pour la topologie etale, isomorphe au sous-schema ferme de A^ = Spec k[xi^..., x^] d'equation xi -- -Xr = 0 (1 < r < A^). Par exemple, si S est un trait de point ferme s et X un 5-schema semi-stable (1.7*) ^^\ la fibre speciale Xg est de type cnr. Le produit d'un ^-schema de type cnr par un Ar-schema lisse est de type cnr, mais on prendra garde que le produit de deux fc-schemas de type cnr n'est pas en general de type cnr. Pour une meilleure notion, stable par produits, voir [11, IV]; nous n'aurons toutefois pas besoin de cette generalisation. 1.2. Fixons, pour toute la suite de ce numero, un corps separablement clos k de caracteristique p. Soit Y un Ar-schema de type cnr. Notons TT : y —> y son normalise. Pour n entier > 0, soit Yn = {yeY
I card 7r-^(y) > n},
et notons Yn le normalise de Yn. Si Y est reunion des coordonnees D^ d'un espace affine A]^,y„ est reunion des a n des Df, et Yn somme disjointe de ces intersections. localement pour la topologie etale, lisse sur un Ar-schema
hyperplans de intersections n Comme Y est, de ce type, on
(1) Les numeros affectes d'un asterisque renvoient a Particle principal.
394
LUC ILLUSIE
voit que, pour tout n,Yn est lisse. Les Y^ sont des sous-schemas fermes de Y formant une stratification de y , Y,=YDY2D---DYnD"', a strates Yn — Fn+i lisses. Nous noterons TTnlYn-^Y la projection. Suivant Deligne [5, 3.1.4], on definit un systeme local (etale) En d'ensembles a n elements sur Yn, dont la fibre en y est I'ensemble des n branches contenant I'image dans Y d'un voisinage etale de y dans Yn. Notons k{En) le Of -module de base En^ et Sn son determinant. Notons Q"y (Sn) le complexe de de Rham de Yn/h a coefficients dans £:„, et posons (1.2.1)
Ln-.^l^n.^'y^iSn).
Si Y est reunion d'hyperplans de coordonnees Di{\ < i < r) d'un espace affine A ^ , on a ^" ~
vly
^ ^t^n nD,.
Dans le cas general, Ln s'identifie a un sous-complexe de 7r*Q'/y/yw, ou (Y/Y)'' :=YXY'XYY (n facteurs) et TT : ( 7 / 7 ) " ^ Y est la projection: une section cj de TT^Q^s'ecrit localement comme une (n — l)-cochaine {s : An_i -^ T^'^iy)) ^ tos e^% /n^n • DP / IN' ^^ '^~^{y) est I'ensemble des branches passant par y et Di designe la branche correspondant a f G 7r~^{y) ; elle est dans Ln si et seulement si elle est alternee, i.e. u;^ = 0 si 5 est non injective, et tOg-a = sgn{a)ujs pour s injective et a une permutation de A n - i . Pour n variable, les L„ forment un sous-complexe (1.2.2)
L. =
{Li^L2--'---^Ln---")
(avec Li en degre 1) du complexe de Cech TT^ii y
>• ^ * ^ ^ c y / y N 2
^' ' '
^ TT+iZ fY/Y)^
^* * '
(la differentielle d \ L^ -^ ^n-fi associant localement a la cochaine u la cochaine (—1)^E(—l)*5iu;). Le quotient est acyclique. L e m m e 1.3. Si Y est reunion des hyperplans de coordonnees Di d'un espace affine X = fX]^, le complexe augmente 0 ^ nx{\ogY){-Y)
^n-x-*Li-^L2^---^Ln^---,
ORDINARITE
395
avec ^'X piace en degre 0 et Cl^^ —>• L\ donne par to i-^ (—l)^Eco'|Dj^ est acyclique. En pariiculier, le complexe L. est acyclique en degre > 1. On se ramene, par Kiinneth, a r = 1, cf. [9, 4.2 (c)]. 1.4. L'isomorphisme de Cartier usuel C~^ : Qy,^W*Qy^ (le "prime" designant comme d'habitude le pull-back par le Frobenius de la base) donne, par torsion par Sn, un isomorphisme (1.4.1)
C-' :
(LiY^WLn
(noter que WLn = Tr„^W{^^ ® Cn) = TTn*{Ti'Q^ ® €n))- Posons, pour n > I, (1.4.2)
K„ = Kex Ln ^
Ln+i.
D'apres 1.3, on a done des suites exactes (1.4.3)
0 -* Kn -^ i n ^ Ln+l ^
. I ^ ^ ••• ,
(1.4.4)
0 ^ A'„ - - L„ - . A'„+i ^ 0.
L e m m e 1.5. Les suites dedmtes de (1.4.4) par application de 7i^ sont exactes, et C"^ (1-4.1) induit des isomorphismes de suites exactes
(1.5.1)
{Ki,y —. (4)'
iK^.Y
— 0
I 1 (i)n 1 0 —^ n^Kn --> n'Ln —> WKn+1
-^ 0.
0 —
-.
On procede par recurrence descendante sur n(L„ = 0) pour n » 0). Notons Un (resp. Vn) la fleche verticale de gauche (resp. droite) de (1.5.1). Supposons Um et Vm definies et bijectives pour m > n -h 1, avec v^ = Um+i' Alors Vn := tin+1 rend commutatif (1)^. Done, pour tout i^WLn —^ WKn+1 est surjectif, done la suite inferieure est exacte, et (1.4.1) induit un isomorphisme ti„, d'ou la conclusion. Corollaire 1.6. Les suites deduites de (1.4.3) et (1.4.4) par application de W (resp. B\ resp. Z^) sont exactes.
396
LUC ILLUSIE
Les assertions relatives a (1.4.4) et Z^, B^ resultent de celles relatives a (1-4.4) et W (1-5) par recurrence (croissante) sur i. Celles relatives a (1.4.3) en decoulent. On pourrait aussi verifier 1.5 et 1.6 de la fagon suivante (Deligne). On pent supposer Y reunion des hyperplans de coordonnees Dj d'un espace affine X = fK^.. Les isomorphismes de Cartier (1.4.1), C~^ : fi^/—^Ti^Q'xj et C - i : Q)^,{\ogY'){-Y')^n'{n'x{\ogY){-Y)) [9, (4.2.1.3)] fournissent un isomorphisme du complexe (1.3.1)* deduit de (1.3.1) par application terme a terme de (—)* sur celui deduit par application de 7i\ Ce dernier est done acyclique, et par recurrence sur f, on en deduit qu'il en est de meme de celui deduit par application de Z^ (resp. B^). Le complexe (1.3.1)* est done acyclique en tant que complexe filtre par (—)* D Z* D B* D 0, done se coupe en suites exaetes courtes filtrees (1.4.4)*. C o r o l l a i r e 1.7. Les conditions
suivantes
soni equivalentes
:
(i) H^ {Y, BKl^) — 0 pour tout i, tout j et tout n ; (ii) H^ (Yj ^^li)
— 0 pour tout i, tout j et tout n .
Les suites deduites de (1.4.4) par applications de J5* etant exaetes, (i) entraine (ii). La suite deduite de (1.4.3) par application de 5* etant exaete, ^ * ( y , 5 / \ ^ ) est, pour i et n fixes, Taboutissement d'une suite spectrale de termes initiaux H^{Y,BU^){r > 7i), done (ii) entraine (i). Remarque 1.8. Si Y est de dimension 1, L^ = 0 pour n > 2,1/2 est a support ponetuel et BL2 = 0 pour tout i, done la condition (ii) de 1.7 signifie simplement que H^(Y, 5 Q ^ ) = 0 pour tout i et tout j , i.e. que Y est ordinaire. 2. R e s i d u s d e P o i n c a r e et c r i t e r e d e H y o d o 2.1. Soient S un trait, de point ferme s separablement elos, et point generique ry, et X un 5-sehema semi-stable (1.7*). Apres extension etale de S et localisation etale sur X , X est done S'-isomorphe au sous-schema de A^^ = S[xi,..., XN] d'equation xi - - - Xr = TT, o\x TT est une uniformisante, et si j designe alors I'inclusion de la fibre generique X^j dans ^ , ^ x / 5 ^^^ ^^ sous-eomplexe de j*^'Xr)/j] engendre par les formes du type a dxi^/xi^ A • •' Adxi /xi^ pour a dans Timage de ^'x/s et 1 < z'l < • • • < i^ < r. Pour n entier > 0, notons Pn^x/s ^^ sous-eomplexe de (^'xis engendre loealement par les formes du type ci-dessus, avee q < n (on verifie que cette definition ne depend pas du choix des "coordonnees locales" Xi : si ( t / i , . . . , t/jv) est un autre choix, apres localisation et une eventuelle permutation, on a yi — UiXi
ORDINARITE
397
pour i < r, ou Ui est une unite sur X, et dyjyi = dui/ui Pn^x/s forment une filtration croissante de (J^X/S (2.1.1)
0 C PoOJx/s C'"C
-f dxi/xi).
Les
PnOJxis C • • •
(avec Pn^xis — ^'xis po^r '^ > > 0)? analogue a celle consideree par Deligne dans [5, 3.1.5] (a cela pres qu'on travaille ici avec un complexe de de R h a m logarithmique relatif, ou Ton a la relation "^KKr^^i/^i — ^)- ^^ choix de la notation P„ de preference a la notation Wn de (loc. cit.) est fait pour eviter des confusions possibles avec des complexes de de R h a m - W i t t , cf. [16]. Notons Y = Xs \^ fibre speciale de X, posons co^ — <^'xjs ®^s k {k — k{s)) (1.7.2*) et designons par Pn<^ I'image de Pn^x/s dans u y . On obtient done une filtration croissante de uy (2.1.2)
0 C P o ^ y C • • • C Pn^Y
C •• •
(avec Pn(^Y — ^Y pour n > > 0), et PQUJ'Y egal a Timage de Pour n > 1, une section LJ de Pn^xls haut, a; =
^ i
«2i-in A dxijxi^
^yik-
^'^^^^^ localement, comme plus A • • • A dxi^/xi^
-f b
•
avec ai^ .^^ dans Pimage de ^ ^ / ^ et 6 dans Pn-i ; les images inverses des ai^ .-i^ dans Q^^ n • nz) (^^ ^ i ^^^ ^^ diviseur (x^) de A ^ ) forment une section de L^ (1.2.1), et Ton verifie que son image dans I/^^j par la differentielle d de (1.2.2) ne depend pas du choix de Pecriture ci-dessus. En re collant ces definitions locales, on obtient une application de Pn^x/s dans 1/^4.1 ; pour q variable, ces apphcations forment un morphisme de complexes, qui se factorise en (2.1.3)
/i;:gr^uv+"^i:„+i.
On etend la definition de R an cas n = 0 en associant a une section LU de PQLOY la section de L\ ayant localement pour valeur la cochaine i i—» (—1)^ (Pimage inverse de uj sur D^). L'enonce suivant est decalque de Steenbrink [34, 4.15] : L e n i m e 2 . 2 . La fleche R est injechve
et a pour image /v„+i(1.4.2).
Par analogie avec la construction de [5,3.1.5], nous appellerons residu de Poincare Pisomorphisme (2.1.4)
R :
gi^Wy+"^Kn+i
398
LUC ILLUSIE
deduit de (2.1.3). Prouvons 2.2. La verification est locale, done on pent supposer que X = V{xi ' • • Xr — TT) C S[xiy..., XN] OU TT est une uniformisante. Posons Z = V(xi' " Xr — t) C T[xiy..., Xiv], T = S[t] ou t est une indeterminee. Alors X/S se deduit de Z/T par le changement de base 5 —^ T,t »—> TT, done ^x/s — ^z/T ^OT ^S^ OVL U)Z/T ^^^ ^^ complexe de de Rham logarithmique relatif, defini par (*)
^ Z / T — ^^^Z/T
et la suite exacte (**)
0 -> r^^T/si^ogTo)
-^ Q^/5(logZo) -^a;^/T -^ 0,
oil To (resp. ZQ) est le diviseur {t = 0) de T (resp. Z) et / : Z -^ T la projection. Par suite, si k designe le corps residuel de S et fs : Zs = Spec k[xi,...,
ar^v] -^ Tg = Spec k[t],t i-^ xi - - • Xr
le morphisme deduit de / par le changement de base s —^ S, on a. LJ'Y — ^ZslTs ^fc[t] ^(^ ^"^ 0)7 ^vec ^ZslTs — ^z/T '^OT ^TS decrit par des formules analogues a (*) et (**). Notons A I'espace affine Zg = Spec k[xi,... ,Xjv]Pour n > 0, soit PnQ^{\ogY) le sous-complexe de fi^(logy) engendre localement par des formes du type a dxi^/xi^ A • • • A dxi^/xi^ avec a dans ^ A ' ^ — ^1 ^ • • • < « ? < ^, et g < n. Ces sous-complexes forment une filtration croissante de Q^(logy) et Ton a, comme dans [5, 3.1.5], un isomorphisme residu de Poincare (2.2.1)
Res : gr^Q7^(logy)-^L„ = e^p,^n.-ni^,, ,
envoyant la classe de a dxi^/xi^ A ••• A dx^^/xi^ sur I'image de a dans ^D n- nD '^* designant Thyperplan {xi = 0) de A. De plus, Pn^y defini en (2.1.2) est I'image de P„fi^(logy) dans cjy, et les fleches (2.1.3) et (2.2.1) donnent lieu, pour n > 1, a un carre commutatif gr^fi-+"(logy)
(2.2.2) Posons, pour abreger.
^^^ 1 -
.
gr^av+-
1^
ORDINARITE
399
et considerons le complexe (2.2.3)
Mi-^M2-^
^ Mq^ • • •,
avec Ml place en degre 1, la difFerentielle 0 etant le produit exterieur gauche par rimage de dt/t, i.e. Yli 2. II verifie en fait une propriete plus forte. Par definition, 6 envoie Pn dans Pn+i, de sorte que Ton deduit de (2.2.3) un complexe (2.2.4)
grf Mi-tgr^M2 ^ • • • -^ grf M, ^ • • •.
On verifie que les residus de Poincare (2.2.1) definissent un isomorphisme de (2.2.3) sur le complexe L. (1.2.2). Get isomorphisme se prolonge en un isomorphisme de complexes (2.2.5) 0 - ^ fi'^(log y ) ( - y ) —.fiU - ^ grfMi - ^ grfM2 ^ • • .
o _
fi-^(logy)(-y)
-^QA —
Li
—^
L2
—
••
avec ^A place en degre 0, et Q'^ -^ Li compose de R : PQUJY —^ ^ i et de la projection ^A -^ i^o^V• D'apres 1.3, le complexe augmente inferieur est acychque. Par suite, dans le complexe filtre Q'^(logy)^Mi-lM2-.-(qui prolonge (2.2.3)), la diff'erentielle est strictement compatible a la filtration, i.e. on a, pour tout n > 0,
^1^,1 (logy) n p„i};Ji(iogy) = ePn-iQiA{\ogY) {Pn-1 = 0 si 71 = 0). Comme, par definition,
Ljy=nA{\ogY)/{eQ--\logY)^Q'A{logY){-Y)) et PnUj'y
= : I m P n f i " ^ ( l o g y ) -^ Uy ,
il en resulte que (pour tout n > 0) on a
PnUy =
PnQA{l0gY)/(9Pn-lQA\\0gY)-\-QA(Hy){-Y)).
400
LUC ILLUSIE
On a en particulier une suite exacte
ce qui, vu I'exactitude de la ligne inferieure de (2.2.5), prouve 2.2 pour n = 0. On en deduit d'autre part que, pour n > 1, on a une suite exacte g r ^ _ i M n - i - ^ g r ^ M n — ^ gr^cjy —> 0, ce qui grace a (2.2.2) et (2.2.5), prouve 2.2 pour n > 1. L e m m e 2 . 3 . Soii X/S comme en 2.1. Supposons le point ferme s de caraciertsiique p. Soti F : Y ^>- Y' le Frobemus relaiif de la fibre speciale Xs — Y. Pour tout i^WF^uj'y est localement libre de type fim, et SI V : V '-^ Y designe Vouvert de lissite de Y/S, Visomorphisme de Cartier C~^ : v^^\^,^v^l-L'^F^V[y induit un isomorphisme cfeu;y/(:= u ; y 0 O y / ) sur WF^uSy La question etant locale, la meme reduction qu'au debut de la demonstration de 2.2 ramene au cas ou Y/s est deduit par reduction modulo t de A / S p e c ^[t], A = Spec Ar[a^i,... ,X]^],t H^ Xi" -Xr.k — k{s). On applique alors (1.7.5*). L e m m e 2 . 4 . Sous les hypotheses de 2.3, pour tout n et tout i, la fleche WPn —^ Woj^ est injective, et Visomorphisme de Cartier C~^ : LOYI-^WCOY ^'^duit un isomorphisme C~^ : Pn(^Y,^WPnUj'Y (ou Pn<-OYi '-— PnU^ir^OY'). On procede par recurrence descendante sur n (PnOJy = coy po^r n » 0). Fixons n et supposons la conclusion etablie pour tout m > n. Considerons le diagramme
WP„oj-+"^
Wgv^uj-+"
^
WKn+i
- ^
WL„+i
oil le carre de droite est celui de gauche dans (1.5.1). II suffit de prouver la commutativite, pour tout i, de (2.4.1). En effet, celle-ci entrainera la surjectivite (pour tout i) de W P „ W y " ' ^ Ti'gr^w'jt^donc I'exactitude de
ORDINARITE
401
et C~^ induira un isomorphisme de P„_iw^" (resp. gr^w^") sur WPnUi'^'^ (resp. W g r ^ w ^ " ) . Pour verifier la commutativite de (2.4.1), on peut supposer, comme plus haut, que Y/s est la reduction modulo t de A/Spec kit]. Avec les notations de la preuve 2.2, on dispose de Tisomorphisme de Cartier usuel C - i : ^\,(logY')^^W^A(^ogY) [21, 7.2] et des isomorphismes residus de Poincare (2.2.1). Admettons provisoirement I'enonce suivant: (2.4.2) Pour tout m et tout i, la fleche Ti'Pm^AilogY) ->• W'f2'^(logY) est injective, et C~^ induit un isomorphisme de Pm^A'i^ogY') sur W P ^ ^ A (logy). De plus, les carres p„fi^,(iogy')
Pm+in'^.\\0gY')
w'p^+iQ'+i(iogy)
n'Pm^-AilogY) et p^n'+niogY')
c WPm^2
{LLY
-1
C-1 (logy)
-^
(1.4.1)
n'L„
sont commutatifs. Le pourtour de (2.4.1) est alors le carre central du diagramme suivant: p„n^+"(iogy')
P„+lfi^+"+^(l0gy') (1) Pn<^Y'
c
-1
(3)
c-
* i^li+lY
~ (5)
WPni^y"
^'^'Ln
~
c-
(4)
~
c- 1
+l
(2) ?^'p„fi7"(iogy)
7i^p„+ifi;;"+'(iogy).
Les carres (1) et (2) commutent, grace a (2.2.5), par definition de R. Le carre (3) commute, par definition de C~^. Le carre exterieur et le carre (4) commutent d'apres (2.4.2), done le carre (5), et par suite, (2.4.1) commutent. II reste a etablir (2.4.2). On le fait encore par recurrence descendante sur m. Si les deux premieres assertions de (2.4.2) sont vraies pour m' >m,
402
LUC ILLUSIE
le second carre de (2.4.2) est defini, et commute, car C~^(dxi/x^) =^ dxi/x^. LaflecheTi^Py^Q^ ""(logY) -^ H^gr^Q^ "" (logY) est alors surjective (pour tout i), et ceci entraine, comme plus haut, que la suite 0 ^ •H'Pn.-iQ'^ilogY) ^ W'P„Q;,(logy) ^ n'gT^Q'A'^ogY) -^ 0 est exacte, et que C~^ induit un isomorphisme de Pm-i sur WPm-iLa commutativite du premier carre de (2.4.2) resulte encore de ce que C~^{dxi/xi) = dxi/xi. Les assertions de (2.4.2) sont done etablies, et ceci acheve la demonstration de 2.4. Corollaire 2.5. Sous les hypotheses de 2.3, pour tout n et tout i, les suites deduites de 0-^Pn-lU^Y
par application de 7i^,Z^,B^
-^ Pn^Y
-^ gi^tOy
-^ 0
sont exactes.
Les assertions relatives k Z\B^ se deduisent en efFet de celles relatives a W (2.4) par recurrence (croissante) sur i. Proposition 2.6. Soient S un trait de point ferme s separablement clos de caracteristique p et X un S-schema propre et semi-stahle (1.7*), de fibre speciale Y = Xg. Avec les notations de (L2.1) et (2.L2), considerons les conditions suivantes : (i) H^{Y,BL\^) — 0 pour tout i, tout j , et tout n ; (ii) W {Y^Bgr^LOy) — 0 pour tout i, tout j , et tout n ; (iii) H^{Y^BPnOJY) = 0 pour tout i, tout j , et tout n ; (iv) H^{Y,BUJY) = 0 pour tout i et tout j (i.e. X a reduction ordinaire (1.8*)). Alors, (i), (ii), et (iii) sont equivalentes, et (iii) implique (iv). Si dimY < 1, /e5 conditions (i) a (iv) sont equivalentes, et signifient encore queY est ordinaire. D'apres 1.7 et grace a (2.1.4) (i) et (ii) sont equivalentes. D'apres 2.5 applique a B, (ii) et (iii) sont equivalentes. L'implication (iii) => (iv) est triviale, et il en est de meme de (iv) => (iii) quand Y est courbe, j5grf cjy etant a support ponctuel (cf. 1.8). Le critere de Hyodo 1.10* resulte aussitot de 2.6, par passage au localise strict de S. Dans cette situation, le systeme local Sn est trivial.
ORDINARITE et le complexe L„ s'identifie a la somme directe des Q ^ 1 < ii < • • • < i„ < r.
403 n - nz>
pour
Remarque 2.7. Pour Y de dimension > 1, il n'est pas vrai en general que (iv) implique (i). Voici un exemple que m ' a communique Hyodo. Soient C une courbe elliptique supersinguliere dans P^ et X I'eclate de C dans P | . Alors X est semi-stable sur 5 , la fibre speciale Y est reunion du diviseur exceptionnel E et de P^, qui se coupent transversalement suivant C. Ni E ni C ne sont ordinaires {E = P(A^), ou N est le conormal de C dans P | ) (cf. 1.4). Toutefois X a reduction ordinaire. On verifie en efFet que, si / : y —> Z := P^ est la projection, on a BQ^^^Rf^Buy pour tout z, et par suite H*{Y, Bu\r) = H\Z, BQ'z) = 0Remarque 2.8. Soient S un trait de point ferme s parfait de caracteristique p et X un 5-schema semi-stable, de fibre speciale Y. Le complexe de de R h a m - W i t t Wcoy de Hyodo [15], cf. aussi [20], permet de donner une caracterisation de "reduction ordinaire" analogue a celle expliquee en 1.3*. Plus precisement, supposons X propre. Alors X a reduction ordinaire (au sens de 1.8*) si et seulement si Pendomorphisme (cr-lineaire) F de H^(Y^Wajy) est bijectif pour tout i et tout j . Quand H*{Y,WUJ'Y) est sans torsion, il revient au meme de dire que le polygone de Newton defini par les pentes de Frobenius agissant sur H^(Y,WLJ'Y) coincide, pour tout n, avec le polygone de Hodge defini par les nombres de Hodge /i*'"~* =: d i m i J " ~ * ( y , cjy). Supposons de plus que la fibre speciale Y soit un diviseur a croisements normaux somme de diviseurs lisses. On pent alors esperer construire une suite spectrale analogue a celle de Steenbrink [34, 4.20], aboutissant a H*(Y, Wix^y) convenablement filtre, de terme initial somme de cohomologies de Hodge-Witt d'intersections k kk des composantes de Y, et equivariante sous Taction de F . La bijectivite de F sur le terme initial entrainant sa bijectivite sur Taboutissement, on obtiendrait ainsi une seconde demonstration de 1.10* (du moins pour s parfait).
BIBLIOGRAPHIE [1] Berthelot, P., Cohomologie cristalhne des schemas de caracierisiique p > 0, SLN in M a t h . 4 0 7 1974. [2] Berthelot, P. et Dlusie, L., Classes de Chem en cohomologie cristalline, C. R. Acad. Sc. Paris, t. 2 7 0 , (1970), 1695-1697 et 1750-1752. [3] Bloch, S. et Kato, K., p-adic etale cohomology, P u b . IHES 5 2 (1986), 137-252. [4] Crew, R., Specialization of crystalline cohomology^ Duke M a t h . J., 5 3
404
[5] [6] [7] [8] [9]
LUC ILLUSIE 1986, 749-757. Deligne, P., Theorie de Hodge II, P u b . IHES 4 0 , (1972), 5-57. Deligne, P., La conjecture de Weil I, P u b . IHES 4 3 , (1974), 273-307. Deligne, P. La conjecture de Weil II, P u b . IHES 5 2 , (1980), 137-252. Deligne, P., Lettre a L. lUusie, 4. 11. 76. Deligne, P. et Illusie, L., Relevement modulo p^ et decomposition du
complexe de de Rham, Inv. Math., 89 (1987), 247-270. [10] Faltings, G., Crystalline cohomology andp-adic Galois representations^ preprint, Princeton University, 1988. [11] Faltings, G., F-isocrystals on open varieties, results and conjectures, preprint, Princeton University, 1988. [12] Gros, M., Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Memoire SMF 21, Gauthiers-Villars, 1985. [13] Grothendieck, A., Formule de Lefschetz et ratwnalite des fonctions L, Seminaire Bourbaki, 2 7 9 , decembre 1964, Benjamin. [14] Grothendieck, A,, Croupes de Barsotti-Tate et cristaux de Dieudonne, Presses de PUniv, de Montreal, 1974. [15] Hyodo, 0 . , ^ note on p-adic Stale cohomology in the semi-stable reduction case, Inv. Math., 91(1988), 543-557. [16] Hyodo, O., On the de Rham-Witt complex attached to a semi-stable family, preprint, 1988. [17] Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline, Ann. ENS 4^ serie, t. 12 (1979), 501-661. [18] Illusie, L., Reduction semi-stable et decomposition de complexes de de Rham, preprint, Orsay 1988. [19] Illusie, L. et Raynaud, M., Les suites spectrales associees au complexe de de Rham-Witt, P u b . IHES 5 7 , (1983), 73-212. [20] Kato, K., The limit Hodge structure in the mixed characteristic case, preprint, 1988. [21] Katz, N., Nilpotent connexions and the monodromy theorem, P u b . IHES 19, (1970), 175-232. [22] Katz, N., Slope filtration of F-crystals, dans Journees de Geometric Algebrique, Rennes 1978, Asterisque 6 3 (1979), 113-164. [23] Katz, N. et Messing, W., Some consequences of the Riemann hypothesis for varieties over finite fields, Inv. Math. 2 3 (1974), 73-77. [24] Koblitz, N., p-adic variation of the zeta function over families of varieties defined over finite fields, Comp. m a t h . 3 1 (1975), 119-218. [25] Mazur, B., Frobenius and the Hodge filtration. Bull. Amer. M a t h . Soc. 78 (1972), 653-667. [26] Mazur, B., Frobenius and the Hodge filtration, estimates, Ann. of Math., 9 8 (1973), 58-95.
ORDINARITE
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[21 Miller, L., Curves over a finite field with invertihle Hasse-Witt matrix^ Math. Ann. 197 (1972), 123-127. [28; Miller, L., Uher Gewohnliche Hyperfldchen, I, II, J. Reine Angew. Math. 282 (1976), 96-113; 283 (1976), 402-420. [29; Mumford, D., Bi-extensions of formal groups, Algebraic Geometry, Bombay Colloquium 1968, 307-322 Tata Inst. Fund. Research and Oxford Univ. Press, 1969. [3o; Norman, P. et Oort, F., Moduli of abelian varieties, Ann. of Math. 112 (1980), 413-439. [31 Ogus, A., Frobenius and the Hodge filtration, dans Berthelot, P. et Ogus, A., Notes on crystalline cohomology, Math. Notes, Princeton Univ. Press, 1978. [32 Ogus, A., Supersingular K3 crystals, dans Joumees de Geometric Algebrique de Rennes 1978, Asterisque 64 (1979), 3-86. [33; Rapoport, M. et Zink, Th., Uber die lokale Zetafunktion von Shimuravarietdten, Monodromiefiltration und verschwindende Zyklen in ungleicher Charaktenstik, Inv. Math., 68 (1982), 21-101. [34; Steenbrink, J., Limits of Hodge structures, Inv. Math. 31 (1976), 229-257. [35; Steenbrink, J., Mixed Hodge structure on the vanishing cohomology, Symp. in Math. Oslo, (1976), 525-563. [36; Steenbrink, J. et Zucker, S., Variation of mixed Hodge structures, I., Inv. Math. 80 (1985), 489-542. [37; Suwa, N. et Yui, N., Arithmetic of Fermat varieties I, Fermat motives and p-adic cohomologies, preprint, 1988. SGA 5 Cohomologie i-adique et fonctions L, Seminaire de Geometrie Algebrique du Bois-Marie 65-66, par A. Grothendieck, SLN 589, 1977. SGA 6 Theone des intersections et theoreme de Riemann-Roch, Seminaire de Geometrie Algebrique du Bois Marie 66-67, par P. Berthelot, A. Grothendieck, L. lUusie, SLN 225, 1971. SGA 7 Croupes de monodromie en geometrie algebrique, Seminaire de Geometrie Algebrique du Bois-Marie 67-69, I par A. Grothendieck, SLN 288, 1972, II par P. Deligne et N. Katz, SLN 340, 1973.
Universite de Paris Sud Centre d'Orsay Mathematique - Bat. 425 91405 Orsay Cedex France
Kazhdan-Lusztig Conjecture for A Symnietrizable Kac-Moody Lie Algebra
MASAKI KASHIWARA dedicated to Professor
Alexandre Grothendieck on his sixtieth birthday
0. I n t r o d u c t i o n In recent years, with the progress of mathematical physics, it becomes more and more i m p o r t a n t to study systems with infinite freedom. In [K], we studied the flag variety of Kac-Moody Lie algebras, as a typical case of an infinite-dimensional manifold. By t h a t study, it is revealed t h a t the most natural language of scheme introduced by A. Grothendieck is again the most appropriate algebraic tool to deal with an infinite-dimensional manifold. T h e purpose of this paper is to generalize the result of BrylinskiKashiwara [B-K] and Beihnson-Bernstein [B-B] to the K a c - M o o d y Lie algebra case by using the flag manifold. T h e main result is the following one. T h e o r e m . The generalization of the Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras (proposed e.g. by [D-G-K]) is true. T h e proof of this theorem is not much different from the one of [B-K]. However, since the flag variety is an infinite-dimensional variety in general, we need special care to prove it. We win explain the contents. Let g be a symmetrizable K a c - M o o d y Lie algebra, and X its flag variety. For an integral dominant weight A, let 0{\) be the corresponding hne bundle over X (see §3 for the precise definition) and let 2)A be the twisted ring of differential operators 0 ( A ) 0 S x 0 0(-A)onX . We restricted to the symmetrizable case because we use the following statement t h a t is proven by D e o d h a r - G a b b e r - K a c [ D - G - K ] by using the result of K a c - K a z h d a n [K-K]: any highest weight of the Verma module with a dominant integral weight X as a highest weight is in the orbit of X by the
408
MASAKI KASHIWARA
Weyl group. We prove in this paper the following theorem. T h e o r e m . Let DJl be an admissible B-equivariant we have (i) 5 ^ ( X ; 9Jt) m 0 for any n^{). (ii) 5 ) x 0 r ( X ; 9 J l ) —> 9JZ is an
'Dx -module.
Then
isomorphism.
For the precise statement see Theorem 5.2.1 and for the notations see the contents. This theorem shows t h a t the irreducible g-module L{w o A) with highest weight w o \^ where w G W, is obtained as V{X\Cxu{y<)) for the irreducible 5-equivariant ^A-module £iu(A) whose support is the Schubert cell BWXQ. By the Riemann-Hilbert corresp on dance, this 'X)x -module corresponds to the perverse sheaf supported on BWXQ. This cohomology groups at points in X can be described in terms of Kazhdan-Lusztig polynomial as in the finite-dimensional case (Kazhdan-Lusztig[K-L]). This geometric part will appear elsewhere. (0.1) In this paper Hom means the outer Hom and Hom means the inner Hom (or the sheaf of homomorphisms). If we say positive, it includes 0. N means set of positive integers. Z>n (resp. Z < „ ) is the set of integers more t h a n (resp. less than) or equal to n. 1.
^-modules
(1.1) In this section, we define the sheaf of differential operators X)x and a special class of X)x-niodules for a scheme X over C of countable type (see [K]). As in [EGA], we define the subsheaf FmC^x) of Endc{Ox) by (1.1.1)
i^m(^x) = 0
for
m<0,
and (1.1.2) Fmi^x)
= {Pe
EndciOx);[Ox,P]
C fm-i(2)x)} M
m > 0.
T h e n , we have
Fii^x)
= 0x®
HomoA^x.Ox)-
Remark t h a t Q,]^ is a quasi-coherent sheaf but F i ( D x ) is not quasi-coherent in general. But F m ( ® x ) has a following finiteness property. For an affine open subset {/, T{U]Fm{'^x)) has a complete linear topology where {PenU;F,n{^x));
P{fj)^Oforj=l,.--,N}
SYMMETRIZABLE KAC-MOODY LIE ALGEBRA
409
form a neighborhood system of 0 for / i , • • • ,/AT G T{U]OU)By this, FmC^x) becomes a sheaf of complete linear topological spaces on X. Note that, for an afRne open set U^ FmC^x) \u is the projective limit of coherent Of/-modules. Set D x = U ^ m ( S x ) C End{Ox)' Then S x is an 0x-ring. Definition l . L l . We say that a left S^-i^odule 9Jl is admissible if it satisfies (i) DJl is quasi-coherent over Oxj (ii) 9Jl is a topological S x -module with discrete topology. The condition (ii) is equivalent to saying that, for any affine open subset U of X, any s in r([/;9}Z) and an integer m, there exist finitely many firJN e T{U] Ox) such that Ps-O for any P G r ( ^ ; FmC^x)) with P{fi) = Q{oT i = lr" ,N. The following lemma is obvious. L e m m a 1.1.2. The admissible 'Dx-'fnodules form an ahelian category. (1.2) For a morphism / : X ^ y from a scheme X of countable type to a finite-dimensional scheme y , set S)x-^F = Ox ^j-iOy '^y- Then it is easy to see that 'Sx-^Y C Endcif'^Oy ,0x) and it has a structure of (S)x,/"^2)y)-module. Hence, for a S)y-module 9Jl, /*9n = 3 ) x - . y 0 S ) ^ dXl has a structure of S x -module. (1.3)
Now, let us assume X affine and X ~ lim "<
5n where Sn are n6N
smooth and affine over C and Sn-\-i —^ Sn is also smooth for any n. Let fn '• X ^>' Sn he the projection. Then we have S)x = lim Ox 0Oq ^ 5 • We can prove easily L e m m a 1.3.1. IfdJl is an admissible "S^x-f^odule, then 9Jt = lim f^^n ^n where ^ „ are quasi-coherent S)5^-morfw/e5. Definition 1.3.2. We call an admissible 2)x-module 9Jt holonomic (resp. regular holo- nomic) if there exist n and a holonomic (resp. regular holonomic) S s ^ - module ^ such that 9Jl 2:^ fn^(1.4) This definition 1.3.2 can be generalized for S)x-module if X is covered by open affine subsets satisfying the conditions in (1.3). For a holonomic 2)x -module 9Jt, we can define its dual 9Jl* by 971* = /*9^* for an ^ as in Definition 1.3.2. This does not depend on the choice and we can define 9Jl* globally. We have the similar properties as in the finite-
410
MASAKI KASHIWARA
dimensional case. (1.4.1)
dJlh-^ dJl* is an exact contravariant
functor
(1.4.2)
9Jt** :^ m.
(1.4.3)
Rom^^{m,m') - Eom^ jm'\m*).
in 9Jl.
2. K a c - M o o d y Lie a l g e b r a (2.1)
We shall review the definition of Kac-Moody Lie algebras. Let / be a finite set (the index set for simple roots) and let (] be a finite-dimensional vector space. Assume the following conditions. (2.1.1)
(2.1.2) (2.1.3) (2.1.4) (2.1.5) (2.1.6) (2.1.7) (2.1.8) (2.1.9)
f) 3 /ij, f)* 3 ai {i G / ) and < hi,aj > is a generalized Cartan matrix (i.e. < hi,a^ >— 2, < hi^a^ >G Z= 0 < ^ < hj.ai >= 0). {ai} are linearly independent. P C ^* is a lattice such that P 3 ai and < P, hi > C Z. g is a Lie algebra satisfying the conditions (2.1.5)-(2.1.9). f) is contained in g as an abelian Lie subalgebra, g is generated by f) and e^ and fi, [h,ei] = ai{h)ei and [/i,/t] = —Cii{h)fi for any /i G ^ and i G / , [e^Jj]=^Sijhi, iadeiy-<^^^^^>ej = (adfiy-<^^'^^> fj = 0 for i ^ j .
We use the following notations. (2.1.10) n (resp.
n—) 25 the Lie subalgebra generated
by the ei
(resp.
fi) {i € / ) • (2.1.11) For a G i)*, we set g^ = {x G 0; [h, x] — a{h)x for any h G f)}. (2.1.12) A = { a G r \ {0}; 0« 7^ 0}, A+ = {a G A; g , C n } , A _ = {a G A;g«C n_}. (2.1.13) n(i) =z 0 g^an(/n_(z)= 0 g^. a€A+\{aJ
(2.1.14) (2.1.15) (2.1.16) (2.1.17) (2.1.18)
aGA_\{-aJ
g, = f ) 0 C e , 0 C / , - . T = 5pec(C[P]). Gj 25 the group with Lie algebra g^ and Cartan subgroup xxk - {adn)^~^n and n~ = {adn_)^-^n_ for k >1. U = \im exp(n/njt), 6^_ = lim e x p ( n _ / n ; ^ ) , 6^,-= lim
T. {n{i)/nk)
C U andUl_ — lim e x p ( n _ ( i ) / n ^ ) ^ where exp denotes the corresponding unipotent algebraic group. (2.1.19) B = TxU, B.=TxU., P^ = d
x (/,, p-
= d
T h e n Pi D B D Ui and Ui is a normal subgroup of Pi.
x
U'.
SYMMETRIZABLE KAC-MOODY LIE ALGEBRA
411
Remark 2.1.1. In [K], we assumed that (2.1.7)-(2.1.9) are the fundamental relations of g, but the results there can be generalized in this framework without modification. (2.2)
We define
(2.2.1)
g = lim0/njk, k
(2.2.2)
UmiQ) =
lmiUm{Q)/Um-i{Q)nk, k
m
(2.3) For A G {)*, let M(A) (resp. M_(A)) be the Verma module with highest (resp. lowest) weight A, namely, M(A) (resp. M_(A)) is the /7(g)module generated by an element ux (resp. u^) with the fundamental relations (2.3.1) hux =< h, X > Ux (resp. hu^ =< h, X > u^) for any /i G ^, (2.3.2) eiUx = 0 (resp. fiU^ = 0) for any i G / . (2.4) (2.4.1)
For an (^-module M, we set, for A G i^* Mx = {ue M; {h - A(/i))^w = 0 for any h e h.rn:> 0}.
We set, for a locally f)-finite [)-module M, (2.4.2)D(M) = {fe
Hom(M;C);dim(/7(f))/) < oo} = eAHom(M,C)A.
For a locally f)-finite g-module M with dim MA < oo for any A G b*, we define its character by (2.4.3) (2.5) (2.5.1)
chM = ^ ( d i m M A ) e ^ . For A G f)*, we set M*(A) = D(M_(-A)).
Then M*(A) is a C/(g)-module. (2.6) For a g-module M on which f) acts semisimply, a highest weight vector u with highest weight X is, by definition, an element of Mx such that
412
MASAKI KASHIWARA
nu = 0. Then M*(X) has a unique highest weight vector up to constant multiple and its weight is A. Hence there exists a non-zero homomorphism M(A) -^ M*(A) unique up to constant multiple. T h e image is denoted by L(A). This is an irreducible g-module with highest weight A. (2.7) Let W be the Weyl group of 9, namely W is the subgroup generated by the simple reflections 5j, where (2.7.1)
oiAut{\)*)
SiX = X- < hi,X > ai.
Now we define the other action o of VF on I)* by (2.7.2)
SioX
= X-{
-f l ) a i .
If we choose p E ^* with < p^hi >= 1 for any i G / , we have (2.7.3)
woX
= w{X-\-p)-p
for
w e W.
We have also (2.7.4)
woX
For w eW, (2.8)
= wX-
^
a.
let i{w) be the length of w. We have i{w) = # ( A + D
wA-).
T h e following lemma is well-known. Leniina 2.8.1.
For A G ^* and i E I such that < hi^X >G
Tjy^i,
Homg(M(5i o A), M(A)) = C .
T h e homomorphism is given by 1/5^0A '-^ / ^ *' ^^ ^AP r o p o s i t i o n 2 . 8 . 2 . For A,/i G f)* and i G I such that < hi^X >G Z > _ i and < hi,fi > ^ Z>o, we have (2.8.1)
H o m 0 ( M ( / i ) , M{X)/M{si
o A)) = 0,
(2.8.2)
H o m g ( M ( ; i ) , M{siO X)) ^EomQ{M{fi),
M(A)).
Proof. In fact, M{X)/M{sioX) is locally g^-finite. Hence any highest weight (, of M{X)/M{si o A) satisfies < /i,,<^ >G Z>o. Q.E.D.
SYMMETRIZABLE KAC-MOODY LIE ALGEBRA
413
L e m m a 2.8.3. For X,/Ji E: ^* and i E I such that < hi,fi >G Z>_i and < /zjjA > ^ Z<_i, we have Ext^(g)(M(/i)/M(5,- O ^ ) , M(A)) ::. 0.
Proof. (2.8.3)
It is enough to show that any exact sequence 0 —^ M(A) — . M - ^ M{^)/M{si
o //) — . 0
splits. Since M(A) and M(/i)/M(5iO/i) are locally l)-finite, so is M. Hence there exists v G M^ such that <^(i^) = u^ mod M(5j o fi). Setting i =< hi.ii > + 1 , M{si o/i) = U{^)f-;u^, and hence (2.8.4)
/ / « e M(A).
Since e^i/^ = 0, we have (2.8.5)
CiV e M(A).
The relation e,// = ffei+£{hi+ll)ff-^ implies / . e j / ^ = //^^e,-^ + i{hi -\-i-\- l)f[v. Since / / i ; G M(A)^_^or., (/^i + ^ + 1 ) / / ^ = 0 and hence fii^tfiV — fiCiv) — 0. Since ji acts injectively on M(A), we deduce (2.8.6)
ti![v = ffe,v.
Set f[v = Pux with P e t/(n_). Since U{n_) = C[fi] ® C/(n_(z)), we can write P in a unique way
Then, we have (2.8.8)
[e„P] = ^ / r [ e . - , P ] + nfr^k
-n+
1)P„
= E / " [ ^ - ^ ] + ^f?~'Pn{hi- +nThus we obtain (2.8.9)
eJ[v = Y,if"[e^,Pn]
+
n{n-£)fr'Pn)ux
e).
414
MASAKI KASHIWARA
By (2.8.6), e^fiV belongs to fiM{\) (2.8.10)
= ffU(n_)ux.
Hence we obtain
h , Pn-i] + n{n - ()Pn = 0 for 1 < n < i,
because {adei)U{r\_(i)) C C/(n_(i)). This shows (2.8.11)
Pn e C(adei)"Po for 0 < n< i
and (2.8.11)
{ade^YPo = 0.
On the other hand, PQ G U{n_{i))^_x-ia, and < hi.fi — X — £ai > = — 1— < /ijjA > —£ ^ Z>_£. Therefore, the representation theory of s/2 implies PQ ^ 0. Thus P^ = 0 for n < £. This implies f^v G fiM{X). Hence, we may assume, from the beginning, ffv =• 0. Since fiCiV = eiffv rr 0 by (2.8.6), we have eiv = 0. For j / 2, flejV = ejf[v = 0 and hence ejV = 0. For /i G i^*, fii^" < h,fi > v = (h— < hjfj, > -\-£ < h,ai >)fiV = 0 and hence (h— < h^fi >)v — 0. Therefore li^ H^ t; defines M{^)lM{si o fi) ^^ M. Thus the exact sequence (2.8.3) splits. Q.E.D. Corollary 2.8.4. For X,fi G f)* and i E I such that < /ij,/i >G Z>_i a7i{/ < /ii,A > ^ Z<_i, w;e /iai;e (2.8.12)
Hom(M(/i), M(A)) -^ Hom(M(5, o fi), M(A)).
By this corollary , we have Corollary 2.8.5. For A G P+ and w,w'
eW
(2.8.13)
Hom(M(w; o A), M(A)) = C.
(2.8.14)
dimHom(M(ti; o A), M{w' o A)) < 1.
Proof. The first statement follows easily from the preceding corollary or Proposition 2.8.2 and (2.8.14) follows from Hom(M(w; o A), M{w' o A)) C Eom{M{w o A), M(A)). Q.E.D.
S Y M M E T R I Z A B L E K A C - M O O D Y LIE A L G E B R A
415
3. F l a g v a r i e t y o f a K a c - M o o d y Lie a l g e b r a (3.1) For the definition of the flag variety X of (g, f),P), we refer to [K]. There, two constructions of X are given. One is to construct X as a subscheme of Grass (§). The other is t o get X as G/B_. Here G is a scheme on which B- acts locally freely (see [K]). Let XQ € X be the 1 mod 5 _ . Then we have the Birkhoff-Bruhat decomposition
(3.1.1)
^=
U ^^^0wew
(3.2) For i £ / , we set Xi — G/P^. Since P~ acts locally freely on G, this quotient exists as a scheme in the Zariski topology. Then qi : X ^ G/B- —^ Xi - G/Pr is a P i -bundle since i ^ " / B - = Gi/Gi fl 5 _ ^ P ^ Since qi is universally closed and surjective, Xi is separated. Set qi{xQ) — Xi. T h e n q^^{xi) = GiXo = {XQ} U (Gi n B^)siXo. Let p : G -^ X and pi : G -^ X^ be the projections. L e m m a 3 . 2 . 1 . Assume l{wsi) < l{w). (i) q^^{Bwxi) = BWXQ U BwSiXQ. (ii) BWXQ — qJ^{BwXi) fl wsiBsiXQ. (iii) BwSiXo — q^^{BwXi) fl wsiBxo. (iv) BWXQ - ^ Bwxi
Then
by qi.
Proof. Since q~^{Xi) = GiXo = {XQ} U ( 5 _ fl GI)SIXQ, we have q~^{BwXt) = BwxoUBw(BnGi)siXo. Then, wat £ A _ implies w(B- O Gi)w~^ C B, we obtain (i). (iii) follows from (i), BwsiBxQ f] BWXQ = ({) and wStBxo D BwsiXQ. (ii) follows from (i), (iii) and qf^Bwxi = q~^qi{BwSiXo) C qf^qt{wSiBxo) C wSiq^^qi{Bxo) = wsiBxQ U wSiBsiXQ. (iv) follows from 5 u ; a r o ^ ^ " V f l ^ and t / n " ' " ' t / n / ^ - C G i f l ^ ' ' ^ / C {1}. Q.E.D. By this we have P r o p o s i t i o n 3.2.2. l{wsi) > l{w).
Xi
= UBwxi
where w ranges over W
with
(3.3) Let p : G -> X ^ G / 5 . be the projection. For A G P , we define the Ox -module 0{X) by V{U-0{X))
= W£
T{p-'U-OG)\^{9h.)
= hz\(g)
for {g,h^)ep-\U)
x
B.)
416
MASAKI KASHIWARA
for any open subset U of X. Here 61 is t h e homomorphism
where the last arrow is the homomorphism whose differential is —A. T h e n 0{X) is an invertible Ox -module, and 0{X) is clearly Pi -equivariant. We define the ring 2) A by (3.3.1)
S)A = OiX) 0ox ® x ^Ox
0{-X).
Since g acts infinitesimally on X and 0{X), we have (3.3.2)
g-.r(X;DA).
This can be extended t o a continuous ring homomorphism (3.3.3)
U{Q)^T{X-^X).
(3.4) Let M be a discrete U{Q) -module. This is equivalent t o saying t h a t M is a {I -module such t h a t , for any element u of M, ixj^u =: 0 for k ^ 0. We say such a g -module is admissible. Definition- Proposition 3.4.1. (i) For X £ P and an admissible 0 -module M , 9Jl i-^ H o m 0 ( M , T{X; DJl)) is represeniable on the category of admissible ^ A -modules. We denote by ^ A ® ^ ihe admissible S)A -module that represents the functor above. (ii) M y-^ X)A(8)M is a right exact functor in admissible 0 -module M. (iii) X)A 0 M - ^ D A 0 M IS
(iv) M \-^ "Dx^M commutes
Proof.
surjective.
with inductive
Assume first M = U{g)/U{Q)nk.
limit.
We shall show t h a t
9JT ^ Homc7(g)(M,T{Bxo;M))
^
Eomu(n){U{n)/U{n)nk,T(Bxo'.m)){3.4.1) is representable in the category of admissible ^ A \BXO -modules. By identifying BXQ = Uxo and trivializing (9(A) by the section s given by s{ub-) — 61 (w £ U^b_ G B-), we can identify D A \BXO and S)^/- T h e n the homomorphism n —^ r ( f / ; S f / ) induced by g —> T{X]'Sx) is nothing b u t t h e one induced by the left infinitesimal action of n on U. Set ^ — "Su-rUk
S Y M M E T R I Z A B L E K A C - M O O D Y LIE A L G E B R A
417
where Uk is t h e unipotent group with t h e Lie algebra n/tijk. T h e n for any admissible 1>u -module 3Jl H o m s ) ^ ( ^ , im) = {f€
T{U; DJl); n ^ / = 0 } .
Hence t h e functor (3.4.1) is represented by 01 in t h e category of admissible (^A \BXO) -module. We denote by (5)A \BXO)^^ its representative. Then for an index set J and kj G Z > i ( j G J ) , set A^ = ®j£j U{^)/^kj' Then m ^ Eomu(g){M,T{Bxo]M)) is represented by 0 ^ - ^ J ( S A \BXO)
M' (SA
Now any admissible g -module M has a resolution 0 <— M <— i— M", where M' and M" have the form similar to N. Then I B X ' O ) ^ ^ is represented by the cokernel of S A ^ M ' <—
S)A§M''.
Now, the notion of admissible module is invariant by the action of the Weyl group. Hence a n I—> Homt;(g)(M,
T{wBxo]m))
is representable in the category of admissible (S)A \WBXO) -modules. We shall denote by (2)A \WBXO)'^M its representative. Now, we can easily patch { ( S ) A \WBXO)'^^}^ ^ G W, together, and the ^ A -module obtained by patching t h e m clearly represents the functor in (i). T h e statements (ii) and (iv) follow from the definition and (iii) follows from the corresponding statement on (S)A \WBXO)^^, t h a t is obvious from the construction. Q.E.D. (3.5) For an admissible Vx -module 5Jl, we can define the notion of B -equivariance, e.g. as in [ K']. T h e detailed care will be left to t h e reader. Note t h a t this is given by an isomorphism (3.5.1)
^*9JT^pr*D7t
as ^BxA" -module with chain conditions as in [ K']. Here pr, jj, : B x X —^ X are given by pr(b,x) — x and fi{bjx) = bx. We say an admissible X^A -module dJl is B -equivariant^ if so is the 'Dx -module 0{—X) ^ 9Jl. Lemma 3.5.1. For any admissible B -equivanani 5)A -module dJl and any open quasi-compact B-stable subset Q of X^ i7"(fi;OT) has a structure of B -module.
Proof.
We shall define the co-multiplication / f " ( Q ; 9Jt) — . 0(B) (8) i/"(12; M)
418
MASAKI KASHIWARA
as follows: i7^(Q;9Jt) -> ^"(/i-^Q;/i*9Jl) = i/"(Q; 0(B) (g)c 5n) ^ 0 ( 5 ) 0 H'^iQ] m). Here the last isomorphism follows from the quasi-compactness of Q. It is a routine to check that this satisfies the axioms for co-multiplication. Q.E.D. Remark 3.5.2. In the lemma above, we cannot drop the quasi-compactness of Q. For example, r(X;0u;^u;(A)) = f j ^ M*{w o A) is not a B -module in general (See Proposition 3.6.7 and Definition 3.6.8). This difficulty is overcome in §4. The following lemma is easy to prove. Lemma 3.5.3. If M is a ( 0 , 5 ) -module, then 'Dx^M equivariant J)^ -module. (3.6)
is a B -
In this section, we shall study a special type of S)^ -module. Lemma 3.6.1.
Lei w eW
(i) ^ k . o ( ^ W ) = 0
and \£P.
fork^Kw).
(li) H'{X;n'^Zl{0{X))) = H'/J^^^ (iii) HL.o(^-^O(X)) = 0 forki^l{w)
Proof.
Then for any k.
and
We have a commutative diagram ^ 5 x 0 D Bwx^ \\\ \\\
and ^C/ = ( ^ [ / n t / ) X i^U^US), Since ^?7nC/- ^ C^(^) and wBx^ is an affine open subset of X, we obtain (i) and (ii). Let s be a section of C)(A) over WBXQ given by WBXQ ^ wU "—^ G, i.e, s{wub^) = bZ^ foruEU and 6_ G B-. Then t?(A) = O. Moreover the weight of s with respect to
SYMMETRIZABLE KAC-MOODY LIE ALGEBRA T -action is wX, because s{t~^wu) = s{w(w~^t~^w)u) We have as a T -module H'^ZIO(^'^<^W)
= OrUnU)®
urn
=
419
{w~^tw)'^s(wu).
n n_) 0 (5 (8) 5,
where 8 is the delta function and its weight is T 3 t H-^ dei{Ad{t) :^ n n n _ ) , which is J2aewA^nA. ^' By (2-7.4), we have (3.6.1)
w\-\-
^
a-wo\,
and hence the weight of 6 <S> s is w o X. On the other hand, as a T -module 0(^/7 n [/) ^ 0 ( ^ n n n) ^ ^aewA^nA^S{Q%) and C/(^n fi n_) ^ ^ ( ^ n n n_) = 0aGt^A+nA_5(g^). Hence the character of 0(^C/nt/)(2)t/(^nnn_) is
'-
n
n
'-
This gives the last statement of (iii). P r o p o s i t i o n 3.6.2.
Q.E.D.
For any A G P V{Bxo\0{X))^M^{X),
Proof. (3.6.2)
By the preceeding lemma, we have diT{Bxo\ (9(A)) = chM*(A).
Using the notation (2.4.2), —A is a lowest weight of D{T{Bxo\0{X))) hence we obtain a homomorphism (3.6.3)
V{BXQ',0{X))
and
— . M*(A).
By (3.6.2), it is enough to show that (3.6.3) is injective. In order to see this, we shall show that there is no highest weight of V[BXQ]0{X)) other than A. This follows immediately from the following fact. (3.6.4)
An element ofV(U\Ojj) invariant by the left infinitesimal action of n IS a constant function. Q.E.D.
420
MASAKI KASHIWARA P r o p o s i t i o n 3.6.3.
Assume < hi,X >>—I.
Then
Proof. Since chHQ^^^^{X;0{X)) = chM*(5j o A), by arguing as in the proof of the preceeding proposition , it is enough to show that any highest weight of H^^^^^{X;0{X)) is 5^ o A. We have BSiXQ C
U^
StBxo
C ''U
Hence i/i,,^^^(X;(9(A)) S r{''U\Ui;0.,u)/Ti''U;0.,u){exptf,;t G C} S [7, X C. Then
//^,.,„(X;(9(A)) - 0(U,) ® iC[t,t-']/C[t-'])
Set '^U = Ui x
= 0(t/,) ®
cr']r\
( 3.6.5) Let / G T{'-U\U,;Os,u)/Oi''U;Ouu) = OiUi) ® C[t-^]t-^ be a highest weight vector. Then / is invariant by the left infinitesimal action of Ui and hence f{ue^^*Si) = /(e^'^'s,) for u 6 Ut,t G C. Therefore / G C[t~^]t~^. We shall calculate the infinitesimal action of e^. Let e be a quantity with e"^ = 0. Then f{e~^^'ue*^'Si) = /(e~^^'ue^*'e~^^'e*-^>Si) = /(e~^^'e*-^'s,), because [/, is invariant by Pj. On the other hand, we have e-".e*^-s, = e*^-e-^(^-+"''-*'-^')si =
e'^'+''^^'s,e-'^^'-"''\
This shows / ( e - " ' e ' ^ ' s O = /(e«^'+^''^'Si)(l - £ < /^i, A X ) = fie'^'Si) + £ ( < 2 l - < hi, A > <)/(e*^-sOHence (f{t) = /(e*-^*^^) satisfies the differential eciuation: t{t -—
< h^,X >)(p = 0
mod
C[t].
Since < h^^X >> —1, v? is a constant multiple of/~^. Thus, there is a unique highest weight vector of HQ^ ^ {X; 0{X)) up to constant multiple. Q.E.D. P r o p o s i t i o n 3.6.4.
If < hi^X > > —1, then
Hhs.roi^; 0{X)) = H\Bxo;0{s,
o A))
SYMMETRIZABLE KAC-MOODY LIE ALGEBRA as an (OxX^^^)^^)
Proof. phism
421
-'module.
Since they are isomorphic to M*(A), there exists an isomor>p : //^,.,„(X;0(A)) ^
H\Bxo;0{si
o A))
as g -module. This is unique up to constant multiple. We shall show that this is OxX^^i) -linear. Consider the two homomorphisms fuh-
Ox.iBxi)
® /f^,.,„(X;0(A)) — H°iBxo;Oisi
o X))
obtained by the multiplication of OxX^^i) ^^ ^hs^xoi-^'^^W) ^^^ H^{BxQ\0\s^oX)). We have to show fi = f2- As a g-module Ox^Bxi) C OX{BXQ) = M*(0), and / i and /2 are g-linear. If / i / /2, then the image of / i — /2 has a highest weight, that must be weight A. The weight space of Oxt{Bxi)<S^H^g^^^{X]0{X)) with weight A is contained in C(g)iJ^^ ^ {X]0{X)) and / i = /2 on this space, which contradicts / i 7^/2. Q.E.D. Corollary 3.6.5.
For w e W and X e P with < h^,X >> -I,
we
have
as a Q -module.
l{w),
L e m m a 3.6.6. For w E W, i E I and X E P such thai l{wsi) < < hi,X >> —1, we have, as g -module
H'^:l(^-> OW) = Htl'rM; Oisi o A)). Proof. Since BWXQ = qf^{Bwxi) q~^{Bwxi) n wsiBxo, we have
= H'^:l:\Xi>'ii.K>.B.oiOisi
Proposition 3.6.7.
fl wsiBsiXo
and BwsiXQ =
o A))) = H'^Zl-^iX-Ois, o A)). Q.E.D.
//A € P^, then for any w £W
nX;n'^lliOiX)))^M*iwoX).
422
MASAKI KASHIWARA
This follows immediately from the preceeding lemma and Proposition 3.6.2. This proposition is proved by G. Kempf [Kempf] in the finite-dimensional case, and S. Kumar obtained similar results in the Kac-Moody case ([S.K]2). Definition 3.6.8. For A G P and w; G H^, we set »^(A) = W^^],^(0(A)). Then 55iy(A) is a 5 -equivariant holonomic 5)A -module and satisfies
(3.6.8)
//^(X;«,(A))=r I
M*(i/;o A) for n =: 0, 0
forn ^ 0.
Proposition 3.6.9. Lei dJl he a B -equivariant admissible S)A module and w G W. Assume (3.6.9)
Supp
971 C
BWXQ
on a neighborhood of
BWXQ.
Then there exist an index set J and a B -equivariant homomorphism (p : 971 —r 93iy(A)®*^ such that (p is an isomorphism on a neighborhood of BWXQ.
Proof. This follows from the following two lemmas that are more or less well-known. Lemma 3.6.10. Let Y be a closed subscheme of X such that locally X = Y X C^. Then for any admissible S)x -module dJl with Supp(dyi) C Y^Hom^ ( S x ^ y ? ^ ) ^^ ^^ admissible ^ y -module and 9Jl ^ ^Dx^y 03)^ iTomj) ( 2 ) x ^ y , 9 ^ ) .
Lemma 3.6.11. Let X be a homogeneous space of an affine group scheme G. If DJl is an admissible G -equivariant © x -module, then 9JI = ^eJX Here X is a homogeneous space ofG ifG acts on X and for a C-valued point X of X, p : G —^ X given by g \-^ gx is faithfully flat. Then Lemma 3.6.11 follows from the faithfully flat descent ([SOA]).
S Y M M E T R I Z A B L E K A C - M O O D Y LIE A L G E B R A
423
4. S ) - m o d u l e s o n t h e flag v a r i e t y (4.1) In this section, we shall study j9-e qui variant ^ - m o d u l e s on the flag variety X for an arbitrary Kac-Moody Lie algebra. Let $ be a finite subset of W such t h a t
(4.1.1)
liw'
£ ^
and \i w ^W
satisfies w < w', then ^ 3 w.
For such a $ , let fi(^) be the open subset |Ju;e$ BWXQ = [J^je^ Note t h a t any 5-stable quasi-compact open subset has this form.
IVBXQ.
Let A G P + , and set P A ( ^ ) = U t . ^ * ( ^ ^ ^ + Q-)We denote by Mod{'Dx,B)the abelian category of P-equivariant admissible 3)A-inodules, and Mod{Q,B) the abelian category of (g, 5 ) m o d u l e s . Lenima 4.1.1. Let U be a quasi-compact B-stable X such that U D Q ( ^ ) . Then for any dJl in Mod{'Dx,B), Hu\m)^U;OJl) belongs to P^i^).
open subset of any weight of
Proof. By the condition, U can be written as Q ( $ ' ) with $ ' D <^. We shall prove the statement by the induction of #(<^' \ $ ) . When ^' ^ ^ , let us take a maximal it; in <^' \ $ . Set U' = fi(^' \ {w}). T h e n we have a long exact sequence
By the hypothesis of the induction, ^ t / ' \ a r * ) ( ^ ' ' ^ ) satisfies the property. Hence it is enough to show the property for H^ijj,(U',dyi). We have a spectral sequence
S"^ce nl^^JdJl) is in Adod{^x,B), n%^^^{m) \Bru.o is a direct s u m of copies of St^(A) by Proposition 3.6.9. Hence £2^ is also a direct s u m of copies of H^{U] ^iy(A)), which is isomorphic to 0 or M'^(w o A). Since the weights of M*(w o A) are contained in tt; o A -f Q - , we are done. Q.E.D. Corollary 4.1.2. H''{U;On),
Under the same assumption ^ H"{Q(^y,m),
as m Lemma
for A« ^ P A ( $ ) .
4-^-^
424
MASAKI KASHIWARA
Corollary 4.1.3. For any fi £ P, there exists a quasi-compact Bstable open subset U such that for any quasi-compact B-stable open subset U' D U, H''{U']m)y, -^ ^"([/;9Jl)^ 25 an isomorphism. Definition 4.1.4. For 9Jl G Mod{'Six^B), we set
u where U ranges over quasi-compact 5-stable open subsets. We set r(X;OT) = H^{X-m).
Then Corollary 4.1.3 implies
V{X',m) = 0 ^ e p r ( X ; 9 n ) ^ -> r(X;9Jl).
By the above discussion, for an exact sequence in
Mod(^x^B)
we have a long exact sequence
We have also (4.1.2)
^"(X;S„(A))
M*{wn o7^A) t Oior 0. for n = 0
(4.2) Let <^ be a finite subset of W satisfying (4.1.1). Let C be the thick subcategory of Mod{Q^B) consisting of (g, jB)-modules whose weights are in Px{^). We set (4.2.1)
Mod{Q, ^) = Mod{g, B)/C.
This is an abeUan category because C is thick. For any 5Jl in Mod^'Dx, B), 5^(X;9Jt) and i/"(Q(^); 9Jl) are isomorphic in Mod{g,^) by Corollary 4.1.2. Definition 4.2.1.
For weW.we
set 9Jt^(A) = 0(A) (g) 93^.
For the notation, see §1.4. This is a 5-equivariant holonomic S);\-module whose support is the closure of BWXQ.
S Y M M E T R I Z A B L E KAC-MOODY LIE A L G E B R A
425
Moreover, we have (4.2.2)
OT^(A)
I wBxo :^ ^^.(A) |
WBXQ.
Since for any holonomic S)x-"module 5Jl, (4.2.3)
Homj)^(9Jt,^,)^Homj)^(9JlU,,,^^
U , J ,
we have, by the duality (1.4.3), for any holonomic ©A-module 9JT, (4.2.4)
Hom5)^(OT^(A),9Jl) ^ Homj)^(9Jl^(A) U ^ ^ , , ^ ^ B ^ J .
Also by the duahty we have (4.2.5)
/ / 9Jii^(A) —^^isa surjeciive -modules and Supp{^)f]Bwxo
homomorphism of admissible ^ A = (j), then 01 == 0.
By (4.2.2) and (4.2.3), we have a homomorphism (4.2.6)
9n«,(A)->Q3«,(A).
This is an isomorphism on WBXQ. Let Cu>{^) be the image of this homomorphism. Proposition 4.2.2. Lei M he in Mod{g,B). Then the support of "Dx^M IS contained in the union of the closure of BWXQ such thai Homg(M,M*(ii;oA)) ^ 0 .
Proof. T h e support S of 'Dx'S'M is invariant by the action of B. If Bwxo is open in 5 , then 'Dx<S>M \BWXO is a direct sum of copies of ^«;(A) by Proposition 3.6.9, and hence Eom^^{'Dx<S>M
\BWXO^ ^W>(A) \BWXO)
H o m ^ ^ ( X i A ® M , ^ « ; ( A ) ) = H o m g ( M , M * ( u ; o A ) ) ^ 0. Theorem 4.2.3.
1)x'S>M{w o X) —>• 9Jlu;(A)
is an
We shall first prove the following lemma. L e m m a 4.2.4. (i) Supp{'Sx^M{w (ii)
o A)) C
BWXQ.
5)A§M(I/;OA)|5^^,^«^(A).
(iii) Eom{Tix§>M{woX)^'^^,{X))
= 0
for
w':^ w.
=
Q.E.D. isomorphism.
426
MASAKI KASHIWARA
P r o o f . Since 'Dx^M{w o A) is in Mod{Q^ B) and generated by one element, its support 5 is a closed B-stable subset. Let w' E W he such t h a t BW'XQ is open in S. Then 'Sx^M{w o A) \BW'XO is a direct s u m of copies of ^w'{^) and hence Rom{'Sx§>M{w o X),^w'{^)) 7^ 0. O n t h e other hand, Eom{'Dx<S>M{w o A),Q3u;/(A)) is isomorphic to Eomg{M{w o A), T(X] 53^'(A))) = H o m g ( M ( u ; o A), M*(it;' o A)). Hence, if it is not zero, then w — w'^ and in this case, this dimension is 1. This implies S)A§M(t(; o A) \BWXO-
®to(A) \BWXO • Q'E.D.
of Lemma
4.2.4.
Now, we shall prove Theorem 4.2.3. Since Muj{\) :^ 'Dx§>M{w o A) on BWXQ, this isomorphism extends to a homomorphism (p : Oyi^{X) —> DA§)M(t(;oA) by (4.2.4). On the other hand, by Corollary 4.1.2, we have ior fi> w o X, f{X-m^{X)),
c^ T{Q{{w^;i{w')
<
i{w)})-m^{X)),
Hence r(X;9Jt^(A))^oA ^ M*{w o X)^oX and r ( X ; 9Jl«;(A))^oA+c.. ^ M''{w o X)u}oX+a, — 0. Hence r(X;9Jlu;(A)) contains a highest weight vector u with weight it? o A. T h u s we obtain a homomorphism M{w o A) -> r(X;9J{^(A)) and hence xp : 2)A§M(ii; o A) - ^ 9^«;(A). Note t h a t (/? and tp are inverse to each other on BWXQ. Since ip o (p \BWXO— '^•d, (4.2.4) implies xp o (p = id. Now, we shall show Coker (p = 0. Since Supp Coker (p^ BIUXQ — , if Coker <^ 7^ 0, then there exists a n o n - zero homomorphism Coker (p -^ ^«;'(A) for w' ^ w. This contradicts Lemma 4.2.4 (iii). This shows Coker p = 0, and hence tl> and (p are isomorphisms. This completes t h e proof of Theorem 4.2.3. Q.E.D. (4.3)
Let weWJel
(4.3.1)
satisfy e{siw) > £{w).
T h e n we have (4.3.2)
PIWXQ = BWXQ U BsiWXQ.
Hence we obtain an exact sequence
(4.3.3)
SYMMETRIZABLE KAC-MOODY LIE ALGEBRA Lemma 4.3.1.
7^p,^^,(C?) = Q forni^
427
£{w).
Proof. 'H^^^^^{0) \BWXO= 0 for n ^ £{w). By the equivariance by Pi, we have the desired result. Q.E.D. Thus we obtain Lemma 4.3.2.
(4.3.4)
There is an exact sequence
0 - 7^S„(0(A)) - Q3„(A) - <8,.,(A) - . 0.
We have Lemma 4.3.3.
Proof.
Eom^^{^^{\),^s,wW)
= C.
In order to prove this, we note the following lemma.
Lemma 4.3.4.
'Dx^M*{w o A) -^ ^wW
^^ surjeciive.
In fact, this follows from the surjectivity oiOx ^M*{wo\) —> ^«;(A), which is a consequence of the fact that BWXQ <—^ X is an affine morphism. By this lemma, we have (4.3.5)
Hom^^(
This proves Lemma 4.3.3.
Q.E.D.
Taking the dual we have the following proposition corresponding to Lemma 4.3.2 and 4.3.3. Proposition 4.3.5. Assume i{siw) > £{w). (i) We have an exact sequence (4.3.6)
0 -^ 9Jl,,«.(A) ^ 9Jl^(A) ^ m^{\)/m,^^{X)
and ^w{X)/^s,w{X) 25 Pi-equivariant (ii) Homj)jan,.^(A),9H„(A)) = C.
-^ 0
428
MASAKI KASHIWARA
Proposition 4.3.6. Let w, w' e W,X £ P^ and let cp : M{w'oX) —> M{woX) be a non-zero Q-homomorphism. Then the induced homomorphism, T^x^M^w' o A) -> X)A0M(W; O A)
is injective.
Proof. We may assume without loss of generality that w' •= SiW and £{s^w) > £(w). Since we know in this case Homj) (9^«;'(A), 9K^(A)) = C, and the non-trivial homomorphism from OJlu)'{X) into 971^^^(A) is injective, it is enough to show that S^^V^ is non-zero. Let w be a non-zero section of 'Sx^M{w o A) with weight A. Setting £ —< hi,\ > -fl, it is enough to show f^u :^ 0. Now, we have Piwxo = Bwxo U Bsiwxo, and
U
U
u
u
Bs^wBxo n SiBwxo^^^
'^"^Un U^.
Now, we identify **^[/ - c X (^*^/7 n ^ 1 ) X C^^^t/ n Ui) by C X ( ^ ' ^ ^ n C / i ) X ('^""U nUi) 3 {t,ui,U2) ^ e^^^uiU2. Let 5 be the section of 0{X) over SIWBXQ given by s(siwu) = 1 for u £ U. Then 5 has weight siwX. Let S(ui) be the delta function. Then u — t^~^Y{t)S{ui)s, where Y(t) is the Heaviside function. Let us denote by L the homomorphism g —>• r{X\1)x). Then with the coordinates (/, 1/1,1/2), L(fi) - d/dt. Hence, {d/dt)Y{t) - 6{t) gives {d/dtfu
= {i-^ l)\S{t)S{ui)s ^ 0. Q.E.D.
Corollary 4.3.7. If M{w' o A) -^ M{w o A) is a non-zero g-homomorphism, then the composition M{w' o A) —^ M{w o A) —» r ( X ; 5Jl^^(A)) 25 non-zero.
SYMMETRIZABLE KAC-MOODY LIE ALGEBRA
429
5. S y m m e t r i z a b l e case (5.1) Hereafter, we assume that g is a symmetrizable Kac-Moody Lie algebra. In this case, V. Deodhar, 0 . Gabber and V. Kac [D-G-K] prove the following fact : (5.1.1)
For A G P-i-, the highest weight of any irreducible suhquoiient of M[X) has the form w o\ for some w G W.
L e m m a 5.1.1.
M{wo A) —> r(X;97lu;(A)) is injective.
This follows immediately from Corollary 4.3.7 and (5.1.1). (5.2) Let Mod{^, B, A) be the full subcategory of Mod{g, B) consisting of objects M such that the highest weight of any irreducible subquotient is of the form w o \. Then by (5.1.1), M(A) as well as M*(A) belongs to Mod{^,B,\). The main theorem of this paper is the following. T h e o r e m 5.2.1. Let A G P+. (i) For any 9Jt in Mod{Ttx,B), ^ " ( X ; a n ) = 0 / o r n / 0 and r ( X ; W ) belongs to Mod(g,B,X). (ii) For any M m Mod{1)x,B), S)A0r(X;5Jl) -^ M is an isomorphism. (iii) M{woX)^f{X;dn^{X)), (iv) r ( X ; £ ^ ( A ) ) - L ( ^ o A ) . (5.3) In order to prove this, let us consider the following statements for finite subsets $ and ^ ' satisfying (4.1.1) and ^ D $', (i,^, ^')
(ii,^,^')
For any M m Mod{Vx,B) with Suppmf]Q{^') = (f),5"(X;9JZ) = 0 in Mod(g,^) for any n :^ 0 and any highest weight irreducible subquotient o/r(X;9Jl) has the form w o X with w G
For any M in Mod^Dx^B) with Suppmf]Q{^') morphism M —^T{X\dy()which is an isomorphism "Dx^)^ -^ ^^ 25 an isomorphism on ^ ( $ ) . (iii,^,^') M{w o X) —r r ( X ; 9Jtu;(A)) IS an isomorphism in any w e ^ \ ^ ' . (iv,<^,^') L{w o X) —^ T{X; Cuj{X)) is an isomorphism in any w E ^\(^'.
= (j), and a on Mod{Q, <^), Mod{Q,^) for Mod(Q,^) for
If (i,^, <^')-(iv,^,<^') are true for any <^ and ^', then Theorem 5.2.1 is also true.
430
MASAKI KASHIWARA
(5.4) We shall prove ( i , $ , $ ' ) - ( i v , $ , $ ' ) by the induction of # ( $ \ $'). Since they are trivial if ^ = ^ ' (see Corollary 4.1.2 and Proposition 4.2.2), let us assume # ( $ \ $ ' ) > 1. Let us take a minimal element WQ in $ \ ^ ' and set ^ " = ^'[J{WQ}. By the hypothesis of induction, (i,^,<^")-(iv,<^, <^") are true. Let us consider the long exact sequence of global cohomology groups associated with (5.4.1)
0 - . £„„(A) - . *8,„(A) - . !B^„(A)/£„„(A) -^ 0.
Since Supp{^^,iX)/Cn,oi^)) n Bwoxo = 4>, ^"(X;S^„(A)/£^„(A)) = 0 in Mod{Q,^) by (i,$,$"), and we obtain (5.4.2)
^"(X;£„„(A)) = 0,
for
n 7^ 0,1
and an exact sequence r(X;
- 0.
Hence by tensoring J)A? we obtain an exact sequence 2)A®r(X; 25„o(A)) ^ S)A®r(X; 55^„(A)/£^„(A)) ^
S)A®H1(X;
r„„(A)) -^ 0.
Since 5),®r(X; «^„(A)/£^„(A)) ^ S„„(A)/£„„(A) is an isomorphism on ^ ( ^ ) by (ii, ^,<^''), we have 'Dx^H\X',C^,{X))
=^
on
Q(^).
Assume that H^{X\CU}Q{)^)) does not vanish in MCK/(0,$). Since 5H-'^;>Ct.o(^)) is a quotient of f (X; ^«;,(A)/£^,(A)), (i,<^,<^'') implies there exists a surjective morphism H^{X\Cu)o{y<)) ~^ ^ ( ^ ° ^) ^^"^ some u; G ^ \ ^ " . Hence S ) A § L ( U ; O A) = 0 on fi($). Since 'S\^L{w o A) is isomorphic to Cw{^) by (ii,,$") and (iv,$,$''), this is a contradiction. This shows H^{X]C^S\)) = 0. Thus we established (5.4.3)
5 " ( X ; £ ^ , ( A ) ) = 0 in
Mod{Q,^)
for
n / 0.
Now consider the exact sequence 0 — ^ — 9n^,(A) -^
£,,(A) — 0.
SYMMETRIZABLE KAC-MOODY LIE ALGEBRA Since 5iipp07 Pi Q($'') =: (j), ( i , ^ , ^ ' 0 implies 5"(X;0^) = 0 in for any n ^ 0. Then, together with (5.4.3) we obtain (5.4.4)
^"(X;Dn^,(A)) = 0 in
Mod{Q,^)
for
431 Mod{^,^)
n ^ 0.
(5.5) P r o o f of (i,<^,$'). Let 9Jl be as in ( i , $ , ^ ' ) . Then there exists a morphism (f> : 971®"^ —>• 5Jl which is an isomorphism on BWQXQ. Hence, m = Ker(p and 0^'= Coiber ^ satisfy 5uppmnQ(<^") = Suppm'nCt{^") = 0. By ( i , ^ , ^ " ) . H'^iX'^m) = 5 " ( X ; m ' ) = 0 in Mod{^,^) for any 7i / 0. Moreover (5.4.4) implies 5"(X;9;T®^^) = 0 in Mod{g,^) for n ^ 0. This shows 5"(X;9Ji) = 0 in Mo(/(g,<^) for n 7^ 0. The last statement in (i,<^, ^') is easily reduced to the case where 9Jl = ^w{^) for w; G VF \ $'. (5.6) Proof of (iii, $ , ^ ' ) . following lemma. Lemraa 5.6.1.
In order to prove this, we shall note the
For a B-equivariant holonomic 5)x-i^odule 9Jt,
^(-fch H%x;m{\)) = Y,{-T ch H"{x-m-{\)). In fact, we can easily reduce this to the case where DJt = Cyj. In this case, 9JI = 9Jt*. Q.E.D. We shall apply this lemma for 9Jt = S^; for li; E ^ \ ^ ' . Then we obtain dimf (X; 9Jt^(A))^ = dimM*(ix; o A);, = dimM(w; o A)^ for fi G Px{^)(5.6.1.) Since M{w o A) —^ T{X : 9Jliy(A)) is injective, it is an isomorphism in Mod{Q,^). (5.7) P r o o f of ( i i , ^ , $ 0 - Let dJl and M -^ f{X;m) be as in (ii,^,^0Let us take (f : DJt®*^ -> dJt which is an isomorphism on BW^XQ. Then 91 — Ker ^ and 91' = Coktr ^ satisfy 51/^^91 fl Q(^") = 5wpp91' fl Q(^'') = c;^. Let £ be the image of (p. Let N be the fiber product of M and r ( X ; £ ) over r(X;9Jt). Then we have the commutative diagram S A ^ A T —> £ ) A 0 M — ^ 'Sx®{M/N)
0 —^
JC
—^
m
—>
91'
—> 0
—.0.
432
MASAKI KASHIWARA
By ( i i , $ , ^ ' ' ) , ^A<§(M/Ar) -^ 9T' being an isomorphism on Q ( $ ) , it is enough to show that X)A(8)A^ —^ £ is an isomorphism on Q(<^). Let A^' be the fiber product of N and f(X;m®'^) over f{X]C). T h e n we obtain the commutative diagram
;0T)
2),
i 0Since 'Dx'S)T{X; 01) morphism on ^ ( ^ ) , on Q ( ^ ) . Since N' Mod{Q,^), {N')u^o\
m
* 'Dx®N' — ' ,
i
mf'
—».
—* 0
I C
—^0
-^ 01 is an isomorphism on f2(<^), S ) A 0 A ^ —^ £ is an isoonce we prove t h a t 2 ) ^ ^ ^ ' —^ ^ ^ o ^^ ^^^ isomorphism -> f{X ; Wt®;^) ^ M{wo o A)®-^ are isomorphisms in - C®-^ and {N')u;oX+a,- 0. Hence we obtain M{wooXf'
-.N'
-^f{X;m^'j
in which morphisms are isomorphisms in Mod{^,^). Since 'X)x0M{wo o A)®-^ - ^ ^A§)A^' is surjective on ^ ( ^ ) by Proposition 4.2.2 and Vx^M{iuoo X^^J _^ dJl^-^ is an isomorphism, Tlx^N' —^ 9Jl(u;o)®'^ is an isomorphism on Q(<^). This completes the proof of (ii,<^,$'), (5.8) P r o o f of (iv,^,<^')- Since £^(A) is the image of 9}t^(A) - ^ «^^(A), r ( X ; £ ^ ( A ) ) is the image of r ( X ; O T ^ ( A ) ) - ^ r ( X ; ^ ^ , ( A ) ) in Morf(0,^). Since L(w o A) is also the image of M(w o A) —^ M*(i6' o A), we obtain We have completed the proof of (i,<^, ^ ' ) - ( i v , $ , $ ' ) and thus the proof of Theorem 5.2.1. (5.9)
I don't know if the following statement is true.
(5.9.1)
For any object M in Mod{g,B,X), isomorphism.
M -^ r(A:;X)A§'M) ts an
This is proven in the finite-dimensional case ([B-B], [B-K]).
REFERENCES [EGA] [SGA] [B-B] [B-K]
A. Grothendieck and J. Dieudonne, Elements de Geometrie Algehrique, I-IV, Publ. Math. I.H.E.S. A. Grothendieck et al, Seminaire de Geometrie algebrique. A. Beilinson and J. Bernstein, Localisation de g- modules^ C. R. Acad. Sci. Paris 2 9 2 (1981), 15-18. J - L . Bryhnski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic system^ Invent. Math. 6 4 (1981), 387-410.
S Y M M E T R I Z A B L E KAC-MOODY LIE A L G E B R A
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[ D - G - K ] V. Deodhar, O. Gabber and V. Kac, Structure of some categories of infinite-dimensional Lie algebras^ Adv. in Math. 45(1982), 92-116. [K-K] V. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. in Math. 3 4 (1979), 97-108. [K] M. Kashiwara, The flag manifold of Kac-Moody Lie algebra, Amer. J. of Math. I l l (1989). [K'] M. Kashiwara, Representation theory and ^-modules on flag varieties, PTVC. of Orbites unipotentes . . . (to appear in Asterisque) and R.I.M.S. preprint 622. [K-L] D. A. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 5 3 (1979), 165-184. [Kempf] G. Kempf, The Grothendieck-Cousin complex of an induced representation, Adv. m Math. 29 (1978), 310-396. [S.K] S. Kumar, 1. Demazure character formula m arbitrary KacMoody setting. Invent. Math. 89 (1987), 395-423. 2. BernsteinGelfand-Gelfand resolution for arbitrary Kac-Moody setting, preprint. [M] 0 . Mathieu, Formule de caracteres pour les algebres de KacMoody generales, Asterisque (1988)159-160.
Masaki Kashiwara R.I.M.S. Kyoto University Kyoto, J a p a n
Euler Systems V.A. KOLYVAGIN
Dedicated to A. Grothendieck on his 60th birthday
Introduction In this paper we study Euler systems defined by the characterizing condition AXl, perhaps with the addition of other conditions (AX2 and AX3 systems, see §1). Our main purpose is to apply them to determine the structure of the class groups of certain algebraic number fields i?, and the Mordell-Weil groups and Shafarevich-Tate groups of Weil curves. In the case of the class group CI of a field R^ Theorem 7 of §2 says that, if the Galois group G of i^ is annihilated by /— 1, where / is a rational prime, and if V* is a homomorphism from G to the group of (/ — l)-th roots of unity in Z/, then (under certain conditions on R and ^p) any Euler system associated to R which is non-degenerate (in its (/,i/')-component) determines the structure of the V'-component of C/ 0 Z/, i.e., it determines the set of integers nj, n, > n,^_i, such that (CI 0 Zi)^ ~ YlT=i Z//"* as an abelian group. Theorem 7 also shows how the Euler system determines bases of (CI 0 Z/)^ consisting of prime divisor classes, the expansions of certain prime divisor classes in these bases, and also certain representations of primary numbers. For example, this holds for the cyclotomic field Ki = Q ( 0 ) (see below) with odd characters ip and the system of Gauss sums, or with even characters I/J and the system of cyclotomic units. As a corollary we find that the order of X = ( C / 0 Zi)^ is bounded from above by the predicted explicit order [X]?; and this, along with formulas for the class number, enables us in several cases (cyclotomic fields, fields which are abelian extensions of an imaginary quadratic field) to prove that [X] and [X]? are equal. Theorem 7 gives us a prototype for a theory which describes G((K''^Y^/K) (where K is an algebraic number field and (A'^*)^^ is the
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maximal abelian extension of the maximal abelian extension of/<') in terms oi K in much the same way as abelian class field theory describes C{K^^/K) in terms of K. We have proved a similar theorem for the Shafarevich-Tate groups 111(74,/C) of a Weil curve A over Q , where K is an imaginary quadratic field which is associated to A in a certain way. Here we use the Euler system of Heegner points over ring class fields of K^ and we compute the invariants n,- > nj+i in the decomposition III = ^ (Z/Z'^^a,--h Z//"»6i) in which {ai^bi} = /""», {cti^aj} = 0 for all i,j, and {ai^bj} = 0 for i ^ 2^ where { } : III x III ^^ Q / Z is the Cassels pairing. T h e complete statement and proof of this theorem will be published in another article, while the present article contains the technically somewhat simpler proof of t h e "approximate" divisibihty of [III]? by [III], where [III]? is the order of III predicted by the Birch-Swinnerton-Dyer conjecture. Among other results, we find t h a t , if the L-function of the Weil curve A has a simple zero at 5 = 1, then A ( Q ) has rank 1 and I I I ( A , Q ) is finite. For more details, see below. For further discussion of Theorem 7 we refer the reader to the text. T h e rest of the introduction will be devoted to a detailed description of t h e results relating to bounds for the rank of A ( Q ) for Weil curves and divisibihty of [(C/(8) Z/)^]? and [III]? by [ ( ( 7 / 0 Z)^] and [III], respectively. We first note t h a t the module of Euler systems which is associated with an algebraic group A and a tower of extensions of the algebraic number field A', apparently has canonical generators (cyclotomic units, elliptic units, Heegner points, etc.) which determine the arithmetic nature of t h e C^function. It would be of interest to find canonical generators and work out the theory for a complete adelic version of Euler systems. We start by considering the most classical case of cyclotomic fields. T h u s , suppose that / > 3 is a rational prime; K\ — Q(0)5 where 0 is a primitive /-th root of 1; G = G a l ( A f / Q ) with canonical isomorphism /> : G - ( Z / / ) * given by Cf = Cf^^^ ^^r all geG.Y^e we let j =
set ga = />"H«). and
j^ag„Az[G]\nZ[G]
denote the Stickelberger ideal. Let 6a € Zjf denote the (/ — l ) - t h root of 1 for which Oa = a {mod I). For odd integers i, 1 < i
Let ordi : QJ' —^ Z be the /-adic ordinal, and let yi = ordi{Bi^i), Bi^i = l^'Ui, where Ui is an /-adic unit.
so t h a t
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If J4 is a Z([G]-module, then we define Ai = {aeA\
a"" = 6>> Va € (Z//)*}.
A is the direct sum of the Ai, where 0 < i < / — 2 (we have / \
J
1-1
\
1-2
a=l
/
1=0
J
i-i a=l
Let CI be the class group of Kj, let A be the /-component of C/, and let [Ai] = P* (the order of a finite group B will be denoted by [B]). Let /^/+ = Q ( 0 + Cr^)' ^^^ '^^ ^ ' + '^^ ^'^^ ^^^^ group of i^/4,. From classical formulas for [CI] and [C/4.] (see [1], Ch. 5) it follows that ^ x , - = YlVi^ where z is odd, 1 < z < / — 2. The classical Stickelberger relation JA_ = 0 (where we define A± = {a £ A \ a^-^ = ± a } ) gives the relation l^'Ai = 0. In particular, this means that y, = 0 => Xi = 0 . If A^ is a monogenic G-module {Ai ~ Z//^»), then obviously x,- < y,-, and since Yl^i ~ Si/* it then follows that Xi = yi. A- is monogenic if A^ = 0 (Iwasawa). In the general case the basic conjecture that Xi = yi for a long time appeared hopeless. Finally, Ribet [2], using ^-adic representations associated to the jacobians of modular curves, proved that x,- = 0 => yi = 0. More precisely, under the assumption that /|i3i,j, Ribet was able to construct a nontrivial /extension of Ki with a suitable condition on the action of the Galois group. Continuing in this direction, Wiles [3] proved that ^4,- Ci^ Z/P* =^ Xj = yi. Using the philosophy of modular constructions of unramified extensions, Mazur and Wiles [4] proved the analogue of Xi = yi for towers LKin, where L is an abelian extension of Q (the "main conjecture" for abelian extensions of Q, or more precisely, the imaginary component of the main conjecture). Further generalizations, with L replaced by a totally real extension of Q, were obtained by Wiles in [5]. In the present paper Euler systems are used to give a direct proof of inequalities of the type Xi < yi. In the case of cyclotomic fields, when combined with the equality Yl^i — Z^2/«J ^^^^ gives a new (and fairly elementary) proof of the equality x,- = ?/,•, and, it seems, of the main conjecture. The -h-version of x,- = yi consists in the following. Let U denote the group of units of Kj^, and let UQ C U he the subgroup of cyclotomic units: UQ is the Z[G4.]-submodule of U (where G-j. = Gal(A'/-|./Q)) generated by the units (Cf - l)(Cr^ " 1)/(C/ - l)(Cr^ - 1), ^ ^ (Z//)*. It is well known (see [1], Ch. 5) that U/UQ is a finite group. Let B be the /-component of U/UQ, and set [Bi] = l^\ where i is even, 2 < e < / —3. The formula for [C/+] implies the equahty Yl^i — ^ Vi (^^^ summations are over 2 < 2 = 2j < / — 3). Using class field theory, from the minus-results of
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Mazur-Wiles one can prove that Xi = yi. Recently Thaine [6] gave a direct proof that l^'Ai = 0, which is the -f-analogue of the Stickelberger relation. We shall prove that Xi < yi directly, and this will also give a new proof that Xi = yi. Of course, regularities of the type x,- = yi extend far beyond the realm of cyclotomic fields. In essence they form a branch emanating from the trunk of abelian class field theory and extending into the foliage of (highly ramified) nonabelian class field theory. Rubin [7-8] used an analogue of the elliptic and cyclotomic units, and, generalizing Thaine's proof, obtained annihilator relations of the form l^'Ai = 0 for class groups of abelian extensions of a quadratic imaginary field k. Using the natural connection between class groups and the Selmer groups of an elliptic curve E over fc, End(£') 0 Q = fc, Rubin proved that the Shafarevich-Tate group UI{E^k) of E over k is finite if L{E,k,l) ^ 0, where L{E^k^s) is the canonical L-function of E over k. Here \M{E^k) is annihilated by Uk\ill{E,k)]l, where [III(£', ib)]? is the order o{m{E,k) predicted by the Birch-Swinnerton-Dyer conjecture, and Uk is the number of units in k. We note that [III(£', A:)] is the square of a natural number (if it is finite), because of the existence of the non-degenerate alternating Cassels pairing UI x UI -^ Q/Z, and we obviously have [Ull^/^m _ Q. if £• is defined over Q, then III(£', Q) is also finite provided that L(£', Q , l ) ^ 0, and 1II(£', Q) is annihilated by iijt[III(£', Q)]?. Rubin gave the examples: m{E, Q) = 0 for £• : y2 = X^ - X\ m(J?, Q) - Z/2 -h Z/2 for ^ : y2 = X3-f 1 7 X ; m ( £ ' , Q ) ~ Z / 3 - h Z / 3 for ^ : Y^ = X^ ~ 2^3^52. The Birch-Swinnerton-Dyer conjecture holds for these curves. Rubin further proved that L'{E, Q, 1) 9^ 0 => rank E'(Q) < 1 => rank £'(Q) = 1 (see the discussion below of the results of Gross and Zagier). Earlier Coates and Wiles [9] had proved that X(£',Q, 1) / 0 => £'(Q) is finite. Coates and Wiles used the orthogonality of the elliptic units and points P G ^ ( Q ) under the analogue of the Hilbert symbol. In the present paper we prove inequalities of the form Xi < yi for class groups of certain abehan extensions of Ar (see §2). Combined with the class number formula, this must imply that x,- = y,-. Apparently it will also lead to a proof of a "main conjecture" which, when applied to the elliptic curve £", enables one to prove that the dimension of the finite-dimensional Q/-space (where / is a rational prime) limS'/n(£',Q)0Q/ (where Sjn is the /"-Selmer group of E over Q) is equal to the multiplicity of zero at s = 1 of the /-adic L-function of E over Q (and also prove the corresponding equalities for abelian extensions of k). If L{E^ Q, 1) ^
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0 or L'(E', Q , l ) ^ 0, this must give us a complete proof of the BirchSwinnerton-Dyer conjecture for E: \ni{E, Q)] = \ni{E, Q)]?. Until now the algebraic group A was taken to be the multiplicative group. Now let ^ be a Weil elliptic curve over Q. (According to the Weil-Taniyama conjecture, every elliptic curve over Q is a Weil curve; this is the case if A has complex multiplication or if A has the form of a Weil curve over a quadratic extension of Q.) Let N be the conductor of Ay K he the quadratic imaginary extension of Q with discriminant D = square {mod 4Ar), and r G A{K) be a Heegner point (see §3). Gross and Zagier [10] proved that for (D, 2A^) = 1 one has: height(r) = ciL\A, K, 1), where Ci is an expHcit nonzero constant and height(r) is the archimedian height of the point r. In particular, this result implies that L'(Aj K,l) ^ 0 => TSinkA{K) > 1. We have: L{A,K,s) = L{A,Q,s)L{A,Q,XD,s), where XD is the quadratic character associated to K and L(i4, Q,Xi>) ^) = E " = i XD{n)ann'', where L(AM.s) = = ^ ^ = 1 « n n - ^ If L ' ( ^ , Q , 1) # 0, then L'{A,K, 1) = V{A,Q, !)• 'L(A, Q,XD, 1) (here the condition on Z) implies that the multiplicity of zero at s = 1 of L{A, Q, s) and L{Aj Q , X D ) ^) have different parity (see below); in particular, L{AyKyl) := 0). From the connection between the values of L(^,Q,X£), 1) as D varies and the coefficients of a certain modular form of weight 3/2 (Waldspurger) it follows (see[10]) that there exists K such that L(yi, Q,xi>, 1) / 0. Since 7-[>i(Q)tor] ^ A{Q) and r has infinite order (r has infinite order ^ height (r) ^ 0), it then follows that L'(A, Q, 1) ^ 0 => rank A(Q) > 1. If P G ^ ( Q ) and d G H\Q,A)M, then E,;(^.^)M,t; = 0 by the reciprocity law. Here { , )M,V ' A{Q{v))/M x H^{Q{V),A)M -^ 1/M is the local Tate pairing, and v runs through the places of Q. The search for "explicit" classes of homogeneous spaces d to use with the above orthogonality relation led the author in [11] to construct classes of homogeneous spaces by means of Heegner points. By interpreting the orthogonality relation expHcitly, in [11] we proved that A(Q) and in(A, Q) are finite if L{A, Q, s) has an even order zero at 5 = 1 and r has infinite order. Here in(A, Q) is annihilated by the natural number C2CD1 where CD is the largest natural number dividing the image of r in A(K)/A{K)tor, and C2 G N is a number determined from the parameters associated to the bad reduction of A and the action of Gal(Q/Q) on i4(Q)tor (Q is the algebraic closure of Q). The number C2CD is closely related to \ni{A,K)]?^/'^ (see below). In [12] the results of [11] were extended to the curves yl(/)) : DY'^ = 4X^ — g2X — gs, where A : Y^ = 4X^ — g2X — gs and L(yl, Q, s) has an odd order zero at s = 1. More precise information was also obtained about C2, for example, ordi(c2) = 0 if ^ does not have complex multiplication and the natural homomorphism Gal(Q/Q) -^ Autzt{\iinAin) :^ GL2(Z/) is surjective (as is the case for almost all /, by Serre). For several curves B = A or A(^j))
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with L ( 5 , Q , 1 ) 7^ 0 it was proved that \ni{B,Q)]?^^^UI{B,Q) = 0 (here [ m ( 5 , Q)]?^/2 ^ N) If [ u i ( 5 , Q)]?^/2 ^ 1^ this gives m ( B , Q) = 0. Thus, the triviahty of in(^(i)), Q) was proved for A : Y'^ = 4X^ - 4X + 1 and 23 values of D = —7, — 1 1 , . . . , and the Birch-Swinnerton-Dyer conjecture was verified for 5 values of D. A comparison of the approach of Thaine and Rubin to proving the annihilating relations in the case of the multiphcative group and that of the author in the case of elliptic curves suggested the possibility in [12] of combining them into a single general framework. Namely, the cyclotomic units, the elliptic units and the Heegner points are all Euler systems with congruence (see §2, 3). A crucial point in our derivation of inequalities of the form Xi < yi is the possibility (see §1) of passing from the original Euler system T to its derivative T', which is also an Euler system with congruence. Repeating this process, we obtain a sequence of Euler systems {T(^'), i > 0 } . At each step new annihilating relations arise (in H^{K\^A) if A is an elliptic curve, in the class group of K\ if A is the multiplicative group; here Kx is an abelian extension of the ground field or of a quadratic extension of the ground field). From all of these annihilating relations one eventually obtains inequahties of the form x,- < y,-. In the case of the Euler system of Gauss sums, the Euler system of cyclotomic units, and the Euler system of Heegner points, the relations that arise at the first stage of this process are, respectively, the relations of Stickelberger, Thaine and Rubin, and the relations obtained by the author for Weil curves. We now describe some consequences relating to Weil curves of the results of the present paper. Corollaries 11-13 of §3 imply: Theorem A. Let r he a point of infinite order (which for (D, 2N) = 1 15 equivalent to the condition that L'{A,K, 1) / 0, hy Gross-Zagier). Then rank^(ii:) = 1, M1{A,K) is finite, C^CD^{A,K) = 0, and \UI{A,K)] divides C4CJ). Here C3 and 04 are defined in the same way as c^ (see ^3). Let 7 : XN —>> ^ be a Weil parametrization such that 7 o WN = 67 + 7(0), where e = i t l , w^ : X^ -^ ^jv is the principal involution of the modular curve, and 7(0) G ^(Q)tor is the image of the cusp of Xjsf corresponding to 2: = 0 under the natural covering i/UQU{00} -^ XN (here H is the complex upper half-plane). Then A{s) = {27ryN''^T{s)L{A,Q.s) satisfies the functional equation A(2 — s) = (—€)A{s); in particular, (~e) = (—1)^, where g is the multiplicity of zero of L{A,Q^s) at s = I. Let {D^2N) = 1. From the condition D =square (mod 4N) it follows that A(c)(2 -s) = eA(p)(5), where A(o) = (27r)-'(Ar£)2)'/2r(s)L(^(D),Q,s). In particular, e = (—1)^(^), where y(£)) is the multiplicity of zero of L(^(£>),Q,5) = L{A,Q,XD,S) at s = 1; and L'{A,K,l) =
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L{A, Q, l ) i ' ( ^ ( D ) , Q, 1) for £ = - 1 , L'{A, K, 1) = L'iA, Q, l)LiA^D)M, for 6 = 1. We have the following
1)
Corollary B . Lei r be a point of infinite order. Then i n ( ^ , Q) and III(A(£)),Q) are finite, are annihilated by 2C^CDJ and have orders dividing C42°'Cj), where a is the 2-rank ofUI{A^Q) andUI{A(^D^,Q), respectively; rankA(£>)(Q) = 1 and rankA(Q) = 0 if e = —1; and rankA(£>)(Q) = 0 and rankA(Q) = 1 if e = I. Proof. T satisfies the condition r^ = er + ^7(0), where a is the automorphism of complex conjugation and h is the class number of K (see [11], §3). We have the A'-isomorphism t : A —^ ^(D) given by ( X , y ) ^ (X,y/D^/2) Then A^D){Q) = t'i^ e A{K) \ x^ = ~ x } , so that the claim about the ranks obviously follows from the equality A{K) = 1. Furthermore, I1I(A, Q) and III(A(£)),Q) map naturally to UI{A,K) with kernels in H^{G{K/Q),A{K)) and H\G{K/Q),Ai^D){K)). respectively Since these cohomology groups are finite and 2-periodic, it follows that III(A,Q) and III(74(£>), Q) are finite, and the annihilation and divisibility relations also hold. Corollary C. Let L'(A,Q,1) ^ UI{A, Q) is finite.
0.
Then rankA(Q) = 1 and
Proof.
By Waldspurger (see above), there exists D such that ^ 0; then L ' ( A A , 1 ) = i:'(^, Q, 1)L(^, Q , X D , 1) i^ •^ 0 = ^ r is a point of infinite order. L{A,Q,XDA)
Let g be a rational number dividing AT, let m^ be the number of connected components in the fibre of the Neron model of A which are rational over Z/g, and let rrifq — WaX^'^q- Let c G Q* be such that 7*(a;) — cujj ^ where a; is the Neron differential form on A, LJJ = 27r\/—1 X ^ ^ i fln* •exp(27rv^z)(fz, z e H. If (D, 27V) = 1 and V{A,K,l) / 0, then, because A{I{) has rank 1, the Gross-Zagier formula for height(r) (see [10]) is equivalent to the formula
, _mlc'\UliA,K)]? ^^-
[A{K),or?
... •
^'
Let k = End(^) (g) Q, Tai = lim^/n, and let />/oo : Gal(^/ik) —^ AutziiTai) be the natural homomorphism. The map pjoo is surjective for almost all / (see §3). Let D{0) denote the discriminant of O = End(^).
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Corollary D . Suppose that r is a point of infinite order; / / 2; / ^iD(0) in the case when A has complex multiplication; and p/oo is surjective. Then oTdi{\ni{A,K)]) < ord/(C|^). If {D,2N) = 1, then oTdii\ni(A, K)]) < oTdi{\m{A, K)]?) + oTdiiicrriNy). Proof. In this case ord/(c4) = ord/([yi(X)tor]) = 0 (see §3). We apply Theorem A and the formula (1). Let 1^2. Then UI{AjK)ioo ^ in(A,Q)/oo X III(A(x)), Q)/oo (the first factor can be defined as the ainvariant subgroup, and the second factor can be defined as the subgroup on which a acts as the inverse). Let {D,2N) = 1, L'{A,K, 1) ^ 0. Then m{A,K) is finite, and ord/[m(^. A')] = ord/[in(^,Q] + ord/[III(A(^),Q)]. By formulas (1) and (2), we also have ord/[III(A, K)]l = ord/[m(yl, Q)]? + ord/[m(^(Z)),Q)]?. Thus, Corollary D implies Corollary E. Suppose that T has infinite order, and I is as in Corollary D, Then ord/[m(A, Q)]-f ord/[m(yl(i)), Q)] < ordf(C|>). Now let {D,2N) = 1, so that V{A,K,l) ^ 0. //ord/|in(A(p),Q)]? = 0, then oidi\UI{A,Q)] < ord/[m(^,Q)]? + ord/((cm^)2). / / o r d / [ m ( ^ , Q)]? = 0, then OTdi\ni{A^D)M)] < ord/[III(A(z)),Q)]? + ord/((cmAr)2). We have the following plausible conjecture. Conjecture F . Suppose that L{A, Q, 1) ^ 0 or V{A, Q, 1) ^ 0. Then there exists D such that, respectively, L'{A,Q,XD,^) ^ 0 or L{A,Q,XDA) 7^0, and such /Aa< ord/[m(A(£>),Q)]? = 0. In the case L'(A, Q, 1) ^ 0, a natural way to prove Conjecture F is to prove (if this has not yet been done) the nontriviality (mod I) of the Waldspurger modular form associated to A. In the case L(A,Q,1) ^ 0, the author does not know whether it has been proved that there exists D such that L'{AJQ,XDA) T^ 0 (o^ course, in reality most D have this property). Corollary G. Suppose that L{A, Q, 1) ^ 0 or L\A, Q, 1) ^ 0, / 25 as in Corollary D, and A and I satisfy Conjecture F. Then ord/[111(^1, Q)] < ord/[m(^,Q)]? + ord/((cm^)2). \{ B = A or ^(D), then, as noted above, the system of Heegner points and its derivatives determine the structure of UI(5, A'). The simplest examples will be discussed below. If we know the estimate ord/[111(5, Q)] < ord/[III(5,Q)]?, then we can determine the structure of 111(5, Q)ioo (and, in particular, prove the /-component of the Birch-Swinnerton-Dyer conjecture: ord/[III(jB,Q)] = ord/[II[(5,Q)]?), by exhibiting a subgroup X C
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UI{B,Q) of order Z^^, where 2x = ord/[m(5,Q)]?, after which we have 111(5, Q) = X. If any of the exphcit elements of 111(5, Q) has period /^, then 111(5, Q)/oo c^ Z//^ -|- Z / P , because of the existence of the Cassels pairing. Namely, if M = /", then we have a large number of elements z = T^x A ( ^ ) ^ ^^(^>^Af) (see §3) whose localizations z{v) € H^{K(V)^AM)I where i; is a place of A', have been computed (see Theorem 9 of §3). Under certain conditions the image (z) of z in H^{K,A)M is in UI{K,A)M^ Here z^ = ^(-1)'*'^. If / 7^ 2, then (z) 6 UI{A,Q)M if e("-l)'- = 1, and t{{z)) G m ( A p ) , Q ) M if e{-iy = - 1 . We shall give examples. Suppose that (/, [A(/<')tor]) = 1 and M = /^, where y = ord/(C£)). From Theorem 9 (see also the subsequent remarks in §3) it follows that T^ (M) G UI{KJA)M if Ai = (p), where p is a rational prime not dividing DN such that ly^^^ divides (p - (j^)) and ly^y^ divides (a^ - 1 - (;^)). Here /^° is the largest of the powers of / annihilating the /-components of the finite unramified cohomology groups H^{K{v)^A)unr of ^ for v\N (t/o|ord,(m^)if(D,iV) = l). We give one other way to construct in(A,X)/oo. For simplicity, suppose that {D,N) = 1, {l,[A{K)tor]) = 1, and (/,2mAr) = 1. Let Ku be the maximal unramified abelian /-extension of AT, and let Gu = GB1(KII/K)] then Gu c:± the /-component of the class group oiK. Let K\ be the maximal unramified abelian extension of A, let ri G ^ ( ^ i ) be the Heegner point associated with the ideal i of K (see §3), and let TU = Nxi/Kui'^i)' Then (see [11], §3) T^j = erf/'V[Ai/Ai/]T(0), where ^(0 G Gu is the image of i under the Artin map. Let cr' = cr6{i). We let Z denote the Gi/-submodule of Q[Gi/] generated by the identity element zx of the group Gu and by ^2 = ] ^ i^geGu 9) • We define a Gi/-homomorphism fiiZ-^ A{Ku) by setting f\{z2) = w/ and fi{z\) = {a' + €)TU, where ui G ^ ( A ) is such that r ' = NKii/K(fi{zi)) = Mti/ (the existence of w/ follows from the definition of CD)' The exact sequence 0 —> Z[Gi/] ^- Z —>• ]^Z/Z —>- 0 implies that we have an isomorphism /2 : H^{Gu^Z) ^ Hom(Gi/, ;j^Z/Z). We have the homomorphism /a = /{ o /2"^ : Hom(Gi/, ]^Z/Z) -^ 5^(Gi/, yl(Ai/)), whose image lies in 1II(A, A)_cAf > i-^? the subgroup of 111(^1, A)Af on which ) : Hom(Gi/, j g Z / Z ) ~> UI(5, Q). For e = —1 and / as in Corollary E, with {I^CTYIN) = 1, perhaps it is always possible to choose D which satisfies Conjecture F and the relation ord,[/m f^o)] = ord/[m(yl, Q)]? = 2a. Then [Gu/G[]] = P^ and Ul(AyQ) ^ Gu/G[i. If this is the case, it would be interesting to find an
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V. A. KOLYVAGIN
upper bound for the smallest such D. Let y2 -h aiXY + Q3Y = X^ + Q2X'^ + ^ 4 ^ 4- ae be the minimal equation of A, and let A be the discriminant of this equation. Suppose that {D,N) - 1. We set m(i>) - 2'-i(^)+2r2(i))^ ^^ere ri(£)) is the number of prime divisors q oi D for which f y 1 =: —1 and r2{D) is the number of prime divisors q of D for which (jj = 1, 2|a^. If and V{A^K,l) ^ 0, then, using the Gross-Zagier formula tion L{AjK^s) = L(^,Q,s)L(7l(j9),Q,s) (and taking into T3iikA(K) = 1, see above and also §3 of [12]), we derive the
Ch = \[A{n)/AiRf]m%m^D)c'miA,
(Z),2iV) = 1 and the relaaccount that formula
Q)]?[in( V ) , Q)]?.
(2)
Here R is the field of real numbers and A(R)^ is the connected component ofyl(R). Corollary H. Suppose thai A does not have complex multiplication, pioo is surjective V/, L'{A,Q,l) 9^ 0, [m(yl,Q)]? = 1, {D,2N) = 1, and L{A, Q, XDA)^ 0. Then [III(A(£>), Q)] divides Cl = -[^(R)M(R)«]m^m(^)c2[m(^(^),Q)]?,
if CD is odd. In particular, [L[I(i4(/)), Q)] divides [III(yl(2)), Q)]? if ^[yl(R)M(R)°]m^m(i))c2 = 1 and [m(A(z)), Q)]? is odd. Proof. In Theorem Bi of [12] it is proven that Cz?in(A(p), Q) = 0. Hence, the 2-component of III(A(2)), Q) is trivial. We then apply Corollary E. Suppose that A is the elliptic curve of conductor 37 with equation y2 + y = x ^ _ X, A = 37, e = 1, c = 1, m^v = 1, [A{Il)/A{Kf] = 2, pjoo is surjective V/ (see [12]), L'{A, Q, 1) / 0, \UI{A, Q)]? = 1 by [13]. In [14] there is a table of C^ for {D,N) = I, -D < 500 (recall that D is a fundamental discriminant, and D = square {mod 4A^)). We have Corollary I. rank^(Q) = 1, 111(74, Q) 25 trivial, and A satisfies the Birch-Swinnerton-Dyer conjecture. Proof. From [14] we have L ( A , Q , x - 7 , l ) 7^ 0 and Clj = 1. Then rankA(Q) = 1 by Corollary B (this is a classical fact: A(Q) = Z P ,
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where Y(P) = X{P) = 0). By Corollary D we have: OTdi\ni{A, K)] = ord,[m(A,Q)] -f ord/[m(A(_7),Q)] < 0 for / / 2. Hence, ord,[m(A, Q)] = ord/|I[I(A(_7), Q)] = 0. In addition, we know from [15] that the 2component of III(yl, Q) is trivial. Corollary H also holds for A when {D,N) = 1, 8/D, since (2) holds in this case (see formula (6) of the introduction to [12]). In the table [14] one has Cj) = 1 for 23 values of £>, from which we find (see [12]) that in(yl(/)),Q) is trivial for these values D — —7, — 1 1 , . . . , —471. Furthermore, using the table and Corollary H, we have: [III(i4(£)), Q)]|3^ for —D = 123,132,211,212,223,419 (Cl = 9); \UI{A^D),Q)]\b'^ for -D = 307,451 {Cl = 25); [m(A(^),Q)]|72 for -D = 379 {Cl = 49). Since C^ = m(£>)[III(^(/>), Q)]?, m(£)) is a power of 2, and [III(yl(£)), Q)] belongs to N , it follows that in these cases m(£)) must equal 1 and C^ = [III(A(£>), Q)]?. The groups III(yl(x)),Q) must be isomorphic, respectively, to Z/3 + Z / 3 , Z/5 -f Z/5 and Z/7 -h Z/7, as can be proved using the procedure described above. For example, for D = —132 we have T L J 3 ) G I I I ( A ( _ I 3 2 ) , Q ) ; for D = —223 we have TLJS) G III(A(_223)5 Q ) - If these elements are nontrivial we have UI{A^D), Q ) - Z/3 -f Z/3 for D = -132, - 2 2 3 . Another example: for D - - 2 1 1 , - 2 1 2 , - 4 1 3 the class number of K = QiVO) is equal to 3, 6, 9, respectively, and h = 15 for Z) = —451, so that in these situations the subgroups Im f^jy-^ C III(A(£>), Q) may be nontrivial. Theorem 9 of §3 gives a large number of relations in H^{Kr, A), where Kr is the ring class field of K of conductor r. These relations can be used to prove analogues of Theorems 6 and 7, generahzing the results of §3; in particular, they give < relations in the sense of the "main conjecture," applied to the Mazur G(A'/A')-modules in towers of cyclotomic fields and Heegner points. The relation with the /-adic analytic L-function can be obtained, for example, through the /-adic analogues of the Gross-Zagier formulas obtained by Perrin-Riou in [16] for Ki and their generalizations to arbitrary ring class fields of K. Combining the < relations with the Birch-Swinnerton-Dyer conjecture (which is in some sense an analogue of the class number formulas), one obtains a proof of the main conjecture. We now list some general notation used in the paper. N = {1, 2 , . . . } is the set of natural numbers and Z^_ is the set of nonnegative integers. If A is an abelian group and M G Z^., then AM and A/M denote the kernel and cokernel, respectively, of multiplication by M. If 72 is a ring, then R denotes its algebraic closure. If L/R is a Galois extension, then G{L/R) denotes the Galois group of L over R. We shall write H^{G{R/R),A) more briefly as H^{R,A), where ^ is a G{R/R)-Tnodu\e. "For almost all" means "for all except possibly finitely many." If O is a commutative ring with unit, then O* denotes its group of invertible elements. The field Q
446
V. A. KOLYVAGIN
is assumed to be imbedded in the field of complex numbers C. a denotes the automorphism of complex conjugation; we shall occasionally write a instead of a^ = (ra. If /? is a finite extension of Q, then SP(R) denotes the set of prime divisors of R; if a is a divisor of i?, then SP{R^ a) denotes the subset of SP{R) consisting of divisors prime to a. U v £ SP{R)^ then R{v) denotes the t;-completion of R, If r G H^{R,A), then T{V) G H^{R{V),A) denotes the f-localization of r. If i/; is an integral divisor of R^ then we define H^(R{w)jA) = ^v£SP(R) v\w H^{R{'^),A). When we want to emphasize the dependence of r G H^{R^AM) on M we shall write TM] this will not cause confusion, since M will never be used to denote divisors. If M1IM2 and T(M2) is defined, then r(Mi) denotes the image of r(M2) under the homomorphism H^{R,AM2) -^ H^{R,AMI) which is induced by the homomorphism AM2 '^ ^Mi of multipUcation by M2/M1. The author wishes to express his deep gratitude to I. R. Shafarevich and to Yu. I. Manin, under whose guidance almost 15 years ago he began to work on the questions which form the subject of this article.
1. Euler Systems and Their Derivatives In our definition of an Euler system we shall not strive for maximum generality, but rather shall take into account the scope of the present article and be satisfied that the cases we consider give a fairly complete picture of this type of system. Nor will we adhere to a complete uniformity between the multiplicative group and elliptic curves: essential as this uniformity is, it sometimes obscures the familiar constructions in the case of the multiplicative group. We let Q denote the ground field, which will be either the field of rational numbers Q or else an imaginary quadratic field of class number 1. Let A denote an algebraic group over Q: A is the multiphcative group if Q ^ Q, and A is either the multiphcative group or an elliptic curve if Q = Q, We let K denote the field Q if yl is the multiphcative group, and we let K be an imaginary quadratic extension of Q = Q if A is an elliptic curve. In the latter case we assume that K is not Q ( \ / ^ ) or Q ( v ^ ) . If 5 is a prime divisor of Q at which A has good reduction, then P^(X) G Z[X] will denote the characteristic polynomial of the Frobenius automorphism Q{S)unr/Q{6) on the Tate module Tai = lim^/n, where / is a rational prime, (/, 5) = 1. We have: PsiX) = X — NQ/Q{S) if A is the multiplicative group; and PsiX) = X^ — apX -h p if ^ is an elliptic curve with canonical L-function L{Ay Q,s) = ^ ^ 1 fln^"* ((^n ^ Z).
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In the case when [Q/Q] = 2, we fix an elhptic curve E with e n d o morphism ring Og, where OQ is the ring of integers of Q. Let K : Q\ —> Q* be the homomorphism associated with E (see [17], Ch. 10, §4), let ^A ' Q*A/Q* —^ Q* given by /Cyi(a) = K{a)a^^ be the corresponding adehc quasi character, let /CQO be the archimedian component of /c^, and let / be the conductor of /CQO ( / is an integral divisor of Q). In the case Q = Q we set / = 1. If Q = Q and A is the multipHcative group, then K : Q\ —^ Q* and KA ' Q*A/Q* ~^ QA ^^^ ^^^ analogous homomorphisms associated with A: /c(a) = n, where n is the positive generator of the divisor a. We let / , /o and SP denote, respectively, the set of divisors, the set of integral divisors, and the set of prime divisors of Q, IfrjE /, lo or 5 P , then we let /(y/), ^0(^)5 SP{TJ) denote the subset of J, IQ or 5 P , respectively, consisting of divisors which are prime to T). If A G /o, then we define the abelian extension Kx of K as follows. If A is an elliptic curve, then Kx is the ring class field of K of conductor A; if A is the multipHcative group and A' = Q = Q, then Kx = K{Ax)i where Ax is the group of «(A)-th roots of unity; if, on the other hand, [Q/Q] = 2, then Kx = K{Ex), where Ex = E^^ is the group of points of E (over K) of period x G OK^ i^) = A. We let d denote the dimension of Taj over Z/: we have d = 1 if ^ is the multiplicative group, and d = 2 if ^ is an elliptic curve. In the case when A is an elliptic curve, we let N be the conductor of A] if A is the multipHcative group, we set AT = 1. In what follows we shaU suppose that / is a fixed rational prime. If A G SP(N), then we set xx = [Kx/Ki] and Mx = ord/((xA,XA+ -\-{-lY-^Px{l))). For k G Z+, let E be a subset of the set {[A,7r]} of ordered (fc + l)-tuples of divisors
A G flQ{N) XefloiN),
for A: = 0
(here TT is an "empty index," i.e., [A, TT] = [A]);
7r=[Xu...,Xk]eSP{N)^
fovk>0.
If ni,n2 G N, rii < n2, and Mj = P^, then the homomorphism AM2 ~^ ^Mi given by multiplication by M2/M1 induces a homomorphism H^{KX,AM2)
-^ H^{KX,AMI)'
T = {rx = {rxiM)}
A set of elements
e\imH\Kx,AM)^
M = T , A G E}
if A: = 0, or a set of elements T={Tx,n
= TxA^x.)
e
H\KX^AM,^),
[A,7r] G E}
448
V. A. KOLYVAGIN
for A: > 0, will be called a k-ih order Euler system with index set E if the following condition holds for all 6 G SP{2NfX) and [A,7r] G S such that the prime divisor 6' of 6 in ii' is unramified in Kx and [6A, TT] G S : if ys is defined to be FrJ,^ {xs -f {-iy-^Ps{Frs')) G Z[G(/i'A/AO], then
Here coTj(g^/Kx is the corestriction homomorphism and Fr^' is the image of 6' in G{Kx/K) under the Artin homomorphism (reciprocity map) 6. If A is the multiphcative group, then 6 = S\ since K = Q. Suppose that A is an elliptic curve. We show that ys does not depend on the choice of S'. US is ramified or remains prime in K, then 6' is defined uniquely. Suppose that S sphts in K. Then x^ = K{6) — 1 (see below), ys = FrJ,^ {K{6) - 1 - Frj, -f a^^s)Frs' - «(^)) = a^(6) - Frs. - FrJ,^ = ^K(6) ~ F'^6' — Fr(^s'Y^ since 0{b) = 1 and 6 = 8'{8'Y (cr is the complex conjugation automorphism). Suppose we are given a k-ih order Euler system (T, S). We shall use it to construct a (fc -f l)-th order Euler system (T',E'). If ^ is the multiphcative group, then the representation on Ax and Ex gives an imbedding px : G{Kx/K) "—^ {OK/^YBy definition, PX{0{TJ)) = K{T}) (mod A) if T; G /o and rj is unramified in Kx- If K = Q and A G /o, then px{G{Kx/K)) = (Z/A)*. If [A7Q] = 2, then for /|A G /o we have: Kx is the ray class field of K of conductor A; O]^ is imbedded in [OK/^Y] the composition of 0 : iOK/>^Y/^K ^ {OK/>^Y\ and (OK/^Y is the direct product of (9^^^ and px{G{Kx/K)). On the other hand, if (A,/) = 1, then px{G{Kx/K)) = {OK/>
[L/Kx] = Mxo• Let M l be any power of / iffc= 0 and Ml = Mx^ if ifc > 0. Let MXQ and
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H^{L,AMI) as follows: r l = coTK^^^/LiTXoKwiMl)) - (yAo/^)n,T(Ml). From AXl it follows that cor/;,//<'^(rl) = 0. Let (p : GL -^ AMI be a cocycle representing r l . We fix an element ^fo G GKX whose restriction to L is t\L' By the definition of the restriction homomorphism (see [11], §3), <^OILIK^{T\) can be given by a cocycle <^i : GKX —^ ^Mi such that y?i(^o) = ^{9^) and ipi{h) = Z^j=o 9o^{9o*^9o) ^^ ^ ^i^- Since (pi is a coboundary, it follows that there exists e E AMI such that ^i(^o) = (^^o — 1)^ and (pi{h) = (h — l)e. These conditions determine e up to an element of A{KX)MI' Let e' = {M\/M)e and v?' = (Ml/M)(p. We define a map ^2 * G^;^ —> ^4^ as follows: (p2{id) = 0, ^2(5^^) = ( 4 " ^ 4- • • • + l)e' for 1 < i < M - 1,
Mh) = {M--l)^^{h)-^iM^2)go
—^
H\KX,AM).
Suppose that AQ G SP{2fNX), a = [A,7r] G D, and Ag is a prime divisor of AQ in K. If ^ = 0, then we say that the pair (a, AQ) is admissible if AQA G S and AQ splits in Kx- If Ar > 0, then we say that (a, Ao) is admissible if [AQA, TT] G S , AQ spHts in Kx^ MXQ divides MAI , and We define E' to be the set of sequences [A, AQ, TT], where ([A, TT], AQ) runs through all admissible pairs. If fc = 0, then we set M l equal to any power of / such that M A O I M I and ( M A , / M A J A ( A ' A ) M I C (^AO " 1 ) ^ ( ^ ) M . , . We define Tx^Xo^iriMxo) G H^^KX^AMX^) to be the class of the cocycle <^2 constructed above. We let V denote the system {r^, f3 G E'}. We have: Theorem 1. T' is an Euler system. Proof. Let 6 G SP{2fNX) and [A,Ao,7r] G S' be such that the prime divisor ($' of 6 in K is unramified in Kx and [6A, Ao,7r] G D'. We have the following diagram of fields:
A'AoA
450
V. A. KOLYVAGIN
Here V and L are extensions of degree M of Kf,\ and K\^ respectively. From the structure oiG{K\/K) described above it follows that the parallel lines in this diagram are equal, i.e., the restriction of the Galois group of an extension denoted by one of the lines to the extension denoted by a lower parallel line is an isomorphism. Hence from AXl and the definition of r l and r l ' it follows that cor£,/y2^(rl') = ydT\. We now use the commutativity of the maps cor and cor', which follows from the definition of cor', and the fact that cor' is a G^-module map! If iZ is a local or global field, then U{R) will denote the group of units of iZ if ^ is the multiphcative group and will denote A{R) if A is an elliptic curve. Let v G SP{Kx)y R = Kx{v). We have the natural exact sequence 0 —^ U{R)/r
— . H\R,Aim)
— . H\R,U)im
— . 0.
In the case when A is the multiplicative group, the isomorphism R*/R* -^ i/^(iZ, i4/m) enables us to identify this sequence with the sequence 1 - ^ U{R)/U{Rf^
—^ R*/R*^"^
^
Z/r —^ 0.
We let H^{R^A\my denote the subgroup o{ H^{R,Aim) which is the preimage of (i/^(/Z, ?7)unr)/"»- If (i^5 N) — \ (in particular, for arbitrary v when A is the multiplicative group), we have H^(RjU)unr = 0 and i/^(i?,yl/m)' = UiR)/!"^. IfivJN) = 1, then H^{R,Aimy = U{R)/l'^ = H^{R,Aim)^nr. If A G / / o , S e SP{fNX), and (j^) # 0, then ( ^ ) / 0, and Ksx/Kx is unramified outside 6 and is totally and tamely ramified at the prime divisors of 6 in Kx (see above, §1 of [11], and the proof in §2 below of the surjectivity of NK.JK, ' Chx -^ C / A ) . Let w E SP{Ksx), v E SP{Kx), 6' E SP{K), and w\v\6'\S. The homomorphism UiKxiv))/!"^ -> U{Ksx{w))H'^ is an isomorphism, which we can use to identify H^{K^x{}^)'>Aimy with H\Kx{v),Aimy. We shall call an Euler system (T, E) an AX2-system if the following conditions hold. Let |7r| denote 1 if A: = 0 and denote Ai • • • Ajk iffc> 0, where TT = [Ai,..., Afc]. Then rA,^(Ml,i;) E H\Kx{v),AMiy for all V E SP{Kx^ 1^1) (here M l is any power of / iffc= 0 and is Mx^ if Ar > 0). We shall call an Euler system (T*, E) an AX3-system relative to A C SP if (T, E) is an AX2-system and there exists Na E /o such that, if a = [A,7r] E E, 5 E A n 5P(2/iViV«A|7r|/), ( ^ ) ^ 0, [a,7r] E E, ti; E SP{K8x), V E SP{Kx), 6 E SP{K), and w\v\6'\6, then m,T(Ml,w;) E H^{Ksx{'^)iAMiy and we have the equality AX3:
nxA^) = Fr;\rxA^))'
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Suppose that (T, E) is an Euler system, and D C D. Then clearly T = {rpj ^ G U} is an Euler system, which we call the restriction of T to E. If [A,Ao,7r] G E', vl G SP{L), v G SP{Kx), and t;l|t; (L is the subextension of Kx^x/Kx having degree Mx^ over Kx), then M2\MXQ will denote the number such that t^^ is a generator of the subgroup o{G{L/Kx) leaving vl fixed; M3 = M A O / M 2 ; and L2 is the fixed subfield of ^^^ jj^ ^^ so that L2 is the maximal subfield of L in which v splits. Let £ denote L{vl)j and let /CA denote Kx{v)' C/ICx is a cyclic extension of degree M3, and t^^ is a generator of G{C/ICx)' The map /f^(L(t;),^Mi) -^ {fco,..-,tM2-i}, ti G J ? ^ ( £ , A M I ) , such that 6,- = 6(rt;l) for 6 G i/^(L(t;),i4Mi), identifies i/^(L(f),^Mi) with the direct sum of M2 copies of /f^(£, A M I ) , where t
(6o, .. .,fcM2-l) = {i
The local map cor(t;) : H^{L(V)^AMI) diagram
^M2-l,^05- •• ,tM2-2)-
-^ H^(ICXJAMI)
for which the
cor(t;)
cor H'{KX^AMI)
- ^
H^KX^AMI)
is conunutative, is given by the rule cor(t;) : 6 H-> ^^oiciKxi^i) — ^oTc/KAEi f>i)' We suppose that {MI/MS)A{ICX)MI C (/^^ _ I)A{C)M3^ Then on the kernel of COTC/JCX ' H^{C,AMI) —^ H^ilCx.AMi) we have the map corj^.^^ : ker(cor£/;cx) "^ ^ ^ ( ^ A J ^ M S ) defined above. Let fj,Bi,B2 ' ff^iPy^Bi) -^ H^iR^As^) denote the homomorphism which is induced by multiphcation by B1/B2 from AB^ to AB2 for J = 1, ^ 2 ! ^ ! , and the homomorphism which is induced by the imbedding ABI *—>• AB2 for j = 2, J5I|JB2- We define a map cor'(t;) : ker(cor(t;)) -^ H^{ICxiAMx ) on the kernel of cor(i;) as follows: if cor(t') takes 6 to 0, then we set cor'(i;) : b ^ COTC/K, ( / I . M I , M . , ( ^ I -h 262 + • • • + ( M 2 - 1 ) 6 M 2 - I ) ) + /2,M3,Mxo {^^^^C/K: (^0 "^
^ f>M2-i)]' One verifies that the diagram
ker(cor)
ker(cor(i;))
H\K,,AMJ
HHICX,AMJ
452
V. A. KOLYVAGIN
is commutative. _ Let E' C E'. We say that £ ' is a j AX2-admissible subset of E' if S' is an arbitrary subset for k = 0 and if E' satisfies the following condition for k > 0: if [A, Ao,7r] G E', t; G SP{Kx) and i;|/, then {MXJM3)A{KX{V))M^, C {t^^ — 1)A{L{VI))M3' In particular, there are no restrictions on E' if A is the multiplicative group, since in that case A(L(vl))i = 0. We say that E' is an AX3-admissible subset of E' if it is AX2-admissible, and if it satisfies the following additional condition in the case k > 0: if [A, AQ, TT] G E', 6 G 5P(2/iViV;,AAo|7r|/) H A, (j^) / 0, [5A, Ao,7r] G E', t; G SP{Kx) and v\6, then {MXJM3)A{KX{V))MX, C {t^'^ - 1)A{L{VI))M3. We have the following T h e o r e m 2. 1, Suppose that (T, E) is an AX2-system, and E' is an AX2'admissible subset ofTi'. Then the restriction ofT' to E' is an AX2system. 2. Suppose that (T, E) is an AXS-system relative to A, and E' is an AX3-admissible subset ofT,'. Then the restriction of T' to E' is an AXS-system relative to A. Proof. 1. Let [A,Ao,7r] G E', t; G SP{Kx), {V,\T^\\Q) = 1. Then T1(V) G H^{L(v),AMiy, and L/Kx is unramified at v. Hence it suffices to prove that cor'(t;) takes ker(cor(f))n/f^(L(i;), A M I ) ' to H^{ICxyAMx Y if C/ICx is unramified. From our earlier expression for cor'(t;)(6) it follows that it is enough to prove that cor^/^ takes i/^(>C, Ajvf i)' to ^ H ^ A , ^ M 3 ) ' - We first show that cor'^^^^(ker(cor(i;)) n U{C)/Ml) C C U{ICx)/M3. Now the following properties of cor' follow directly from the definition. Let ti = <^^. If a = {ti — 1)6, then corj^y^ (a) = = cor£/A:^(/i,Mi,M3(6)). If a = (Ml/M3)res(c), where c G H^{ICX,AMI), then cor^/^ (a) = M1^2~^/i^Afi,M3(c)- Furthermore, res(cor^y^ (a)) = ((M3 - 1) + (M3 ~ 2)^1 + • • • +
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Suppose that (T, S) is an AX3-system, a = [A,7r] 6 S, AQ G e SP{2fNNaX\Tr\l), ( ^ ) :^ 0, [AoA,7r] G S, and for i; G SP{Kx) with i;|Ao we have the condition: M A O | M 1 and (MI/M)A{KX(V))MI = 0, where M = Mx^ > 1. In particular, [A,Ao,7r] G E'. The existence of the local map cor'(f), along with the condition {M1/M)A{ICX)MI = 0, enables us to express Tx,Xo,->r{M,v) in terms of Tx^r(M,v). Using AX2 and the definition of r l , we see that the equahty coTc/fCxi'^'^i'^^)) — ^
means that rl(t;l) = {{xxjM)Fr^^
- {VXo/M)) • rxA^hv)
G H\lCx,
^A/i)unr,M- Let US define M ' to be M if / / 2 and to be M/2 if / = 2. The restriction of Tx^Xo,r{M'^ v) to Gc is equal to ((M — 1) + (M — 2)< + • • • . . . + < ^ - 2 ) r l ( i ; l j = M ^ r l ( t ; l ) = 0. Hence, rA,Ao,7r(M', t;) G i/HG(>C//CA),A(£)MO = Eom{GiC/ICx),A{ICx)M') (we have A ( £ ) M = A{)Cx)Mi since (XQ.NI) = 1). Thus, to compute TA^AC^C^'J^') it remains to compute the cocycle 2^^^^'^(p2{v){go{v)) G A{)CX)MI where ^o(^) G G;CA is a section ott = /AQ ^^d <^2(^) is a cocycle whose class is cor'(i;)(rl(i;l)). Let a G U{C)/M\ correspond to T\{V\). The cocycle (p{v) corresponding to a is determined by its value on the Frobenius automorphism F r £ , which is ^ ^ ~ ^ g . ^^oC^) can be chosen in the inertia subgroup of G r , so that {ip{v)){gQ{v)^) = {goi'^) — l)e = 0 Ve G AMI- By definition, then, the value of Tx^Xo^Tr{M\v) at i is equal to ( M l / M ' ) e i , where ei G A M I is such that M(^^-^^a) = (Fr£ — l)ei. We have natural isomorphisms A{ICx)ioo/Ml ^ U{Kx)lM\ ^ A{F)ioo/Ml, where F is the residue field of/CA. Since (Ml/M)yi(/CA)Mi = 0, it follows that (M > 1): A(/CA)/OC = A ( ' ^ A ) M I / M . Thus, we have an isomorphism A{ICX)MI —^ U{ICx)/Ml. Let /?(M1) : U{Kx)lM\ —• vi(/CA)Afi be the inverse isomorphism. Then (/?(Ml))(a) G A(/CA)M and ^ ^ a = ^ ^ ^ ( / ? ( M l ) ) ( a ) , which implies that we may set ei = M^^^^^jj^. Hence, rA,Ao,.(M',i;) : < ^
2^^^^('\/3(Ml)){b),
where 6 = {{xxjM)Fr^^ - (yxo/M)) rA,^(Ml,t;). If A is the multiplicative group, then xx^ = ^A:/Q(AO) — 1, 2/AO = 0? ord/(M) = ord/(xAo), and rA,Ao,.(M',i;)(0 = -2°^^'(2)^^ ^(^^^)(^^-^i) Now let A be an elliptic curve. Then a^Ao == '^(Ao) — {^) and 2/AO = «(^o) - _ ( ^ ) - - ( 1 - a,(A,) + K(AO)) = a,(Ao) - 1 - ( ^ ) . If ^2 G A(F),oo, 63 G A(F)/oo, and Mea = 62, then
((xA„/M)Fr-„i - (yA„/M)) e^ =
454
V. A. KOLYVAGIN
-(-i--(-(#)^ (#))))since Frl^ - Frx^a^^Xo) + '^(^o) = PxoiFrx,) = 0 on A(F)I<^. Since - F r ^ ^ i ( F r A o - l ) ( F r A o ~ ( ^ ) ) = xx^Fr-^px, = 0 on ^ M (because M\xxo and M\yxo), it follows that if ( ^ ) = —1, then AM C ^ ( / C A ) and Tx,XoAM',v){t) = -2°^^'(2)^^-^i ( ^ ^ ^ ( ^ ^ ^ ) ( ^ ^ 2 j ) Qn the other hand, if ( ^ ) = 1, then {Frxo - I?AM = 0 and rA,Ao,T(M',t;)(0 = _2ord,(2)^^-^i(^^^^ - l)(rA,.(M,t;)(FrAo)). Let /C = ^(A'o). We note that ICx = IC. In all cases we define the homomorphism /?1(M, AQ) : H^{IC,AM)unr —^ AM which takes u G U{IC)/M to {Zxo{Frxo)/M)u, where ZxoiFrxo) = l — Fr^^ when ^ is the multiplicative group and ZAO(F^AO) = -Fr^^{Frxo-l){Frxo-{^)) when yl is an elliptic curve. Since Zxo{Frxo) acts trivially on U{IC) and on AM^ it follows that (ZxQ{Frxo)/M)u is correctly defined. Thus, we have proved T h e o r e m 3. In the above sUuation, Tx^Xo,ir{M'^ ,v) G i/^(G(£,/C), A{IC)Mi ), ««^
We fix A e / / o ( ^ ) . We set EQ = Eo(A) = { A l l L i ^ i . ^i ^ e SP{2fNN'^X), the 6,- are distinct, / = 0 , 1 , 2 , . . . } , where N'^ € /o- We define the sets E^ ' = EQ '(A) forfc> 0 inductively as follows: EQ = SO, 4 * " ^ " = {[A,T,Afc+i]}, where [A,,r] € E<*>, A,+i € 5P(2/ArArj;A|,r|), ( ^ )
^ 0, ( ^ ) = 1- For Jb > 0 we have
(2°'-^'(2)MAJ|MA^^..,
and
annihilates ^ ( A ' ( A ; ) ) , ~ and ^(/fj;|,|/^^(i;)),oo, where V 6 5P(/^ji|,|/;^J, v\i, for it > 1 and for 1 < i < ik - 1, MA4+./(2°''^'(2)AfAj also annihilates A(K^^y(^^,^^-^{v))i«., where v € A^A^+./(2°'<^'(2>MAJ
EULER SYSTEMS We obviously have the inclusions E5 "*" ' C (Ef, M .
455 If [A,Ai,...
...,Aik+i] G So*^^^ then [AiA, A2,..., Ajk+i] and [A, A2,..., Afc+i] satisfy Theorem 2. Hence, by induction, if (Ti,Eo) is an AX2-system, then so is the system ( T J ' , I ) Q ' ) , which is defined as the restriction to Eg of the ifc-th derivative of (Ti,Eo). In particular, this is the case if (Ti,E()) = (T^E(,), where (T,Eo) is an AX2-system. Furthermore, if [A,Ai,...,A,+i] e E<'-^'>, then [AAJ^ • • • A^,A,+i,..., A,+i] e 4'^'''K Here ej = 0 or 1. Using Theorem 2 and induction, we see that, if (Ti, Eg) is an AX3-system relative to A, then the restriction of ( T / ~ , EJ, ~ ) to {[AAj^ • • • A^*, A,.|.i,..., Ajk-|.i], ej = 0 or 1} is an AX3-system relative to the set A n { A i , . . . , A,}. We consider two situations. Situation 1: we have an AX3-system (T, Eo) (relative to SP). Situation 2: A is the multipHcative group and we have an AX3-system (Ti,Eo) such that there exists an Euler system (Q,Eo) of elements of Zi[G{Krj/K)], rj £ EQ (i.e., a set of elements u;^ G Zi[G{KjK)] satisfying AXl: coiK.jK.i^Srj) = (1 - Frj')urj for 6 G 5P(2/7;), where COTJC.JK, : UG{KsjK)] ^ Zi[GiKjK)] is the h o momorphism induced by the projection G{Ksri/K) -^ G{Kr)/K)) which has the following property. For [T/,AI] G EQ and l^^^^^^^Mx^^ there exist V = v(rj,Xi) G SP{Krf), v\Xu u = u{r),Xi) G ( Z / M A J * , and x = x(r/, Ai) G Z / M A I such that r,,,,(MA,r-^'(^),t;)=/i,M.,,M^^,_.,(,)((tiCc;,,,,(M,J+
Here C(^'^i) ^ A{Krf{v))Mx is such that a*^i/a = C(^Ai) {Tnod v) for a a uniformizer of the field L{vl) (see above); i^{vX) ^ H^{G{L{vl)/Krj{v))^ A{Kri(v))Mx ) is defined by the condition i/{vX) • ^Ai •-^ Cj ^^^ ^li'^X) ^ H^{Kr}{v)jAMx )unr IS defined by the condition i/i{vX) • Pr H-•
cWe let (fi'*J,Eo ) denote the restriction to EQ of the fc-th derivative of (n,Eo): if [T/JTT] G EQ , then U;,;^^(MAI) is the element of Z/Mx,[G{Kx/K)] which is equal to
E y€G(/C,,,|/Ar)
^^ ( n d e g ( U , , ^ ) I Pr,(i7) (moc/MAj, \i =l
(3)
/
where w^^i^i = ^bgg £ Zi[G{Krj\jr\/K)]y Pr^ig) is the projection of ^f to G{Krj/K), and deg(^Aj,^) € Z/{Nj(/QiXj) - 1) is the element such that
456 t^.
V. A. KOLYVAGIN ^''
= PrAj(flf). Here Prx^ig) is the projection of 51 to
G{KxjK).
We set Ej =:Eo in Situation 1; in Situation 2 we take T\ ' = EQ for fc < 1, and forfc> 2 we take Ej ' to be the subset of EJ, ' which is deterlk—\\ •
Ik)
mined from EQ in the same way as Eg except with MA^.J replaced by P ^ ^ ' ( ^ ) M A ^ „ I . For k > 0, let (T<*^>,E[*'>) denote the restrictions to E [ * ' of the fc-th derivative of (T, EQ) and the {k — l)-th derivative of (Ti, EQ) in def
Situations 1 and 2, respectively. Let MAI (j) = Mx^/M{j), where Ai(j) = 2 if / = 2 and i = 1 and M{j) = 1 otherwise. Using commutative diagranns, Theorem 3, and induction on Ar, we obtain: T h e o r e m 4. (T<*^>,E[*') is an AX2-system, thai is, for TT = [Xi,.., '"^^kl [^,^] G E[**, V e SP{Kr^), (f,|7r|) = 1 one Aas r^,^ G F^(/i:;,(t;),^MAiy- Furthermore, suppose that v G SP{Krj), \j G SP{K), '^IKl^j /^^ 1 < J ^ 2, in Situation 2. Then
r,,,(MA.(i),t;) € i/i(G'(i(t;l)//^,(i;))M(A^,(v))M..O)),
([77,..., Aj,...] Z5 obtained from [77, TT] 6t/ removing Aj.) /n Situation 2 for V G SP{Kij), V = i;(r/, Afc), one Aas
Ajfc
+ x(^,A,K,A„...,A...(MAj)^i(t;,C(A.)H + «r,,A.
X.-A^xM^Xih,)).
Let X ' be a subextension of Kx over K. For [A,7r] G S j we define TK' e limi/H^^'.^/™) for ib = 0 and TK'A^X,) € i/H^^'-^Mx.) for m
fc > 0 to be (^OIKXIK'{T\,'K)' Then clearly {r/^-/,r} satisfies the analogue of Theorem 4 (// = A) with A'A replaced by A'', the divisors of A'A replaced by divisors of A'', and O^A T and a;A AI Afc_i replaced by their images UK' TT and UK'M A._. in z'/M^AGiK'/K)].
EULER SYSTEMS
457
2. The Multiplicative Group In this section A is the multiplicative group. Let A G /o, ^ G S'P(/A), {^) = 1, z = NX/Q{6) — 1. We let ^^ denote the generator of Az determined by the condition a^^/a = ^s (mod w), where w G SP{Ksx)^ w\S^ and a is a uniformizer of Ksx{w). Let U = U{K(S)); we then have isomorphisms Z/z ^ Az ^ U/U^, where the first isomorphism takes 1 to ^^. Let a<j : U/U^ -^ Z/z be the inverse isomorphism. If 6 G /o, then {Kl/Kl')i denotes the subgroup of Kl/Kl' consisting of the classes of elements of K^^ whose divisors are prime to 6. Let V G SP{Kx), v\6. We let a^ : (Kl/Kl')^ -^ Z/z[Gxl where Gx = G{Kx/K)i denote the GA-homomorphism defined by setting a^(c) = Xl(,€Gx ^^(^»^^(^))^' where cr^g(c) = crrj{g~^c) is the image of c in Kx{v3) = K{6). If M\z, then a^ : (A"*//^*^)^ -^ Z/M[Gx] is simply the homomorphism av {mod M). Let A G flo be fixed, and let A G /o, A|A. Let (f.Eo) and (f\,E()) be the Euler systems described at the end of §1. We let T'*^ denote corx«/XA(^'*^^)j and let iKx^^r and UJKX^TT be abbreviated TX^TT and CJA,^, respectively. If r G H^{KX.AM) = Kl/Kl^ and h is the group of divisors of Kx^ then (r) denotes the image of r in hlh^. We let Mj; (Jk) denote 2-ord,(2)j^^^ in Situation 1 and for Jb > 2 in Situation 2, and MA,/-°^^^(^) for A: = 1 in Situation 2. From Theorem 4 at the end of §1 we obtain Theorem 5. Lei [A,7r] G !:[*'>. n e n rA,^(MAj G {KHKl^^^)^^^, and, if Vr G SP{Kx) with Vr\Xr (in Siiuaiion 2, Vk is the divisor which appears in the conditions on (Ti,Eo)), then
{rxA^xA^))^\\^r
(modl^ r= l
in Situation 1;
r=l
in Situation 2.
)
458
V. A. KOLYVAGIN
Let Gi be a subgroup of G{K^{Aioo)/K), where A'' is the compositum of all fields Kj^,, with [A, TT] G I^i , Ar = 0,1,2, We suppose that the period of Gi divides / — 1. We let B denote the subgroup of elements of order / — 1 in Z*, so that Z* = J5 x (1 -h /Z/). 5 is a cycHc group of order / — 1, and it is isomorphic to its reduction modulo /"* in Z//*^ for all m > 1. If Ai = Z/[Gi], ^ is a Ai-module, and tp : Gi -^ B is a, character (i.e., a homomorphism), then we let A^ denote the Z/-submodule of A consisting of all a G ^ for which a^ = '^{g)o> Vy G Gi. A decomposes into the direct sum of the Z/-submodules Axp\ here A^ = j^A^ where j ^ G Ai,
jV = (l/[Gi])E,eG.^(lA)ffWe let V'l denote the character of G\ induced by the action of Gi on A] (the group of/-th roots of unity); then Aim = (yl/m)^^ Vm > 0. Let G2 denote the image of Gi in G{K\/K), We fix a charsicter V' * G2 —• 5 , and let \l) also denote the corresponding character of Gi. We suppose that there exists a subgroup G3 of G{K\/K) such that G{Kx/K) = G2 x G3 (direct product). Let Af = /" and A = Clx/Cljf, where Clx is the divisor class group of Kx. We let A3 denote Z/[G3]. If J is an ideal of A3, then J ' will denote the ideal consisting of all / G A3 for which fJ C MA3. Let [A, TT] G ^W, 2''^MVM\MX,; 2^"^'(2)/ord,(A)^|jvf^^ in situation 2. In Situation 1, given Tx^7r{M) G Kl/Kl^, we define the ideal MA3 C Jx,ir{M,rp) C A3 to be the annihilating ideal of j^Tx,Tr(M). For the rest of §2 we shall assume that {lj2f) = 1, and also that the map N^^/KX • (C''A)/«> —^ (G/A)/«> is surjective, for example, {l,[K^/Kx]) = 1. ve Theorem 6. Lei \p ^ Vi> V'^ ^ ^ i ; [^)^] G DJ . Then in Situation 1, J'^ ^{M^xj^)A^ is contained in the G^-suhmodule of A^p which is the zero suhmodule when ^ = 0 and [A, TT] = [A], and is the module generated by vi"^, \ 1. In Situation 2, {jxp(*^x)Axp = 0, and for k > 0, JXP(J^X,'K{^)A^ is contained in the G^-suhmodule of A^ which is generated by the vl"^. Proof. In Situation 1, from Theorem 5 we see that it suffices to prove that Vj' G Jj;^(M,V'), Va G A^, 3Ait^i G 5P(2/7Va|7r|), { ^ \
= 1,
[A,7r,Ajk+i] G Ej "^ ', 3i;jb-|.i G SP{Kx), Vk^i\Xk+i, such that the class of f^^j is a and avj,^^{jtpTx,ir(M)) = d'j^j' {mod M), where d' G {Z/My. In Situation 2 it suffices to prove that Va G A^p 3Xk-\-i € 5'P(2/A^^A|7r|), I -^z^^ ]) [^)^)^ib+i] G ^1 ^ib+i € SP(Kx)j
i ^he class of f^^^
is equal to a for some
^ib+i|Aib+i. If the other conditions on Ajt+i hold, then
EULER SYSTEMS
459
the condition [A,7r, Ajfc^.i] E Si holds provided that M^^^j = /ordKiVfc/Q(Afc+o-i) is divisible by MA,/"(^''^+^'«-^^°'^'(^) (in Situation io, 2*0 = 1 or 2), where /"('^''^) annihilates all of the groups which occur in the definition of EJ, '*'^'. That is, Ajk^i must split in i('(v4 n'(A,T))'"'(^'^) ^ n(A, TT) + (io - l)ord/(A). We set V = K^^^^{Aj^^ in'(x,rr)) (where Mx^ = M for Jk = 0). We show that the norm map from Cly to CL^ is surjective. More generally, we prove this surjectivity assertion for an extension of the form Khcr^/Kbj where 6 G /o, c G SP{f) {V C /<'A|^j/ni for some ni G N). Here it is enough to consider the case (6,c) = 1, since otherwise we can set 6 = 6'c"**, where (6',c) = 1; and because the norm map from Clbc^^ to C/ft/ is surjective, it follows a fortiori that the same holds for the norm map from Chc^ to C/j, since Kh D Ki,'- Thus, suppose that (6,c) = 1. Let V G SP{Kh)i ^|c. Then, if we identify G{Khc^/Kh) with a subgroup of {OK/C^Y) the local reciprocity map ^t» • ^6(^)* —^ G{Khc^/Kh) looks as follows on the group [/ of units of K},{v): u \-^ K{UI)UI^ {mod c^) =: txj"^ (mod c"*), where iti is the norm of u in the extension Ki,{v)/K(c). Since Kb{v)/K(c) is unramified, we have ^t'(J7) = {OK/C^YHence, the image of G(Kicm/Ki,) in (OK/C'^Y is {OK/C'^Y^ and t; is totally ramified in Kic^. In particular, (-ft'6)unr H/C^c"* = A'^. Surjectivity of the norm map then follows from class field theory and Galois theory. Since ^K'/Kx • {Cl^)joo —4 {Clx)ioo is surjective, it follows that the norm map from (C/v)/oo to ( C / A ) / « is surjective. Thus, Afp may be regarded as G{W2/V)j where W2 is an M-periodic unramified abelian extension of V which is fixed in Vunr relative to the kernel of the homomorphism Cly -^ C/A -> A ^ A^,. Let 52 C V/V^ be the subgroup which corresponds to W2 by Kummer theory. ^From the Gal-property (x^,t/^) = (x.yY Vy G G{V/Q) of the pairing G(W2/V) x 52 —>• AM it follows that Gi acts on 52 as ipiip"^- Since ^p :/: ipi and Gi acts as Vi on H^{G{V/KX).AM) = ker(A^*/7i'*^ -> V/V*^), it follows that (Kl/Kl^)^ ^ (|/*/y*M^G(v/Kx) j ^ ^^ imbedding. Suppose that we are in Situation L Let 5i be the Ga-module generated by X = j^rA,x(M) in V*/V*^. The annihilating ideal of 5i in the A3module (V*/V*^)^^^'^^^ is Jx^M^il^). We have the exact sequence of Ga-modules: 0 -^ JA,T(A/,V') —^^ A3 -* 5 I —>^ 1. We define the character Xi : Si —^ Ajif by setting Xi(jx) = (^^^^ )*, where {jf)i is the coefficient of the identity of G3 in jf G Z/M[G3], and C is a generator of AM- In Situation 2, we define 5i and xi to be trivial. Let X2 • 52 -^ AM be the character corresponding to a G ^ ^ = Hom(52,v4M)- Since ipitp~^ ^ V'l
460
V. A. KOLYVAGIN
by assumption, it follows that 5i and .^2 have trivial intersection. Let X ' Si X S2 -^ AM be given by x = Xi x X2- Let W be the M-periodic abelian extension of V corresponding to S = SiX 82- Let ^fi E G{W/V) = Hom(5,Ajv/) be such that (yi,s) = xi^)- Using the Chebotarev density theorem, we choose Ajk^.i G 5P(2/A'^^A|7r|) so that Fr^ = gi for some w G SP{W)j ^|Aib+i. We show that Ajk^.i satisfies the necessary conditions. Since g = id on F , Xk^i splits in V. Let v^ G SP{V), v G SP{Kx), w\v^\v. Then NYIK^{V') = f, and, by our condition, t;^^ = 1 in A^. Furthermore, in Situation 1 we have, by assumption: (Fr^,jx) = (^^^^ ^^ = C'^ff, i f / = IZgeGsJgd' Hence, av{^) = d'j^j' {mod M), where d' G (Z/M)* is such that ^ ^^«j^/^/Q(^''+0-i)/M)d' ^ j ^ . ^ concludes the proof of Theorem 6. We now suppose that G3 = 1, i.e., G{Kx/K) = G2. In both Situations 1 and 2, if [A, TT] G ^1 , let J\T^{M,II)) denote the annihilating ideal of UTX,^{M) in {Kl/Kl^)^. We note that for V' 7^ V'l and a G [Kl/Kl^)^, the annihilating ideal of a is /"~"^Z/, where 0 < m < n is the largest integer for which a G {Kl/Kl^ Y^. We suppose that xp ^ ipi and V'^ 7^ V'l- In Situation 1, we suppose that jipTx ^ 0. Let mo G Z^. be the smallest integer for which jxpTxH'^'''^^) ^ 0. Then we have Jx{M,tp) = /"""'o for n > mo- In Situation 2, if jtpoJx ^ 0, then we let mo G Z_j. denote the integer such that jxi)U)x = I^^UQH, where wo G Z*. We note that in Situation 1 TX{M) G UX/UX^, where Ux is the group of units of Kx' We suppose that in Situation 1 {Ux 0 Z/)t/. / 0 (otherwise we would have j^rx = 0); then {Ux 0 Z/),/, c^ Z/, since ip ^ tpi. Namely, because of the existence of Minkowski units, our assumption consists in the stipulation that ip / id is an even character when K = Q and that ip ^ id in the case [K/Q] = 2. In Situation 2 we suppose that A' = Q and tp is an odd character, so that (J/A 0 Z/),/, = 0 . If X G Z / M and x G Z is a representative of x, then
{
ord/(M),
ifx = 0,
ord/(x),
\i X ^ 0.
If TT = [Ai,...,Aik] and 0 < i < k, then TT,- denotes the set of divisors obtained by removing Ai+i,... from TT. Let C/A,/OC,^ ^ E f = l ' ( Z / / " 0 ^ - , where ni > • • • > n^,+i = 0. The sequence n i , . . . , njbQ^.i is uniquely determined from Clx,i°°,tp and characterizes it as an abehan group. Here ko is the rank oiClxj^^xp- In particular, ko = O^Clx,i-o,tp = 0.
EULER SYSTEMS
461
We suppose that j^T\ ^ 0 in Situation 1 and jxi^ux / 0 in Situation 2. We set n = 3mo and M = /*^ as usual. We define 11'^' to be the set consisting of the single element A, and we call mo the weight of 11^°'. Suppose that for s G Z+ and for 0 < Ar < s we have sets n<*) C E[*>(A) =^{[A, 7r]|[A, TT] G Ej (A)} with weights mo,...,mjb, where m,- G Z^. and m,-_i > m,- with equality allowed only for i = s. The sets II'*) and the numbers m,- are connected to one another; namely, II'*"*"^' and ruk^i ^-re determined from 11^*' and m o , . . . , mjk as follows. Let 11 be the subset of T\ '(A) consisting of the elements [A,7r, Ait+i] for which: M/°^^'W(i-'i)|Ai; [A,7r] G n<*>; ord/(rA,,r,(M)) > m,- for 1 — 2*1 < i < ^ — n , where ii = 1 in Situation 1 and 2*1 = 0 in Situation 2; and ord/aij^^j(rx , r ( ^ ) ) = f^k in Situation 1. Let 7/ G n^*), and let YhkJ^i{r]) be defined as:
^'l/.^tN
-
<™^
,,^, ord/ao,+,OVr^,Aifc+i,Afc+2(M))
.
We note that, by the Chebotarev density theorem, the inner minimum is simply ord/C' r (M, ^ ) . Then in Situation 1 we have II'*^"^^' = {[^,A;b^.i] G
Il(*''"^)=n
oxAiay^^XH'^r),k^\{M))
=
mk^i(rj),
where
Tnk^i{T]) = rhk-\.i(ri)}. Here nik^i = mk+i{r)) does not depend on rj. In Situation 2, if mjk+i(7/) = ruk Wr) e n(*>, then n(^+^> = H
and
mjk+i = mjfe. Otherwise, let XI^*"'"^^ denote the subset of II ing of elements [rj.Xk+i] for which OTdi(j^uJrj,x^^,) - ord/C^;^ Then
consist(M,^').
n<^+^> = {[^,A,+i] Gff'*+^> I ord,j>,,,,^,(M) =
Again mj^^i = mjk+i(^) does not depend on r). The sets II'*^' and the numbers m o , . . . , m, satisfy the following theorem for 5 > 0.
462
V. A. KOLYVAGIN
Theorem 7. One has ko -\- I > s, and ko-\-I = s if rris = m , _ i ; Hi = rrii^i - rrii for 1 < i < s. If t] = [A,7r] € n<*^>, Vi G SP{Kx), and Vi\Xi, then the homomorphism Zf —^ {Clx 0 Z/)^ for which 1,- i-> 6,(r/) = j,^(class of Vi) has kernel /"*Z/ x • • • x /"*Z/; m particular^ its image is isomorphic to 5Z,_i Z//"*. iTere /or an^^ 5e^ of elements 6 i , . . . ,6jb <>/ (C'/A ^ Z/)^ /or ly/iicfe Me homomorphism Zf -^ {Clx 0 Z/),/,, 1,- i-^ 6,- Aa5 kernel l^^Zi x •• x l^^Zi, there exists rf E II**) such that 6,- = bi(r}). If rus = m , _ i , 5 > 2, and T] £ II'*"^*, then 5 - 1 = fco, {Cl\ 0 ^ih^ViZl'2'/t'''bi{rj), and, i/[r/, A,] G n<*),
1 1 ^i 0<j<x
for i > 0, zo = yo' Then Zi {mod {Ux/Ux^')xi;) = {jtp class of Vi^^*"^, where c = [C'/A]/[C'/A,/«']- {zi} and {y,} are sets of independent elements mod K^^ for 0 < i < s—1 in Situation I and for 1 0 and m,_i > m,, then the set n'*"*"^^ defined from II'*), is nonempty; that m,^.i = m,4.1(7;) does not depend on rf; and that n^*"'"^^ and m,-|.i satisfy the theorem. In this way, using induction, if we start with an Euler system we can successively determine the sets 11^^),..., H^^^"'"^' and the numbers mo^... ^m^^^i, where ki < mo is a number for which m^k^^i = f^ki] then these data determine the structure, bases of prime divisors and the expansions of prime divisors in a given basis, along with representations of primary numbers, since this is what is described in Theorem 7. - { * + !)
Suppose that [ryjA^^i] G 11 . Let m'^^j = m',^i(r/, A5+1) = ord,C;A.^,(M, rP), y,+r € (Kl/iq' )^, and jVr,,,.^.(M) = jfj+r • We let 2/54.1 denote a representative of ys^i in A'^, and for z < s we let t/,denote representatives of y,-. The factorization of T^,A,+I {^) in Theorem 5 gives us the equality:
<+l
1 1 ^»
Psll,i = {Vs+l)
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463
where i^ £ Z[G2],4 =-^^ (mod M), nj G Z+, ^^,-^.l, w,+i,,- G Z, (ti,+i,/) = 1, (us^i^iyl) = 1, and Ps-\-i,i is a divisor of/C^ which is prime to A i , . . . , A54.1. Here j^/"»W5+i,,- {mod M) = «t;.OV^A,...,A, ..X^))* ^'^^^ ^«+i ^^^ {^«} ^^^ distinct, we have m^^.^ < m, and m^^^ < nj. If we take the /"*»+i-th root of both sides of the above equahty, we obtain:
^,+1
[[^i
Psli,i
=(y.+i)-
(4)
We prove that uig — ^s+i — '^5+1' Suppose this is false, i.e., uig — ^54-1 — ^5+1 + »^' where n' > 0. Let d,- G Z satisfy: i;,;^i f]^.^^ t;,.^ * = (w), where a; G A'jJ. If we combine this equaUty with (4) and take into account that, by induction, the v-"^ generate a subgroup of — lCi=i ^Z^'**) w^ obtain the relation ^"S" n 'i' = 2/»+i ('"'"^ ^ A ' ) ,
(C/A,/^
r.- G Z//.
0 Z/)t^ which is
(5)
We show that this relation cannot hold. We note that Theorem 4 implies that ord/(Qf,;,(yj)) > 0 if ordi;^(yjt) > 0 and i < j in Situation 1, i < j in Situation 2. We suppose that r,-^ = 0 for 0 < 7 < jo, where jo > —1. Then, using the relations between y,- and Zi in Theorem 7, we have the equality
f<«-jo~l
Since ord,;,_j _i_, (Vs+i) > 0 (here 2*1 = 1 in Situation 1 and ii = 0 in Situation 2), it follows that ord/at,,_jQ_i^,j (y,+i) > 0- ^y ^^^ definition of n'^'-^o""^), we have ordia^^_^^_^^^ (y,) > 0 for i < 5 — jo — 1, and ord/at,,_j^_i^,^(t/5-jo_i) = 0. Hence Vg-j^-i = 0. Thus, by induction, we have r,- = 0 for t < s. But this is impossible, since ys-^i ^ K^, Hence, 0 < m, - m',^1 < n^+i. Let 6,+i G {Cl\/M)^ (since /'"^ annihilates C/A,/«> by Theorem 6 and ord/(M) > mo, it follows that (Clx/M)^ and (C/A 0 Z/)^ are canonically isomorphic) be an element having period n^+i in the quotient of {Clx/M)^ by the subgroup generated by bi = ^.(ry), i < s. In Situation 2, if n,+i ^ 0, let z^^i G {K^/Kx^ ' ^)tl>i ^'s-\-i = K'+V^' Let zj^j G A'^ be a representative of ^i^.1. The elements yo j • • • > 2/5 in Situation 1, the elements y i , . . . , y.
464
V. A. KOLYVAGIN
in Situation 2 with ris+i = 0, the elements y i , . . . ,2/*, 2:^^-1 in Situation 2 with Tis^i ^ 0 generate independent sets of elements modulo K^ . If we proceed as in the proof of Theorem 6, taking Si to be the subgroup of {V*/V*^),p generated by {t/o, • • • ,^5}, {2/1, • . ,2/5}, and { y i , . . . ,t/,, 2:^+1}, respectively, and choosing suitable characters xi • ^i -^ AM^ we prove that there exists A,^i G SP such that [77, A,^1] is in II
in Situation 1
and in II in Situation 2. Here II is the subset of II consisting of the elements [r/jA^^-i] such that ord/at-.+iC-^a^-i) = 0 when risj^i / 0, and n
= n
when n,+i = 0. In particular, if m'^^i = m, for
all [r/, Aa-i-i] E 11 , then it follows from (4) that ris^i = 0. Since rus — ^5+1 < ^^5+1) it follows that, conversely, if ris^i = 0, then m^^j = 0 for all [?/, Aj-i-i] G n , and (4) gives us the corresponding representation of j^(class of 1^5+1) in the basis {hi{'q)}. Suppose that n^^i > 0. Then rris — m'g^i = ^5+1- From (4) we have the relation
^,+1 =yiVr'^'n^f'"""'"""'"•"•• (mod(C/.M"'-')^), i<s
from which it follows in Situation 2 that, in particular, [y;,A54.i] G 11^*"*"^^, since ord/at.,^j(y,) > 0 for i < s, and hence ord/at;,^j(^,) > 0 for i < s. Thus, ord/at,,^i(y,+i) = ord/ai,,^i(f,+i) = 0. Since m',^i > m, - n,+i, and in the present situation m^^^ = m, — n^^-i, it follows that ma_|.i(r;) = rUs — n,^-i. Finally, we observe that if [77, A,^-i] G II'*"*"^), then ys-\-\ is independent of {yj}, i < s (this is proved in the same way as we proved that (5) could not hold). Hence, Zs^\ is independent of {^j}, i < s; and this implies that the bt([rj,Xs^i]) generate a subgroup of {Clx/M)xf; which is isomorphic to Yll=i Z//^*. This proves Theorem 7. Theorem 7 imphes (mo > TUki) Corollary 8, [C/A,/«,^] = l^o-m^, ^ We now give examples. 2.1. Gauss sums. Gauss sums form an Euler system of the type in Situation 2. The information we need about Gauss sums can be found, for example, in [18]. Properties Bl and B2 (see below) were proved by
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Stickelberger. The field K is Q, and we recall that K{6) G K^ where 5 is a divisor of K^ denotes the positive number whose divisor is 8. Let A G /oLet [y;, Ai] G 5]o(A), i.e., T/ = AJ^^i, where 8i G 5P(2A), the Si are distinct, Ai G SP{2TJ), and «(Ai) = 1 (mod ;;). We let x denote a primitive K:(Ai)-th root of unity. Let x * (Z/Ai)* -^ ^/c(r;) ^^^ ^ character. We set TV,,AI(X) = Ea€Z/A, Xia)x\ Ifve SPiKrj) with t;|Ai, then Xn.v : (Z/Ai)* ^ A,(,) will denote the character for which Xv^vio-) = a"'^^^^^^)""^)/^^'')) {mod v). For all Ai G 5 P we fix an imbedding v{Xi) : Q ^-^ Q(Ai), and we let f(r;,Ai) G SP{Kr)) be the divisor determined by this imbedding. We let Z[G,;]', where G,, = G{Kr)lK)^ denote the subgroup of Z[G,;] consisting of the elements Y^g^o ^gS ^^^ which X^s^ = 0 and Yl^gPio) = 0 (mod ?;), where p : Gn ^ C^/vY is the canonical isomorphism given by C^ = C^^^) VC € A^(^r,)' Let ^^ G Z[G^]', z^ = E«i?i/- We define Trj^xA^v) ^^ ^^ Ug^G.^vMix^ri^.tivM))''' ^^^^ '^vMi^v) ^ K ^^^ ^^^^ ^^^ depend on the choice of x. The divisor of Tr^,\i{zr^) has the form:
51.
(n,,A.(z,)) = i;(/;,Ai)^^''^-e(z/,r<<'>^'/«.
Here (a) G Z is the canonical representative of a {mod rf)^ i.e., 0 < (a) < K{r]) — 1, (a) = a (mod ;/). We fix z\ = YlgeG ^99- ^^ determine Zjj = YlgeG ^99 ^ Z[G,;]' from zx as follows: if p G G,, and g' G G\ is the projection of g^ then 5^ = 0 if p{g) ^ 1 {mod {r]/X)), and Sg = Sgi if />(5f) = 1 {mod {TJ/X)). We set
Ti = {r,,A.(MAj = r,,A,(^,) (mod /^^^^O, [^. ^i] G E[,(A)}. Recall that we have taken a fixed rational prime /, and in our case M^j is equal to P^^'(<^i)-i). We show that (Ti,Eo) is an Euler system satisfying the conditions for Situation 1 (see §1). The property AXl follows from the norm relation NK,JKA^S,,XA^S,))
= r,,A,(z,/'-^'-"\
(6)
which can be derived from Bl as follows. Using Bl, one verifies directly that (6) holds for divisors, i.e., (6) holds up to an element of ^^(77) (we have T^.Ai^^Ai ~ ^' ^^^ li u £ Urj satisfies uu^ = 1, then u G A^(^^^). We similarly obtain:
466
V. A. KOLYVAGIN
where b E (Z/ryAi)*, 6 = 1 {mod Ai), b = K{8) (mod ;/), p{gi/b) = 1/^) C ^ ^/c(»/)) ^'^d X(2),Ai • (Z/Ai)* -^ i42 = { i l } is the nontrivial quadratic character. Now (6) follows from this, if we use the fact that, by our choice of {zp]^ zsij maps to z^ under the natural homomorphism Z[Gsr}Y "^ Z[G^]' and if we recall the properties of Zfj. It further follows from Bl that Ti is an AX2-system. The property AX3 (with A = SP) follows simply from the relation X^^^i(^^,Ai) = XvMvM)'
^^ f^*'
^^en X6rj,v(6rjM){0') =
X J(„ ^ A(0> where ^ G ^/c(5). If we take into account that ^ = 1 (mod u) ifuj £SP{Ksr,) with (jj\6, we obtain AX3. It remains for us to verify the last property of Ti with Urj = ~ ^^7^ Sa€(z/n)'(^)^i/a- We use the second well-known property of Gauss sums: S2.
where
Here ^ : (Z/Ai)* -^ Z/Mx^ is the homomorphism such that CC^Ai)^^"*^ = Q(«(Ai)-i)/AfAj (mo(f Ai). By definition, ^AI is a generator of GAJ satisfying C(tAi) = P(/AI)^''^^'^"^^^^^^ {rnod Ai) (x - 1 is a uniformizer of A'AI;;(^), where u G SP{Kx,rj), u;|Ai, and (x - l ) 7 ( x - 1) = p(t) (mocf a;)). From (3) it follows that
G Z/(MA7/"'''^''^)[G,].
We have:
EULER SYSTEMS
Xa=
Yl
i^hin^i^
467
(rnod Ai)) =
6G(Z/Ai7;)*, b=a (mod rj)
=TI
Yi
6 (mod Ai)(^>^i'^ I =
\ 6 € ( Z / A I T ; ) * , h=a
(mod
r})
^/c(Ai)~l
=^ n
,c+K(Ai)(a-c), I _
c
/K(A.)-1
=/•
JJ
\
c'+<''-'=>'
/ //c(Ai)-l
=^
n \ \
(since «(Ai) = 1 (mod M A J )
\
>
c^-'"=>' (WAi)-!)!)*"*' n
<==!
/
'^"^'^
(c),>(o),
since (a - c)q = (a), - (c), if (a), > (c),,, and (a - c),, = K{r)) + (a),, - {c)„ def
if {a)n < {c)rj. We have U{c)r,>{a)r,^=^ if i^)v = ^iv) - 1. while if (a)^ < K{TJ) — 1, then n
c=
We note that
n
(d+K('7)e) =
468
V. A. KOLYVAGIN
Thus,
K(»7)K(AI)
E
(^(r)+{a),jr((«(Ai)-l)!)-
a€{Z/f,)-
- Kir,){a),J^iK{n)) - K{r^)T ( ( " " ^
^«)'')'))g^
with some r G (Z/Ai)*. If we take into account that (K:(AI) — 1)! = — 1 {mod Ai), T{—1) = 0 (/ is odd), K{\\) = 1 {mod M A J , and Zrj{J29i/a) = 0 by assumption, we finally obtain the equality
which, in conjunction with B2, proves the desired property of Ti. Let / satisfy / = 1 {mod [(Z/A)*]). Then Theorem 7 gives a description of the ^-component of C/A,/«> in terms of Ti and its derivatives, where V' is a homomorphism from G{Kx/K) to Z* such that V^(—1) = —1, V' 7^ V^i? tp ^ ip'^. In particular, if A := (/), / > 3, then an elementary verification shows that for f = 2s -f- 1, 5 = 1 , . . . , ^ , there exists zx G Z [ G A ] ' such that j ^ j [jZx{Ylae(z/xy{^)9i/a)) = l^'udr,^ where Ui G Z; and the pi were defined in the introduction {ipi{g) = ^(^f) {m.od A)). According to Corollary 8, we have x,-=ord/[C/A^/oo ,^.] < jji] and since YlVi — Z^^« ^Y the formula for the class numbers of A'A and A'A+ (see the introduction), it follows that Xi = yi. 2.2. Cyclotomic units. Cyclotomic and elliptic units give us examples of Euler systems in Situation 1. In §2.2 we take /\ = Q, A G /oWe fix a system of primitive /c(7/)-th roots of unity (^^, where r; G Eo(A), such that C,7i,72 = C^'''^ For 6A G ( Z / A ) * we define 6^ G (Z/?/)* by the condition 6^ = 1 (mod (yy/A)), 6^ = 6A (mod A). We set r^ = (C;; — l)(Cf^^ — 1). If A is not a power of a prime divisor, we set r^ = r^; if A is a power of a prime divisor, we set r, = (r;)«N/r^ = ( C ^ - l ) ( C , - ' ' - l ) / ( C , - l ) ( C , - ' - l ) . We show that T = {r^^, rj G Eo(A)} is an Euler system of the type in Situation 1. Using the equality n e r _ i ( X — ^) == X^ — 1, we have the following relation for 8 € SP(2^): NK.JKM',^) = (^n)^''^''"^-. AX2 follows from the equality fl^^zii ^ati(l ~ 0 — ^- AX3 is obvious: ^Fr
C^^-1 = C r ' e - 1 , wheree(^) = 1. Hence, O r ; - l = (Cr;-1)^'* (modu;), where a; G SP{Ksr^), a;|6 (note that ^ = 1 {mod uj) V^ G A^i^^))-
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469
For example, for A = (/), Corollary 8 implies that Xi < yi^ f = 2s, 1 ^ ^ ^ ^ ^ (see the introduction). Hence x,- = i/,-, because Y^Xi = YlVi from the class number formula for K\^. 2.3. Elliptic units. In this example K is an imaginary quadratic extension of Q. We recall that E denotes an elliptic curve with complex multiplication by the ring of integers of K. Let Y^ -f a^XY + a^^Y = X^ -f ot^X'^ + a^X + ocQ be the Neron module of E over K, and let A(£') be the discriminant of this equation. Let a G /o(6/), and let 60(2:, a) be the function on E which is defined over K by the formula 0o(>?,a) = z//c(a)~^ rK^C-^) "" ^(^))~^j where the product is taken over all /? G {Ea \ 0)/ ± 1, 1/^2 ^ A ( £ ; ) ^ ' ^ / Q ( ^ ) - \ 1/ G /C*. We let Jo denote the set of functions ip : /o(6/) —• Z such that <^(m) = 0 for almost all m G /o(6/), and Em€/o(6/)^("^)(^i^/Q(^) - 1) = 0. We define 60(2:, v?) to ^^ Tlmeio(ef) ^o{Zj m)'^^^\ Let \(p\ be the product of those m G /o(6/) for which <^(m) 7^ 0. We fix a system of 0/c-generators (rj of Efj, rj E /7o(|v?|), such that CviV2 = *^{V2)Cvifor(r/2,/) = 1. We set r^ = 0o(Cr;,<^). Let A G //o(|<^|), N'X= \(p\. Then T = {r^, ?; G So(A)} is an Euler system in Situation 1. The norm relation and properties AX2 and AX3 are easily derived from the properties of Qo{Zj(f) and the good reduction properties of E, The proof is actually analogous to the proof for cyclotomic units. For details, see [8]. Suppose that A G 5 P ( 6 / ) , NKfQ{\) is a rational prime, / = NKfQ{\), A = /A. We suppose that (f satisfies (A, |^|) = 1. We let Ell((p) denote the Gx = G(/irA/A')-submodule of Ux (the group of units of Kx) generated by Tx{^) = NKj^/K^ir-x). Then for ^ : GA ^ Z;, ^ # id, Corollary 8 gives us the inequality x^ < yxp{(p), where x^p = ord/[C/A,/~,t/>], Vxpi^p) = oTdi[{{Ux/EU((p)) 0 Z/)-]. Using the class number formula for Kx, here we also show that 3(p G Jo such that ^w^^j-^X/v-Cv^) — Y^tp, from which one obtains the equahty x^p = yxp, where y^p = oidi[{{Ux/Ell) 0 Z/)^], in which Ell C Ux is the subgroup generated by the Ell{(p) when (p runs through
Jo,(M,A) = l. 3. Weil Elliptic Curves Let ^ be a Weil elliptic curve of conductor AT, and let 7 : XN -^ A be a Weil parametrization. Here X^ is the modular curve over Q which parametrizes isomorphism classes of isogenies £" —• £"' of elliptic curves with cyclic kernel of order N. The field K is an imaginary quadratic extension of Q with discriminant D satisfying D = square {mod AN), We
470
V. A. KOLYVAGIN
fix an ideal i of the ring of integers O of K for which O/i 2:^ Z/N. If T] € Io{N) and Orj = Z + K{T])0, then z^ zz i r\Orj is a proper 0,^-ideal. In particular, ii = i. Let K^j be the ring class field of K of conductor rj. We let Zn denote the point of XN over Kj^ corresponding to the isogeny C/Off —^ ^/^^^ (we are supposing that Q is imbedded in C, and that i^^ D Orj is the inverse of irj in the group of proper 0,;-ideals). We set Trj = j(zr,) e A{Krj). Let 6 G SP{Nrj), {j^) / 0, and let 6' be a prime divisor of (5 in K, Since [Ksn/Krj] = [Ks/Ki] = K{6) - {j^) and P^(X) = X^ — asX -f- K{6) {aa = o>K(a) if <^ G /o? where X ^ ^ i ann~^ is the canonical L-series of A over Q), it follows that
= FrJ.'
[K{6) - (^A) - ( F r | , - a , F r , . -f K ( 6 ) ) ) =
= a, - F r , . - (^^ as - Frs'as,
FT-.'
FrJ,\
= if ( ^ ) = 1, if ( ^ ) = - l ,
(in the latter case Frs' = Frs = id). The proof (see [11, 12]) of the norm relations NK^/XI{TS) = FrJ, ([Ks/Ki] — Ps{Frs')) and the relation Ts{iv) = Fr~^Ti(v), where LJ G SP{KS), '^ G SP{Ki), and a;|f |(5, carries over to the extensions Krjs/Krj, and hence T = {rrj, r] ET, = Io{N)} is an AX3-Euler system (A = SP). In particular, if A G Io{N), then (T,So(A)) is an Euler system in Situation 1. As usual, M = l^j where / is a rational prime. If 6 G SP{N)j (j^) ^ 0, is such that 6' is a principal divisor and M\ {K{6) — {j^))j then we define the generator CM,6' of the group fXM of M-th roots of unity by the condition CM,6' = ^s' ^^^ " ' where ^s' is the generator of ^(NK/Q{S'yi) ^^^ which Os'{^s') ~ ^6' Recall that we are supposing that we have a fixed set of generators tj of the cyclic groups G{K^/Ki)y 6 G 5 P , and 9s> is the local reciprocity map. If <5' splits in Kx, as is always the case if [j^) = — 1 (then 6' = 6) and v G SP{K\) with v\6\ then we set ^jvf,t; = CM,6'If/C is a local field, then we let ( , )M,A: * A{IC) x H^{IC,A)M -^ Z/M denote the non-degenerate Tate pairing, and we let [, ]M • AM^AM —^ P M denote the Weil pairing. Let [A,Ai,...,Aib] G ^^Q\\) (see §1 following Theorem 3), M\Mx^ {Mx, = P , where m = ord/(«(Ai) - ( ^ ) , a^(^x^) 1 - ( ^ ) ) ) , and Vr G SP{Kx) with VrlK- If r £ H^{R,A)M, then (r) denotes the image of r in H^{R,A)M' From Theorem 4 and the explicit
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471
formula for the Tate pairing [11], §3, it follows that the following relation holds for ailge G(Kx/Cl) and all s e A(Kxivr))/M: T h e o r e m 9. SP(Kx),v:^\u...,h,
((
€ H\Kx{v),A)^nr
for v G
and
A''XrxM,...,X,(M,vt)))^^,,
_
4 ^ . -^5^(£-i|£-z(i))(,-.,, ,,,.,(«))(„,, We note that (by the analogous theorem for k — \): i9-\
A
(A^))(^r) G A{Kxivr))/M
{iXr,N) = 1).
If ( ^ ) = 1, then jr = 1 and Kxivr) = K(X'^) = Q(Ar); if ( ^ ) = - 1 , then jr = 2 and Kx{vr) = ^(•^J-) is a quadratic extension of Q(Ar) (AJ. splits in For [l,Ai,...,Aifc] G 4 * ' ( 1 ) we define
TX,„.„X, E H\K,A)M,^
to be
In [11] it was proved that r^ = er^^'"^^ + T(0) for rj G SP{N). Here € G { i l } is determined from the condition 7 o ii;7v = 67 + 7(0), where WN ' XN —^ XN is the principal involution of XN\ 7(0) G ^(Q)tor is the image under 7 of the cusp of XN corresponding to z = 0 in the completed upper half-plane; and 0{ir^) G G{Krj/K) is the image of i^j under the isomorphism 0 from the group of proper O^-ideals to G{Krf/K) such that J(by^^^ = J(a''^b), where J is the modular invariant. Similarly, a acts on Trj for all 77 G Io{N) (the proof is similar). Using induction and the definition of r^^Ai,...,Afc) one proves that
l
'^J
Then <,A.
..=e(-l)*rA
x.+h
n
^^^^^^^^^7(0), MA
where h = [Ki/K] is the class number of K. In what follows it will be convenient to use the "empty" index AQ: TAQ = T, TAo,Ai,...,Afc = '''Ai,...,Afc-
472
V. A. KOLYVAGIN
Forfc> 0 we set T' = Ao,...,Afc
{a -f f(-~l)*^"^^)7Ao,...,Afc)
if / is odd or the period of ^7(0) is even,
T'Ao,..,Afc)
if / = 2 and the period of hj{0) is odd.
Then irL,...Mr = <-^yr'xo^...A Theorem 9 implies that for v G SP{K), v ^ XQ,. .. ,Xk, TX^^...^\^(V) H^{K(v),A)»nT, and Vs G yl(/C(A;))/M, ife > r > 0, we have:
G
JM
In what follows we shall limit ourselves to prime divisors Xr which remain prime in K, i.e., ( ^ ) = — 1. It follows from the above that Ko,...,A.M) ^ H'(K{v),AUr for V G SP{K) with i; ^ Ao,...,Afc; ^'..,A;,...('^r) e ^(A"(Ar))/M for ife > 0; and (since FrA.r' ;^,^^.. = = (TT^ y = 6(—l)^r' y ) we have the following relation for all
se
A(k(\r))/M:
>{*'(^Ai,...,Afc('^'-))'^'^r SM,A^
'^'^^^«. M ' <-^)'^%^r'i V / ^ . . , A „ . , (A.) JM
17)
Recall that Am denotes the group of all points of order m on ^ ( Q ) , Am c^ Z / m 4- Z / m . We setrf= 1 if / ^ 2 or if (T acts nontrivially on ^ 2 , and we set d = 2 if / = 2 and a acts trivially on A2. If ^ is a {l,(T}-module, then we denote A± = ker((T ^ 1). If / / 2, then Ain is the direct sum of its cychc submodules Ain^ and Ain_ of order /". We choose a projective system (relative to multiphcation by /) of generators e±/n £ Ainj^. Since the Weil pairing [, ] M • AM X ^ M —>^ /^M is alternating and non-degenerate, it follows that [e^A/i^-MJM = CM is a generator of fiM' Now suppose that 1 = 2, Obviously, AM+ H AM- = ^2-1- = A2-. Hence, if d = 1, then ^2+ ~ Z/2 and AM± are cychc groups. Since 2AM C A M + + AM-, it follows that [AM+][AM-]/2 > 22"-2 On the other hand, [AM+] < 2"* and [AM-] < 2". Hence, [AM±] > 2"^^ 3 ^ ^ then A M ± ~ Z / 2 " , since AM± is the subgroup of elements of order 2^
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473
in Aj^2^±j which is a cycUc group of order > 2". We again choose a projective system {e±M}' Then [ C + M J ^ - M I M = C2*»-i is a generator of //2«-i- In fact, 2''"^[e+M,e_M]M = [e-|.2,e_2]2 = 1, since e^2 = e_2 (a general property of the Weil pairing is that [/ei,/e2]/n-i = [ei,e2]}n). We prove that [e-|.4,e_4]4 ^ 1, in which case [e^.jvf ) € ~ M ] M h^s period 2"~^. If we had [e4.4,e_4]4 — 1, then e^4 and e_4 would generate a subgroup of index 2 in yi4 which is orthogonal to itself with respect to a non-degenerate pairing, and this is impossible. Now let d = 2. Then AM-\- — Z/2^ + Z/2, since otherwise we would have AM-\- D A4] however, a acts nontrivially on A4, since det(/>(o')) = — 1. In the same way as before we prove that A M + ^ Z/2" + Z/2. We also have AM-- :^ Z/2" + Z/2. We choose projective systems {e±M} of elements of period M in AM±' We show that e^M and e_A/ form a basis of AM- It suffices to prove this for €^4 and e-4. Suppose that ae+4 + ^e-.4 = 0. Then 2|a and 2|^. Hence a == ^ = 0, since otherwise we would have 264.4 + 26^4 = 0 => e+4 = —e_4-f-e, where e E A2. Then 64.4 C ^ 4 , which is a contradiction. In particular, [e^M)^-M]M = CM is a primitive M-th root of unity. We set C = C{K) equal to the period of A{K)toT' The properties of the groups G/n = G(Q(i4/n)/Q) (the theory of complex multiphcation and Serre's theory) imply that there exist natural numbers a and 6', not depending on / (see below), such that: ai/^(G(A'(yl/ni+n2)/A'),^/ni+n2) = 0 Vni,n2; and for all n, h'A = 0 for any G/n-submodule A C Ajn for which either {a + 1)^ = 0 or {a - l)A = 0. We set 6 = {2/d)b'. We let MQ denote the least common multiple of the periods of H^{K{v)j A)unr (these groups are finite, and they are trivial for (i;, A^) =: 1). We let M = l^^ and let SM denote the M-th Selmer group of A over K. S±M is the kernel of cr — 1 and (7-1-1, respectively. UI is the ShafarevichTate group of A over /\ , and \11±M is the kernel of cr— 1 and cr-f 1 on U I M • If r G Z, r ^ 0, then Xr will denote a character (homomorphism to /IM) of the group S'sgn(r)M- L^t xo== the trivial character. Let Mi = /ord,(Mo) If t; G SP{K), then H\K,AT)V denotes the subgroup of H\K,AT) consisting of elements r whose projection to H^{K{V),A)T is trivial for all V G SP{K)y V ^ V. If a G Z, then a denotes a if a > 0 and 0 if a < 0. If r ' is a point of infinite order, then we let C denote the largest natural number which divides r' in A{K)/A{K)tor' We have: Theorem 10. Suppose thai r' is a point of infinite order, r G N , n > ord/(C) -f rord/(a6) -f- 2ord/(cd) -h 1, and M = l"^ as usual. Then there exist [1, A i , . . . , Ar] G ^ ^ ' ( 1 ) such that M ^ M I | M A I and characters X{-i)^ek ' S^_i)k,M -^ /^M, 1 < * < r, such that: (1) Let the integers 0 < a o , . . . , a r < n be determined by the condition that / " J Z / is the annihilating ideal of r'^ ^ (M) for 0 < j <
474
V. A. KOLYVAGIN
r. Then the integers rrij = n — aj satisfy the inequality rrik^i — ruk > —ord/(a6). Furthermore, there exist T'^ ^ (M^Mi) G H^(K,AM^MI) such that crj;^ ;^^(M2MI) = I'^^r'^^J^^{M^Mi), As usual, let ^Xo,...,x i^) denote the image of T'^^ ^ (M^Mi) in H^{K,AM) (under the homomorphism induced by multiplication by MMi: AM^MI -^ AM)Then we have rl
^^{M)£H\K,AM)i,
for
(2) Let Xxefc * H^{KiAM)x
v£SP{K),
v^Xo,...,Xj-
—^ f^M f>^ defined by setting
Frl, - 1 M
«) ^±tM M
Then x[^uk€k ^^ ^^^ restriction of x'/^uk^}., ^wrf ahcfq
X ( - I ) - . « , . . . , A . . . ( M ) ) = CM
i-iy-'^ek
where 0 < ord/(g(_i)fc-ifjt) < ord/(a6). (3) Letx[^i>^k-i,k' S(^i)k-i,M -^ f^M be the restriction o/X('_i)fc-ieJk' ^^^ ^-cM = Hom(5_eM, A^M) and S*j^ = Hom(kerXo/iM). We let x^ek e S*^j^ denote the restriction of x'^^ek' Then the period of X{-i)^€k ^^
Zk = 5'('_i)fc,jv^ /
\^{-\Ye{l^iy^{-\Ye{1^2y"-)
is equal to (period of ^jk)/^(-i)fccjb, where 0 < oxAi{q^_iYtk) — ord/(a6). Furthermore, x\.,)k,,
'
^ ^ C
thatr^-^'''^^{ahfcdZk
= 0.
{x^_,),^^^,yX^_,^,^^^,y^^).
so
Before proving the theorem, we use it to derive some corollaries. We observe that if /^^r' = /''r^ with r^ G ^(A"), then r ' is divisible by /"-«o = /^o in A{K)/A{K)toT^ Thus, r ^ l C i.e., mo < ord/(C). We show that if n > ord/(C) + ord/(c) we have mo > ord/(C), and hence mo — ord/(C). In fact, we can write T' = CT[ + e with e G A{K)toT and T[ G A{K). Then CT' = CCT[, Multiplying both sides of the last equality by /n-ordKc)+ord,(C)^ we find that /^-ord,(C)^/ ^ /n^(/^) Eexice, n~ord/(C) > ao, i.e., ord/(C) < n — ao = mo. We further note that when n > ord/(cC) we can take r'^M'^Mi) to be CfCT[, where Q = C/^^^'^^) G Z;.
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Let di denote afc, and let ^2 denote dfcd. We have: Corollary 1 1 .
If r^ is a point of infinite order, then w€ have
Proof. Theorem 10 with r = 1 impUes that l'^''-"^'d2S-cM = 0. Since mo < ord/(C), it follows a fortiori that d2CS-cM = 0. We have the natural exact sequence: 0 — ^ A{K)/M
—^ S M - ^ U I M - ^ 0.
Since 2III_eA/ C (cr — CJIIIA/ = {a — €)(P{SM) C <^(5_cAf)j it follows that 2d2CUI-eM = 0. Hence, 2d2Cin.^c = 0, and this implies, in particular, that III_e is finite, since III_c C I I I T for T = 2d2C^ and U I T is the image of the finite group ST and so is finite. This concludes the proof of the corollary. Corollary 12. If r' is a point of infinite order, then A{K) has rank 1, i n 15 a finite group, and Cdid22abcUI = 0. Proof. Applying Theorem 10 with r = 2, we find that l'^'^'^''d2S*j^ = 0. Since mo - mi > -ord/(di), it follows that /""^ divides I'^^di. Thus, Cdid2S:j^ = 0. Since S*j^ is dual to kerx^, by definition, we have Cdid2 kerxe = 0. We have the sequence of homomorphisms 0 —^ kerxc —^ SeM-^fJ'MSince xdCfCTi) = CM ? it follows that Xe(c^i) has period equal to /n-ord,(ca6)+ord,(^0. Thus, the quotient of 5eM by the subgroup generated by CT[ and kerxe is a cyclic group which is annihilated by cab. Since 2SM C 5 - C M + ScMi it follows that 2{cab)did2C annihilates SM/{CT\)' Hence A{K) = A{K)tov + Zr{. Furthermore, 2 I I I M C fpiS^M + S^-CM), and so 2{cab)d\d2C annihilates UI; in particular, III is a finite group. The corollary is proved. We define R±^i to be dimz//(IIIic)/, i.e., it is the rank of the /component of in±c. Let
{
max(2iZ_,/ + 1, 2R,i + 4),
if / 9^ 2,
max(2i2_e/ + 5, 2R,i -f 6),
if / = 2.
476
V. A. KOLYVAGIN
We have: Corollary 13. Let T' he a point of infinite order. Then [III] divides the natural number Q2 / TT^2il(0ord,(d2) I (ca6)2'*''"2/2^^
Proof. Let B!"^^^ denote the rank of S'j^^j^ (the number of cycUc terms, or equivalently, the dimension over Z// of the subgroup of elements of order /). We set r = max(2i21^j^^ — 1, 2/i*^). We first suppose that r > 0. We apply Theorem 10 with this value of r. We set ;^^ = 0 if \a\ > r. We let Yib denote the subgroup of S1_^YtM S^^^^^*' ated by X(-i)fce(ik+i)) X(-i)'^eifc, X^^^^^^^^y • • • • Then Theorem 10 implies that rank^5(*_j^fc^^/yA:j < max ^0, rank \^(-\YMI^(^^^ dition, r * - — * d 2 n
C y^~^.
- l ) - In ad-
We have 51,M = ^ ( 2 R * ~ - I ) ' '^^'^
rankf S'*,jv//^(2ii^—-i) ) " ^' ^^ similarly have 5*^^^ = ^^KM' Furthermore, \S^^MWM\ divides (p^-^^^^d^f'"{V^^-'^'d^f. Hence, Vp{StM + S'-CM)] divides €'^^2 ohc. Since 2 I I I M C (p{ScM + -S'-CM), it follows that [UIM] divides C'^dl''{ahc)2^''^-^f^^^. Obviously Ri^^ = rankS^eM < /2-e/+rank(5eMnkerv?). Sinceker <^ = yl(/^)/M = (^(/C)tor4Zr{)/M, and {T[Y = €(r{) -h ei with ei G A(/\)tor, it follows that for / 7^ 2 we have rank(5_eA/ H ker^) = rank(74(/<)tor H S-CM) < 1- Thus, ^-cM ^ B-^i + 1* If / = 2, then automatically RL^M ^ ^ - e / + 3, since rank^(/<)2oo < 2. Similarly R\j^ = rank(kerXe) < i^e/ + 2 if / ^ 2, and R*M < i?e/ 4- 3 if / = 2. Hence, r < R{1). If r = 0, i.e., S^eM = kerxe = 0, then Corollary 12 holds because 2IIIjoo c ^{S^cM + SCM) (by Corollary 11, we may assume that M is such that III/oo = <^(5A/))J and rrir-i + ord/(c) -h 1 (which holds because we have m^-i < mo -f (r — l)ord/(a6) by induction; n > ord/(C) -f rord/(a6) + 2ord/(cd) -f- 1 by assumption; and mo = ord/(C), as shown above) and the equality cr';^^ ^^_ (M) — 1^'"^T\Q A ^ _ I ( ^ ) ' '^ obviously follows that rj^'^,...,A.-i(^) ^ ^ Period r-ord,(c)
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{
477
MA _^/«(^i>-^r-i)^
i f r - l > 1,
2OTM'^)M^MU
i f r = 1.
Here n ( A i , . . . , A^_i) G Z^ is such that /»^('^i. ,^^-0 annihilates the groups which occur in the definition of Si (1) in terms of E5 ~ '(1) (see §1 following Theorem 3), so that [1, A i , . . . , Ar] G ^^[\l) if A^ G SP{2N' • n o < i < r - i ^i) and M A , _ I H ^ ^ ' - •'^'•-^^ I M A , (for r > 1). We further note that [1, Ai] G Sj^*(l) if Ai G SP{2N). Recall that we are considering only those prime divisors Xj of Q which remain prime in K. In particular, the Xj split in the ring class fields of K. Let 5(_i)r-ig C i/^(A',i4A/)(-i)'-ic denote the subgroup generated by ^ A o , . . . , A , _ i ( ^ ) ' a n d 1^^
5'(_l)r,=5(_l)r,M.
' Let F = / ^ ( ^ M J . We let S!^ C ^ n ^ ) ^ M ) ± denote the image of 5± under the restriction homomorphism res: H^{K^AM) —^ H^{V^AM)' Let W^(±) be the M-periodic abelian extensions of V corresponding to 5 ^ , let H{±) = G{W{±)/V), let W be the compositum of W(-^) and W{-), and set H = G{W/V). We define homomorphisms (p{±) : H -^ Hom(5j.,//M) by setting (p{±)('q) : s± H^ [s±{T]T]^),e^:M]M' We observe that ^ ( ± ) factors through H(±), and the restriction homomorphism H —+ //^(±) is surjective by Galois theory. We show that (p{±)(H) D 6 Hom(5^,/iM)- It suffices to show that (p{±){H(±)) D 6 Hom(5^,/im)) and, by duality, this is equivalent to showing that bX'{±) — 0, where -^'(ib) is the subgroup of 5 ^ orthogonal to (p{±){H{±)). We have: x{±) G X'{±) ^ VT; G H{±) [x(±)(w^), C^MU = 1If we use the relation x{±){T]r]^) = (dbcr + 1){X{±)(TJ))] the fact that {x{±)){H{±)) is a G(F/Q)-submodule of AM] the impKcation [Z{±), eipAflM = I => {2/d)Z{±) = 0, where Z(±) is any subgroup of A±M (since {2/d)AM C ( C M J ^ - M ) ) ; and the definition of 6, we obtain h{x{±){H{±))) - 0, and hence hx{±) - 0. We further note that, by definition, a annihilates ker(res), and thus a Hom(5±,/iM) C Hom(54-,/iAf )• Hence, 'ip(±)iH) DabS^, where 5 ^ = Hom(S'±,/iM), and rp(±) : H -^ S^ is the homomorphism for which 'ip{±){Tj) : s± —^ (<^(±)(7/))(res(s±)). We let y/_2 C 5'/*_jNr^ denote the preimage of y^_2 C 5'(*_i)r^jv/, so that ^l^iyJV-2 - Sl_^yeMl^r-2 = ^r• We let X{±) C SI denote the image of ^ ( i t ) . Let y;i2 = V-2 ^ ^ ( ^ ) ' where i/ is the sign of (-l)'*e. Let X"(—f/) = X(—I/) (where —i/ is the sign opposite i/), let X"(i/) = X{U)/Y;'_2^ and let ^ " ( ± ) : H -^ X " ( ± ) be the corresponding factorization of V'(i)- We show that 3T] £ H such that the period of V'"(i)(^)
478
V. A. KOLYVAGIN
is equal to the period of X'\±). Since V'"(i) ^^ surjective, this claim is obvious if either of the groups X'^(±) is trivial. Suppose that X " ( + ) and X " ( - ) are nontrivial, and let r ( ^ ) denote the period of X " ( ± ) , n(±) > 1. We let H'\±) C H denote the proper subgroup of H consisting of elements X for which H=*=)-V"(±)(ar) = 0. Then r)eH\ {H'{-\-) U ^ ' ( - ) ) satisfies the necessary condition. Here H \ {H\+) U H'{—)) is nonempty, since a group cannot be the union of two proper subgroups. Since a 6 5 | . C X{±) and (period of 5*^) = (period of 5(_i)r-i,^^) = in-ordi(c)^ j^ follows that (n — ord/(c)) > ord/(period of V'(""^)(^)) ^ ^ "" ord/(a6c), ord/(period of V'(^)(^) (rnod y/_2)) ^ ord/(period of Zr) — ord/(a6). We set x±r = V'(i)(^)- Let ^ G G{W/Q), g = 7/(7. By the Chebotarev density theorem, 3Ar G S'P(2Ar/(Ao • • • Ar_i)) such that Ar is unramified def
in VF and g = F r = Fr^(i;yQ(;^^), where t; is a prime divisor of W which divides Ar. Since Fr\K = it follows that M2 divides MA^. In particular, [1, A i , . . . , Ar] G E j ' ( 1 ) . We note that Ar splits in V/K, since Fr^ = id on V. Since r/r;^ = rjarja = g^ ^ it follows that X±r(s) = [«(w''), er^AfW =
V5 G 5±M-
S, e^:M
M
M
Let ni, n2 G N. We have the exact sequence 0
^ Al^i
)• Alni+n^
yAjnQ
^ 0.
Passing to cohomology, we obtain the sequence 0 - ^ A{K)ln,/r^A{K)ln,^n,
->
(8) Under the homomorphism H^(K,Aimr) —> H^{K,AM), KrO^'^) S^^s to / " " ^ ^ ^ ^ ( M ) = 0, by the definition of rrir (in what follows TTr will denote [Ai,...,Ar], and TTr-i will denote [AQ, . . . , Ar-i]). Thus, CT^^{1^'') = 0,
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and so there exists r^'(M^Afi/""'"•") € H^{K,Aiu2M^in-n,r) which goes to crJ.^(M^Mi) under the homomorphism H^{K,AM3j^^j^-,«r) —* H^iK,''AM^Mi)We then have: CT'^ (M^Afi/""'"'-) = /"*••<' ( M ^ M r •/"-"•O. and c r ^ , ( M 2 M i / ' - ' " 0 € i f H ^ ^ . ^ M ' M W — ) ( - i ) ' . If nx g N and x 6 H^(R, Afi), then we let (x) denote the image of x in H^{R,A)i^x. The equahty cii^{MMil'"^) = 1"'-T^'^{MMII'"^) implies that
K^(MMi)) = (cr;^(MMir'))
(9)
(since /•"(a;(/"'+'")) = (x(/"')) Vm,»ii e N). Let v € 5P(/i:) with (i;,Ai--Ar) = 1. Since (r;^(MAfi/'"%u)) 6 H\Kiv),A)unT (see Theorem 9 and the remarks following it), it follows that also {T"^{MMI,V)) 6 G H\K{v),A)unrBut « ( M ) ) = M i ( r ; ; ( M M i ) ) , and by the choice of Ml it annihilates £r*(/i:(f)'^)unr. Hence, (V;'^(M,t;)) = 0. Suppose that I < k < r, Afi C >l(/!'(Afc)). We let S'^^m : ^/-i -^ i/^(A'(Ajfe),yl);n, denote the isomorphism that is defined by setting
('''^AkV'^^))'"'.^' =
Frl, /»•
V5 G .4(A-(A0)/r^
s, e /'^i
{{Frl^ - l ) / ( / " 0 • ^(^^"(-^fc))//"' -^ ^/'^i is an isomorphism; ( , )ir.^^x^ gives a duality between A{K{\k))ll'^^ and /f^(A'(Ajfc),yi)/ni; since [ , ]/«i is a non-degenerate pairing, ^ ^ \ n i is an imbedding oi Aini in H^{K{Xk)i ^),n,; and finally, [Ajn,] = [A{K{\k))/r^] = [H\K{Xk),Ay,]). We let ^Ajfe,/~i • ^^(A(Ajb),^)/ni -^ ^/ni denote the isomorphism which is the inverse of S^^jm- If ri2 < ni, then ^Xk^i'^^i^^^"^^^) = ^ni-n2^^^^^^^2r), where z E //^^(A(Ajk), yl)/ni. Because of the compatibility of SxkJ^ ^^^ different m, in what follows we shall omit the index l^ and write simply £x^. From (9) it follows that £xr ((r;;(M, A,))) = Sxr ( ( c r ; ^ ( M / - ^ A,))). From (7) we have
£x. ((cr;^(Mr^ A,))) - (-ir^^^^j|^
=
Ml'' since c r ; ^ _ ^ ( M r O = ' " ' ' ^ • " ' ^ . i C ^ ' ^ ' O - Let /?rrfe(_i)r-i^jvi^, where /?r G Z/M. Then Fr'^ - 1
,Fri
-1
d^-^cr;;_^(M) 1 rf
= cM J M
i-l)r
480
V. A. KOLYVAGIN
Here we have used the relation [^M 5^M]A/ = ahcdqT^^y^i^^ {mod M). We further have:
CM
• H^i^c^? 2/?r(—1)'^'"^^ =
o = rcd^A.(K(M,A,))) = Frl - 1 Frl - 1
= ( - i)^e/'>-'"^+'"-. ^ ^ ^
cd r;;_, (A/) =
Since n > ord/(a6cd), it follows that ord/(/?^) < ord/(a6c). Then the previous equality imphes that m^-i — rrir >, —ord/(a6cd). We note that rrir < rrir-i + ord/(a6c(i) < (r — l)ord/(a6) -f ord/(afccd) -f- mo- Hence, n — rrir >_ ord/(cd), since by assumption n > ord/(C) + rord/(a6)-f 2ord/(cd) + l (then ord/(C) = mo). Using the equality 0 = l'^£xX{T'^^{M,\r))) as above, we prove that m^-i — m^ > —ord/(a6). We next have: cahdS\^{{T'^ (M, Ar))) = (-ire(r^—^a6)/?,rfe(^i).-i,M. Let ^A.(crf(<(M,X))) = = a!jferfe(_i)r-iejv/j where I < k < r, Xk £ Z/M. Let CM = CA/.A^J where yjfe G {Z/My. By global class field theory, the following orthogonality relation holds for all s £ SM' r
Y,{s,{T':sMAk)))M,x. = (i. kzzl
Hence, using the definition of £x^ and the relation njk=i Cm = 1, we obtain:
n
Jb =
l
Frl - 1•S, M
""
= 1.
e(_l)r_lfA/
J Af
H e n c e , x p ) t C (X(_.).,(~),X(_i).,(~),...). But
Frl - 1 = r^-'-"''a6(-ire/?,de(_i),-i,MSince ord;((3r) < ord/(a6c), we conclude that (I'"'-l-""-a6)oic
'^
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481
This completes the proof. We now show how a and b can be chosen. Let K' denote the compositum of K and K" = End(^) 0 Q. We let Gjoo denote the group G{K'\Aioo)/K"), where A/oo = Un^'**' ^^^ we let Gin denote the group G{K"{Ain)/K"). If End(A) = 0 is an order in the imaginary quadratic field K" (i.e., A has complex multiplication), then O has class number one (because A is defined over Q), and a choice of a projective system of generators el„ G Ai^ such that Ain = {0/r)eln and leln+i = el„ determines imbeddings p/n : Gjn <-^ (O/l^)* and p/oo : G/oo <^ 0(l)*, where
0{l) =
lhnO/r.
If A does not have complex multiplication, i.e., End(yl) = Z, then the choice of a projective system of generators eln,e2„ G Ajn such that Ajn = (Z//")el„ -f (Z//")e2„ and /ej„-}.i = ejn determines imbeddings p/n : Gin ^ GL2{Z/r) and p/oo : G/oo ^ GL2{Zi). We let ^,n and C?/« denote G{K\Ain)/Kf) and G(A"(^/~)/A^'). respectively. The restriction homomorphism gives imbeddings Qin «—)• Gin and ^/oo <^^ G/oo, so that ^/n and Qioo may be regarded as subgroups of G/n and G/oo, respectively. Here ^/n = Gin <=> / L ' (f /\"(>l/n); if this is not the case, then Qin is a subgroup of index 2 in G/n. Similarly, Qjoo = G/oo O" K^ (^ K"{Aioo)] if this is not the case, then Qioo is a subgroup of index 2 in G/oo. Let End(A) = Z. If/?/oo(G/oo) = GL2{Zi), then we set h"{l) = 1. If p/oo(G/oo) ^ GL2{7ii)^ then we define fc"(/) to be the smallest power Z*^, m > 0, such that p/oo(G/oo) D 7 -f /'"M2(Z/) (here M2(Z/) is the ring of 2 x 2 matrices over Z/, and I is the identity matrix). Now let End(yl) = O. We let D{0) denote the discriminant of O. If p(G/oo) = 0{l)\ then we set h"{l) = 1 if / / 2 or if / = 2 and either 2\D{0) or 2 remains prime in K"\ however, if 2 J[D{0) and 2 splits in /iT", then we set 6"(2) = 2. If p{Gioo) ^ 0(/)*, then we define 6"(/) to be the smallest power /^, m > 0, such that />/oo(G/oo) D l-f- / ^ 0 ( / ) . Next, we let a'{l^K) denote the smallest power /"*, m > 0, such that l^ annihilates H^{Qin^Ain) for all n. It follows from classical results in the case when A has complex multiplication and from results of Serre (see [19], p. 49) in the case when End(yl) = Z that h"[l) and a'{l^K) exist for all / and are equal to 1 for almost all /. We define the natural numbers h" and o!{K) as follows:
b" = X[ih"{l),a'{K) = Y[^a'{l,K). We show that we may take a = c{K)a'{K)[K'/K]. In fact, it follows from the exact sequence (8) that c{K)a'{K)H^(Qin^^n2, Aini) = 0 Vni,n2;
482
V. A. KOLYVAGIN
hence, [KyK]c{K)a\K)H\G{K\Ain,-,n,)/K),Ain,) (Themap res takes H^{G{K'{Ain^^n^)/K),
=0
Vni,n2.
Ain^) to H^{Qin^-\-n^,Aini) with
kernel H^{G{K'/K),A{K')in,), which is annihilated by [G{K'/K)] = [K'/K].) Let End(^) = Z, / 7^ 2. We show that ord/(a) = 0 if h"{l) = 1, i.e., if P]oo{Gioo) = GL2(Z/). In fact, in that case A/00 (A') = 0, since, if we suppose that 3e G A{K)u e / 0, we obtain: pi{gi) C X = | ( ^
J J |nGL2(Z//),
if we take eli = e. But this contradicts the fact that pi{Gi) = GL^ilii), since then [C?/] > [G/]/2 = / ( / - l ) 2 ( / - h l ) / 2 > / ( / - 1 ) ^ = [X]. Furthermore, either pjn{Qin) = /?|n(G/n) = GL2{Z/r)y or else pin{Qin) is the subgroup of GL2{Z/l^) consisting of matrices with discriminant in ((Z//)*)^ (see [12], §4); in either case pin{Qin) 3 0 =: < I
J , a'"^ ~ V ' ^^^^^ implies
triviality of H^{Gin^Ain) if we use the spectral sequence 0 - ^
H\gin/e,Afn
= 0 ) — ^
H\Gln,Aln)
— ^
//^(G, A/a)
~ ^
Q.
From the analogue of Lemma 2 of §4 of [11] that is obtained by replacing mi/ (in the notation of [11]) by ord/(6") and Qjm (denoted Gim in [11]) by Gint (the proof of the modified lemma remains exactly the same) it follows that 6' can be taken equal to d^D{End{A))b'\ where d' = d if End(A) = Z and d' = 1 if A has complex multiplication. Hence, 6 = {2ld)d'D{Exid{A))h". In particular, if / ^ 2, (/, Z)(End(A))) = 1 (which is always the case if A does not have complex multiplication, since then End(A) = Z and D(Z) = 1), and h"{l) = 1, then ord/(6) = 1.
REFERENCES [1] Z. I. Borevich, I. R. Shafarevich, Number Theory, Academic Press, 1966. [2] K. Ribet, "A modular construction of unramified p-extensions of Q(/ip)", Invent. Math., 34 (1976), 151-162. [3] A. Wiles, "Modular curves and the class group of Q(Cp)") Invent. Math., 58 (1980), 1-35. [4] B. Mazur, A. Wiles, "Class fields of abelian extensions of Q", Invent. Math., 76 (1984), 179-330.
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[5] A. Wiles, "On j>-adic representations for totally real fields", Ann. Math., 123 (1986), No. 3, 407-456. [6] F. Thaine, "On the ideal class groups of real abehan number fields", to appear in Ann. of Math. [7] K. Rubin, "Global units and ideal class groups". Invent. Math., 89 (1987), 511-526. [8] K. Rubin, "Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication", Invent. Math., 89 (1987), 527-560. [9] J. Coates, A. Wiles, "On the conjecture of Birch and SwinnertonDyer", Invent. Math., 39 (1977), 223-251. [10] B. H. Gross, D. B. Zagier, "Heegner points and derivatives of L-series", Invent. Math., 84 (1986), 225-320. [11] V. A. Kolyvagin, "Finiteness of £'(Q) and m(jE;, Q) for a subclass of Weil curves", Izvestiya AN SSSR, Ser. Mat., 52 (1988), No. 3, 522-540. [12] V. A. Kolyvagin, "On the Mordell-Weil group and the Shafarevich-Tate group of Weil elliptic curves", Izvestiya AN SSSR, Ser. Mat., 52 (1988), No. 6. [13] D. B. Zagier, "Modular points, modular curves, modular surfaces and modular forms", in Arbeitstagung Bonn 1984, Springer Lecture Notes in Math., 1111 (1985), 225-248. [14] G. Stevens, Arithmetic on Modular Curves, Birkhauser, 1982. [15] B. Mazur, H. Swinnerton-Dyer, "Arithmetic of Weil curves", Invent. Math., 25 (1974), 1-61. [16] B. Perrin-Riou, "Points de Heegner et derivees de fonctions L p-adiques". Invent. Math., 89 (1987), 455-510. [17] S. Lang, Elhptic Functions, Addison-Wesley, 1973. [18] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1982. [19] M. I. Bashmakov, "The cohomology of abelian varieties over a number field", Uspekhi Mat. Nauk, 27 (1972), No. 6, 25-66 [English translation: Russian Math. Surveys, 27 (1972), No. 6, 25-70].
Translated by Neal Kobhtz
V. A. Kolyvagin Steklov Institute Vavilova 42 Moscow, U.S.S.R. 117966 GSPl
Descent for Transfer Factors
R. LANGLANDS and D. SHELSTAD* Dedicated to A. Grothendieck on his 60th Birthday
Introduction In [I] we introduced the notion of transfer from a group over a local field to an associated endoscopic group, but did not prove its existence, nor do we do so in the present paper. Nonetheless we carry out what is probably an unavoidable step in any proof of existence: reduction to a local statement at the identity in the centralizer of a semisimple element, a favorite procedure of Harish Chandra that he referred to as descent. The principal difficulty is to show that the transfer factors of [I] for the original group G are compatible with those on the connected centralizer Ge of the semisimple element e. After some preliminary explanations in Section 1, the compatibility is stated as Theorem 1.6.A. In Section 2 we show that this compatibility indeed reduces the problem of existence to a local problem at the identity on the groups Ge , and in passing we note some other applications. The remaining four sections are devoted to the proof of Theorem 1.6.A. The transfer factors are defined in a rather elaborate manner as the product of five factors that mix group-theoretical data with Galois cohomology. The first steps are to reduce to quasisplit groups and then to discard two of the five factors, leaving only three, one of which is defined in a simple fashion, and two of which involve group-theoretic and cohomological data. It is these two that are difficult to compare for G and G^. The principal tools are the two comparison lemmas of Section 3. The contributions to the factors are labelled by orbits of the Galois group in sets of roots, and the first use made of the comparison lemmas is to deal in Section 4, and rather quickly, with all orbits except those lying outside both Ge and the endoscopic group. This leaves a rather concise but still far from trivial statement that is proved partly by an analysis of the structure of semisimple groups and partly by explicit cohomological calculations. The structural analysis is possible only after the critical lemma of Section 5.1 has been established. •Partially supported by NSF Grant DMS 86-02193
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This lemma allows us to introduce an inductive component into the argument, and then to assume that both the element e and the datum s defining the endoscopic group are essentially of order two, and moreover, that all roots are of the same length. This done, the burden of the rest of the proof is carried by expUcit arguments with the constructs of local class-field theory. They all appear in Section 6. We cannot hope that the groping, pedestrian style of the paper will appeal to Grothendieck, for it lacks the force and penetration that he achieved so readily, like Nietzche's Philosoph der Zukunft, erfinderisch in Schematen, mitunier siolz auf Kaiegorien-Tafeln. Nonetheless, it is a great pleasure for us to express our admiration of his magnificient contributions to the mathematics of our time.
§1. Descent Principles 1.1. Notation 1.2. Images of semisimple elements 1.3. The function ^f 1.4. Descent for endoscopic data 1.5. Descent for ^^ 1.6. Descent for transfer factors 1.7. Final formula x §2. Consequences 2.1. Local transfer 2.2. A characterization lemma 2.3. Reduction to local transfer 2.4. Equisingular transfer 2.5. Regular transfer 2.6. Archimedean transfer §3. Comparison Lemmas 3.1. Reduction to the quasisplit case 3.2. Remarks and notation 3.3. First Lemma of Comparison 3.4. Second Lemma of Comparison 3.5. An apphcation §4. Analysis of 6, b and a Reduction 4.1. Galois action 4.2. Calculation of a coboundary 4.3. Explicit form 4.4. Root types 4.5. Analysis of ^1,^2 §5. Final Reductions
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5.1. Introduction 5.2. Beginning of proof of the critical lemma 5.3. The term 0^2*^^ 5.4. Construction of ^ and <^, 5.5. Reducing the dimension of Gder §6. The Order-Two Case 6.1. Introduction 6.2. Liftings 6.3. Some coboundaries 6.4. Remaining steps 6.5. Symmetric orbits 6.6. Final calculations References
§1. Descent Principles 1.1. Notation We follow closely the notation of [I]. In particular, G is a connected reductive group over a field F of characteristic zero, now assumed local. As in [I, Sect. 1.2.] G*, V* are quasisplit data and ^G = GxWp is the L-group. To conserve notation we fix an F-splitting ( 5 , T, {Xav}) of G and given a class of endoscopic data choose a representative (H^Ti^s,^) with i '.li*-^^G as inclusion and s an element of T . It is also convenient to fix an F-splitting (5^,7if, {y/?v}) of H = Cent(s,G)^ and assume that BH = BnH , TH = T . For the moment we refer directly to [I] for the definition of the factor A . Measures also remain as there. If e G G(F) is semisimple we choose an invariant differential form of highest degree on Cent(e, G)° in order to fix a Haar measure on the F-rational points of this group. We require that differential forms on inner forms be obtained by transport. 1.2. Images of semisimple elements For e in G the identity component of Cent(6,G) will be denoted Ge. If e in G{F) is semisimple then, following [Kl], the stable conjugacy class of € is {9^^eg:ga{g)-^ eGc^aeT}, where F = G a l ( F / F ) . If Cent(e,G) is connected then this coincides with the set of F -rational points in the conjugacy class of e in G{F). In general, an F-rational e' = g~^eg is stably conjugate to e if and only if Intflf : Ge' —* G^ is an inner twist. If G is quasisplit over F then there is an e' stably conjugate to e such that G^i is quasisplit over F
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there is an t' stably conjugate to e such that G^/ is quasispUt over F [Kl, Lemma 3.3]. We now generahze the notion of image from [I, 1.3] (see also [K2]). It is convenient to use the notation 7/f , 7 , 7^ when 7/f is strongly G-regular and e^ , e, 6^ in general. Suppose then that e^ lies in the Cartan subgroup TH{F) of H{F). Then we call e^ a TH-image of e^ in G{F) if for some admissible embedding TH —)• T of Tff in G* carrying e^ to, say, e there exists x in G* , or just as well in G^^ , such that tpx = lntxoxj; has the properties that ^^(e^) = 6 and that both TG = fpx^{T) and ip^ : To -^ T are defined over i^ . In varying TH we obtain all images of e^ . Observe that [K2, Lemma 10.2] shows that all images of a given e^ are obtained by simply fixing one image e^ , if it exists, and then taking all TH -images for some TH fundamental in ^e . 1.3. T h e function ^f Recall that to define transfer factors for (G, H) we may need to pass to a central extension of H . Call a central extension H of H admissible if it is attached to a 2: -extension of G as in [I, 4.4] (although a wider class of extensions could be used [K-S]). The sequence 1 —^ Z{F) —^ H(F) —> H{F) —^ 1 is then exact, where Z is a central torus in H , and combinations of orbital integrals of functions on G(F) are to be matched with those of functions on H(F) that transform under Z(F) according to a certain character A [I, 4.4]. Suppose €fj is semisimple in H(F) and en lies in its preimage in H{F) . The factor A{'yH,7G) has been defined for JH strongly G-regular in H{F) , by which we mean that the image JH of 7H iii ^i^) i^ strongly G-regular. We shall investigate the behavior of ^f{7H)
= ]^A(7if,7G)^(7G,/) 7G
for jH near CH First, following [I, 4.3] it is easy to see that if e^ is G-regular, but not strongly so, then ^f extends continuously to in and that this extension is in fact smooth at €H • Second, if e^ is not the image of any semisimple element in G{F) then no strongly G-regular element in H^ (F) can be the image of an element in G{F) and so ^f vanishes on the strongly G-regular elements in Hi^j^F). In particular <^^ vanishes for all 7ff in a neighborhood of en in H{F). We may then assume that e^ is an image of an element e^ in G{F) . There is an e'n = h~^e^h stably conjugate to e^ such that H^^ is quasisplit over F . If e^ is a TH -image of e^ then we may multiply h by an
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element of H^t to assume that the homomorphism Int/i~^ : TH -^ H^i is defined over F , that is, that h G ^{TH) - Then h acts on the preimage TH of TH in iJ as an element of 2t(Tif) and [I, 4.1.B] implies that
^ f (/^-'T/f/^) = ^ f (Tif ) for all strongly G-regular ^H in TH{F) . Thus we may replace e^ by e'jj and assume from now on that H^ is quasisplit over F . Then H^ is an endoscopic group for G^ as we now explain in detail. We sometimes denote it by H^. 1.4. Descent for endoscopic data We start then with semisimple e^ in H{F) an image of e^ in G{F), and He quasisplit over F. Choose TH such that e^ is a TH -image of €Q . Let e be the image of e^ under some admissible embedding TH -^ T of TH in G* . An argument as in the last paragraph allows us to choose the embedding so that G* is quasisplit over F. We will see that these choices are of no real importance, the essential data being e^ , e and e^ and thus H^ , G* and Ge . To the endoscopic data ( ^ , ? i , s , ^ ) we shall attach an extension H^ of Wp by iff and an admissible embedding (^^ \ Tic ^-^ ^G^ such that He J Tie J s and ^e yield endoscopic data for Ge . Further choices will be made, for example to specify Ge , but again they will not affect the isomorphism class of the endoscopic data. Moreover, all endoscopic data for Ge will be so obtained up to isomorphism. Suppose that BH D TH i^ H and J3 D T in G* are Borel subgroups for which the already chosen TH -^ T IS the attached embedding. Also suppose X G G*c is such that i^xi^a) — ^J ^^^^ both To = ip~^{T) and i^x ' TG -^ T defined over F . Note that G* and x/^x serve as quasisplit data for Ge . G
The embedding TH-^T^^^TG
is by definition dual to a diagram Tif ^ - Tif = T ^
f - ^ fG.
This diagram allows us to identify R{G,TGy R{Ge^, TGY with a subset of R{d, T).
with R{G,T)
and then
Infixing L-data (GePhPt) and ^Ge = Ge^^Wp for Ge or G* we may assume that Ge contains T and that R{Ge,T) coincides with R{GeiTGY as a subset of R{G,T) . We set Be = BnG* and let Be be the Borel subgroup of Ge generated by T and the B -positive roots in T in Ge . The embedding f -^ T of f in Ge provided by Be and Be will be
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identified with T -^T above. The isomorphism TQ —^ T —> T embeds TQ in Ge and extends to an admissible embedding of ^TQ in ^G^ (see, for example, [I, 2.6]). The image, again denoted ^TQ , is independent of the choice of extension. ^ The element s lies in T and thus in G^. Moreover H = Cent(s,G)° . We may take the dual He of He as the subgroup Cent(5,Ge)° of Ge; He is normalized by ^TG • We define He to be the subgroup of ^G generated by He and ^TG , and ^e to be the inclusion map. Observe that there is clearly a split exact sequence i-^H,^—.ne
—
WF—^i
and that {He ,We,s,^c) is a set of endoscopic data for Ge . We also identify the embedding TH -^ T given by BH H He and Bed He with Tjj —>• T above. Then TH —*- T is an admissible embedding for both (GjH) and {Ge ,He ) . Moreover any admissible embedding of a maximal torus of He in G* is admissible as an embedding of a maximal torus of H in G* and carries e^ to e . It remains to examine the effects of our choice. Suppose first that B and BH are changed but that TH —^ T remains fixed. Then {Ge^pe) is replaced by a pair {G'^^p'^) , F-isomorphic to it under a map that carries R{Ge,T) to R{G[,T) and the image of ^TG in ^Ge to its image in ^G[ . This isomorphism further fixes s and carries He and Tie to the new H'^ and H[ . Thus we obtain isomorphic endoscopic data for Ge • With T fixed, the choice of TH and tpx does not affect endoscopic data. Now suppose we replace TH -^ T hy TH —^ T. Then e^ lies in both TH,TH and e in both T,T. We may assume that the new Borel subgroups are obtained from BH and B by conjugation in C e n t ( 6 ^ , ^ ) and Cent(6, G*) . Then again the new data are seen to be isomorphic. Note that if TH —>• T is admissible for {Ge ,He ) then we may use conjugations in He and G* , and the data are unchanged. Finally it is straightforward to check that the choice of {H^Ti, s,^) within its equivalence class does not affect the class of {He , We, s,^e) among data for Ge . Moreover from any class of data for Ge we can recover s £T , H = Cent(s,G)° , H = (H^^TG) contained in ^G and e^ semisimple in H{F) such that {He ,We,s,^e) lies in the class. 1.5. Descent for ^ j ^ Continuing with e^ , e and e^ , we assume that endoscopic data have been fixed once and for all by the choices of the last section. Since these choices will not be mentioned again we reserve no notation for them. In particular, TH —^ T will be an arbitrary embedding of TH in G* which is admissible for (G* jHe ) . To spare notation further we assume that G* ,
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^ are quasisplit data for Ge - If then e,, is a TH -image of e^ there is an X E (G*)sc such that TQ = i^x^{T) and rp^^ :TG -^T are defined over F. For jff strongly G-regular in the preimage of TH{F) in H{F) we calculate ^^{JH)
^
To
where the sum is over representatives JQ for the G{F) -conjugacy classes in the stable conjugacy class of the image JQ of JH under TH-^TH—^T—^TG.
We write JQ as W'^^JQW , w G 2)(Tc?), and now describe a choice of representatives for 2 ) ( T G ) .
Consider first the stable conjugacy class of e^ in G{F). Since Cent(eQ, G) may be disconnected we pass to a z-extension of G and pick e G G{F) mapping to e under G -^ G. Suppose that {ej = wj^ewj : 0 < j < n} is a set of representatatives for the conjugacy classes in the stable class of e, with WQ = 1. Let Wj be the image of Wj in G{F), €j = wJ^c^Wj and Gj — Cent(ej,G)° . Notice that the e^ need not be distinct. We use (G*, t/'j) , where V'j = V' olnti^j , as quasisplit data for Gj . Define a subset S{^H) of {0,1, • • • , n } by j G S(TH) if and only if Cj has e^ as TH -image relative to {Gj^H^ ) , that is, if and only if there exists hj G Gj such that Int hjw~ maps To to Gj over F . Then fix some such hj and set Wj = Wjhj , 7} = W^^TGWJ . Passage to G shows that {wjw' : j G S{TH) and t/;' a representative for S)(7},Gj)} is a set of representatives for S) (To, G ) . Thus ^f(jH) is equal to n
(1.5.1)
Yl E
where JQ = W'^JQWJ
^o')^(Ti/. ^'-VG«^', fmw^-'iw. f)
, 0 < j < n , and 6 is the characteristic function of
SiTjt). From the well-known construction of Harish-Chandra [HCl, Sect. 22; HC2, Part VI] we can find p G C^{Gj{F)), 0 < j < n , such that
^6J) = <^{6J^)
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for all regular semisimple 8 in some neighborhood of EJ in Ge^iF) . It remains to relate transfer factors for (G, H) to those for (Gc j H^ ) . 1.6. Descent for transfer factors To conserve notation 7i will be assumed an L-group, but He must of course remain arbitrary. Then He will be an admissible central extension of He (recall 1.3) and ejj will be an element in the preimage of e^ in jfle (F) . We suppose that e„ is a TH -image as well as a TH -image, and take JH > 7H ^^ ^e (F) with strongly G-regular images JH ^TH , 7H e TH in He^{F). Then the factors
for {G,H)
A =
A{jHnG]lHnG)
Ae =
A{JH^1G]^H^1G)
and
for (Ge , He ) are defined and non-zero whenever JH,7H are images with respect to (Ge ,He ) of JGIJG in Ge (F). That will be our assumption on 7 G J T G throughout this section. Let 0 = A / A e . Then 6 is naturally a product 0161161626^ , the factors corresponding to those of A and A^ [I, Sect. 3]. Keeping TH and TH fixed we consider ^H,7H ^^^r in with JG^JG both near e^ . We will see in 3.1 that 61(7/^,7G;7/f , 7 G ) = 1. A glance at the remaining factors convinces one that 6 extends continuously to i^Hi^a'i^Hi^o) taking a nonzero value there. The extension is then seen to be smooth. Theorem 1.6.A. lim6(7/f,7G;7iy,7G) = 1 iH^in
—'^H
7G,7G
—'^G
The proof will occupy Sections 3 to 6. Suppose F is nonarchimedean. Then the theorem says that 6(7if,7G;7/f,7G) = 1 for JH^IH near in and 7 G J 7 G near e^ . Thus for the absolute factors A(7if,7G), A ( 7 ^ , 7 G ) for G and AeijHHG), A , ( 7 ^ , 7 G ) for G, [I, 3.7] we have A(7ff,7G) _ A(7ff,7G) AeijH^lG) ~ Ae0H.7G) '
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and so we may write A(7^,7G)
-
cAeijHno)
for jH near ejj and yo near e^ , where c is a constant. Observe that for jH near CH the factor AeijHijG) depends only on JH and JQ (see [I, 4.4.A] and 3.5). We emphasize that the assertion of the theorem is that c is independent of the Cartan subgroup TH containing JH • That this is crucial for the transfer of orbital integrals (and therefore characters) will be seen in Section 2. In the archimedean case c is a function. This however presents no difficulties in our applications (2.4, 2.5). 1.7. Final formula For the pair (Gj ,Hc ) we write Cj in place of c . Then n
(1.7.1)
^fi7H) = J2'J^p('f^^
i=o for strongly G-regular JH sufficiently close to CH in He^{F). Note that the characteristic function S(j) of (1.5.1) has disappeared because by definition ^ffijn) = 0 for j ^ S{TH) (recall 1.3).
§2. Consequences 2.1. Local transfer We say that {G,H) admits A-frans/er if for each f £C^{G(F)) there exists f^ e C^{H{F),X) (notation of 1.3) such that / and / ^ have A-matching orbital integrals, that is, (2.1.1)
^''{JHJ")
= Y,A{jH,7GmyG,f)
for all strongly G-regular JH in H{F). It H is the quasisplit form of G, so that A is a constant, we often refer to stable transfer rather than A-transfer. Suppose that F is nonarchimedean. Then for JH near 1 the factor A(7if,7G) depends only on the image JH of JH in H{F) (see [I, 4.4] again); so we may denote it instead by Aioc(7jy,TG) • We say that ( G , / / ) admits local A-transfer at the identity if for any / G C^{G{F)) we can find / ^ G C^{H{F)) such that
(2.1.2)
^''{JHJ'')
=
J2A,OC{7H.1GM7GJ)
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for all strongly G-regular yn near 1 in
H{F).
2.2. A characterization lemma Throughout 2.2 and 2.3 we assume that F is nonarchimedean. Suppose that ^^* is a stably-invariant function on the regular semisimple elements of G{F) that is compactly supported modulo conjugation, viz., that vanishes along all conjugacy classes of regular semisimple elements that do not meet some fixed compact subset of G{F). Then we call $^* a local stable orbital integral if for each semisimple element e in G{F) there exists fe e C^{G{F)) such that ^ ^ ^ ^ T ) = ^^^ilJe) for all regular semisimple 7 near e. Lemma 2.2. A. Let G be a quasisplit group. If ^®* is a local stable orbital integral on G{F) then there exists f G C^{G(F)) such that
for all regular semisimple y in
G{F).
Proof. By a simple passage to a z -extension ([Kl]) we reduce immediately to the case that the derived group of G is simply connected. If c is semisimple in G{F) we denote by Z^ the center of Ge and by Z'^ the set of c' in Z^ at which DQ/DQ^ does not vanish. For e' in Z'^ we have Ge' = Ge and Z^' = Z, , while if e' e Z^ - Z[ then Ge'^G^, so that dimGc/ > dimGc, and Z^f^Ze. Notice that the group Ge is Cent(Ze,G). Thus if g £ G{F) then g^^eg is stably conjugate to c if and only if Int^f""^ : Z^ —> Zg-i^g is defined over F , that is, Ze is stably conjugate to Zg-i^g , There are only finitely many stable conjugacy classes among groups Z^. We label representatives Zo^-^Zr for these classes so that ZQ is the center of G , the group G{£) = Cent(Z^,G) is quasisplit over F for each e (using [Kl, Lemma 3.3]), and so that dimG(^) < dimG(ife) if ik < ^ . Notice that if Zi = Z^ then G{i) = Ge and Z'^ = Z',. It is sufficient to show for each ^ = 0, • • , r that if a local stable orbital integral $^* vanishes on the regular semisimple elements in a neighborhood of U Zk{F) then there exists ft G C^{G{F)) such that (2.2.1)
^^^(7) = ^^*(T,/^)
for all 7 near U Zje{F), for if e' is stably conjugate to e in this set then k
any regular 7' close to e' is stably conjugate to a 7 close to e because Ge is quasisplit. Thus (2.2.1) implies that
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We then proceed inductively, replacing the original ^^^ by
passing to Zi , and so on. Because Gder is simply connected, stable semisimple conjugacy classes are labelled by orbits of the Weyl group in a fixed Cartan subgroup T over F. Of course the orbits lie in T{F), and not all such orbits label stable conjugacy classes. Suppose we are given a Galois-in variant metric on T{F) , one such orbit f = {^Oj^i? • * * )^«} ? 3,nd a 6 > 0 . Then the set of all g G G{F) such that the semisimple part of g is conjugate in G(F) to an s such that \s — ti\ < 6 for some i is an open subset of G{F) that is stably invariant. Multiplying a given function / by the characteristic function of such a set, we concentrate its stable orbital integrals on the set X{T,6) of regular 7 such that 7 is conjugate to s with |s —ti| < 6 for some 6. Thus a stable conjugacy class that meets Z'^(F) will intersect Z[{F) in a finite set { e o , - - , 6 ^ } , and mapping Zi into T over F we send the elements e = eo, • • • , e^ to to, • • ,^r • We complete this set to a full orbit r = {^or * )^»}- Since G^ is quasisplit, it is clear that if 6 is sufficiently small then any stable conjugacy class in X{T^8) meets any given neighborhood of e in Ge. In verifying this, we may suppose that ZiCT so that to = e, A stable conjugacy class in X(r, 6) is then represented by an s E T{F) such that |s — e| < 6 and such that for any a G G a l ( F / F ) there is a u^^ in the normalizer of T in G{F) satisfying (2.2.2)
cr{s) = u'^sua .
Since \(T{S) - (T{e)\ = \(r(s) -€\ = \u-;^su^ - e\ , we can choose 6 sufficiently small that (2.2.2) forces u^ to centralize e. Consequently, by Steinberg's Theorem, s determines a stable conjugacy class in Ge{F). Moreover, if 6 is small enough this class must meet the given neighborhood of e in Ge. By assumption, there is an fi such that ^^^(7) = $^^(7,//) for 7 in a neighborhood of e in G^, and we conclude that this relation is valid on all of X{TJS) . A simple partition-of-unity argument completes the proof. Observe that we could have dispensed with the introduction of the groups Zjt, if we had wished to do so. 2.3. Reduction to local transfer Theorem 2.3.A. Suppose all pairs (Ge , H^ ) have local Ae -transfer at the identity. Then (G, H) has A -transfer.
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Proof. We first observe that if G is replaced by a z-extension G the hypothesis of the theorem remains valid. Further it is then sufficient to prove the theorem for G. Thus we may as well assume H. an L-group. By Lemma 2.2.A we have then just to show that
defined so far for 7/f strongly G-regular in H(F) , is a local stable orbital integral on H{F). We shall use the descent formula (1.7.1). Observe also that local Ac-transfer at the identity for (G^ ,He ) implies local A^transfer at e^ for the same pair (see Lemma 3.5.A). First, if Cfj is regular semisimple in H{F), so that H^ is a Cartan subgroup of H , then local transfer at e^ for (Ge , H^ ) implies by (1-7.1) that ^J extends smoothly to e^ . Thus we have ^f defined on all regular semisimple elements of H(F). It is further locally constant, stable, and compactly supported modulo conjugation. For general e^ we use (1.7.1) to obtain / ' G C^{H^ (F)) and thus / " € C^{H{F)) such that
for jH near e^^ . Thus ^9 is complete.
is a local stable orbital integral and the proof
2.4. Equisingular matching We call Cfj and e^ equisingular if He is an inner form of Ge , that is, if 6^ is (GJ H)-regular in the sense of [K2]. Assume that f,f^ have A-matching orbital integrals. For simplicity, we shall suppose that the derived group of G is simply connected so that neither H itself nor He need be replaced by a central extension in the matchings. Note also that Cent{e^^H) is connected [K2, Lemma 3.2]. Following [K2], and by the homogeneity of germs, we may expect a stable combination of the integrals of f^ along the conjugacy classes in the stable class of e^ to match with some suitable combination of the integrals of / along the classes in the stable class of e^ . We shall show this is true and so verify some conjectures in [K2]. The first step is to define a factor A(e^, e^) . Let e^ be a TH -image of €Q . Then we set ^(^HI^G) - lim A(7i/,7G) , where the limit is taken as JH -^ ^H ^^ TH{F) and JG —^ ^a ^ ^^^ 7^ ^^ an image of JG • Suppose F nonarchimedean. If we consider also JH,7G with jH near e^ in another Cartan subgroup TH{F) of He (F) then
DESCENT FOR TRANSFER FACTORS Ae(7ff,7G ] 7H^7G) = 1 since H^ Thus Theorem 1.6.A asserts that
497
is the quasisplit inner form of G^ .
6 ( T ^ , IG ; 7H, 7G) = A(7i/, 7G ; 7H, 7G) =
A{7H,7G)/^{7H^7G)
= 1 if 7Hi7H near e^ are images of 7GI7G near e. In other words, A(7if,7G) is a constant independent of the Cartan subgroup containing 7H and A{efjye^) equals this constant. If F is archimedean we still have Ac(7/f,7G ; 7HI7G) = 1 but Theorem 1.6.A asserts only that lim A(7if,7G)/A(7iy,7G) = 1 . We conclude nevertheless that A(e^,eQ) is well defined, that is, independent of the choice of TH • We write 0(6^^/) for the integral of / along the conjugacy class of e^ , keeping in mind our convention for measures (1.1). A sign e{G) is defined in [K4]. If we sum over representatives e'^ for the conjugacy classes in the stable conjugacy class of e^ then
is a stable distribution [K3, S5]. Lemma 2.4.A. 5Mppose €„ is {G, H)-regular
(2.4.1)
0^\e,J^)
then
= ^6(G.JA(e^,eJO(£,,/) 'a
where the sum is over representatives e^ for the conjugacy classes in G{F) equisingular with e^ Proof, (i) F nonarchimedean. Suppose 7/f is near c^ in a fundamental Cartan subgroup TH{F) of H^^{F). Then by descent (1.7), YJA{7H,7G)^{7GJ) IG
coincides with
J]A(7ff,Ti)^^'(7i,/^') i and thus with
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The zero-degree term in the ShaUka germ expansion of this expression is A^A(e„,e,)e(G,)/^(e,) J where A is a constant depending only on e^ (see [K3, §3]). Apart from A , this is the right side of (2.4.1). At the same time, ^A(7/y,7G)^(7G,/) 7G
coincides with ^^^{JH,/^) which has AO^^(e^,/^) as zero-degree term in its germ expansion. So (2.4.1) is proved. (ii) F archimedean. Descent again yields ^ A ( 7 ^ , 7 G ) ^ ( 7 G , / ) = ]^A(7^,7G)^'*(7G,/-^') • 7G ;
In place of germ expansions we use Harish-Chandra's limit formula [HC3, Lemma 17.5]. Again TH is taken to be fundamental in H^ . We write IH € TH{F) as Cfj expX and multiply each side of the equation
by n(a(e„)e«(^)/2-e-«(^)/2), the product being over all positive roots of TH in H . We then apply the operator w^ = n'i/cr ? the product being now over positive roots in H^ , and take limits as X —^ 0 . As a first step we obtain on the right side X ; A(e„,e,)lim w,^{U'e'^(^y'
- e-«W/2)$«'(T,.,/i)
(see [S5] for the explicit form of A(7if, 7^), especially the term A2(7ff, 7^), and [W, p. 371] for a similar calculation). We calculate this new hmit by means of Harish-Chandra's formula as in [S5, §2.9], using the results of §37 of [HC3] to keep track of constants. The contribution of the right side is then ^A(e„,e,)Ae(G,)/^(e,) = A^^A(e„,e,)e(G,)0(e,-,/) , A being a constant depending only on He . The left side contributes AO^*(e^, f^) , and so the lemma is proved.
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Observe that from our product formula for A(7if, 7G) [I, §6.4] we obtain a product formula for A(e^,eQ) , as conjectured in [K2, §6.10]. 2.5. Regular matching Suppose 6G is regular in G ( F ) , that is, that Ceni{6G^G) is of minimal dimension. Let 6G = ^Q'^G be the Jordan decomposition; both e^ and UQ belong to G{F) . Assume e^ has image e^ in H{F). As usual, we take /fe quasisplit over F and then choose UH regular unipotent in iJc ( F ) . Thus 8H = ^ffUff is regular in H{F) . We shall use descent and the regular unipotent matching of [I, §5.5] to match integrals over the classes of Sff and SQ . For simplicity of notation we assume W,We are L-groups and that Cent(e^,/f), Cent(eQ,G) are connected. The stable conjugacy classes of SH^SG are then the F-points in their F-classes. Set ^®*(<5if , / ^ ) = Yl^i^Hif^)' where the sum is over representatives 6ff for conjugacy classes in the stable conjugacy class of 6H • Then descent to He and Theorem 5.5.A of [I] show immediately that
lim
^'\6HJ'')=
DeJjHWilHj'')
so that f^ —> ^^^{SH,/^) is a stable distribution. At the same time we set A(e^,e^) =
lim
A{^H,1G)/^eilHno)
,
the limits taken as in 1.6, and
in the notation of [I, 5.5], with A, A calculated with respect to (G^ , ^ e ) . Then we define Suppose / and f^ have A-matching orbital integrals. Descent and the regular-unipotent matching also imply easily that lim
D,JjH)^A(jH,jG)H7G,f)
= Y,Ai6H,6'a)^S'a,f)
,
and so we have the matching ^'\S„,f")
= ^AiSH,6'a)H6'a,f)
.
Here we have dealt with both the nonarchimedean and archimedean cases. We also see that if SH = C^JUH is regular in H{F) and e^ is not an image then
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R. LANGLANDS and D. SHELSTAD
Finally we remark that if / G C^{G{F)) is supported on the full regular set of G{F) then we can find / ^ G C^{H{F)) supported on the full regular set of H{F) with A-matching integrals. 2.6. Archimedean transfer Suppose F is archimedean. If F = C we can define transfer factors for G over C or for the group G = ResjjG over R obtained by restriction of scalars. On identifying G(C) with G{R) in the usual way we see that the transfer factors for a pair (G(C),i7(C)) are the the same as for (G(R), H{R)) (this is a special case of a fact used in 5.5). It is then sufficient to treat the case F = R . Again there will be no harm in taking H to be an L -group. For real groups a transfer factor, that we now denote A^^\'yH,jG), was defined in [S4] using diagrams. It includes moreover an implicitly defined sign, (see [S4, Section 3.5]). Theorem 2.6.A. There is a constant c such that
for all G -regular JH in H{R) . Proof. By continuity we can assume yn strongly G-regular, and because both A and A^"*^ satisfy the Local Hypothesis ([I, 4.2.B], [L2, Lemma 6.17]) we may assume that G is quasisplit over R . For an imaginary root a we may take the x -datum Xa as the character z -^ z/\z\ on C^ and for the remaining roots we take Xa trivial. Then inspection of the terms in each factor shows that A{JH,7G)
=
C(TH)A^^\JH^7G)
where TH = Cent(7H,i^) • Up to a constant independent of TH)C(TH) is either real or purely imaginary. To show that C{TH) is in fact independent of TH , which is what the theorem asserts, we argue as follows. By the matching of orbital integrals for A^*) [S4] there is for each Schwartz function / on G(R) a Schwartz function f^ on H{R) (or an essentially Schwartz function if the embedding ^ : ^H *^^ ^G is not of unitary type [S3]) such that
for all strongly G-regular elements 7ff . We multiply both sides by DH(7H) ' The left side has a limit as JH -^ ^ through any Cart an subgroup and the limit is independent of the choice of Cartan subgroups. This
DESCENT FOR TRANSFER FACTORS
501
follows from applying the Harish-Chandra jump conditions to stable orbital integrals [SI, Section 4]. The right side then has the same property. But so also does
by [I, Theorem 5.5.A]. Because G is quasisplit over R we can choose / such that the limit is nonzero. Thus we get a contradiction unless C{TH) is independent of TH , and the theorem is proved. We conclude now that for each Schwartz function / on G(R) there exists an (essentially) Schwartz function f^ on H{R) with A -matching orbital integrals. Then the Paley-Wiener results of [C-D] allow us to take f" e C^{H{R),KH) if f e C^{G{R),K) , with KHJ< maximal compact subgroups of i7(R),G(R) respectively and the notation indicating bi- KH -finite or bi- K -finite functions. Of course we have not used the Descent Theorem (1.6.A) in this proof of A-transfer. However the proof of A^"^^-transfer that we have used is based on the Harish-Chandra jump conditions and these come from descent to centralizers of semiregular elements. Many of the arguments in [SI—84] for A^"^)-transfer are essentially special cases of results needed for 1.6.A. To prove A-transfer directly we may apply the results of Section 1 to verify the jump conditions of [SI] for ^f . Since we still need some of the arguments from [S2—S4] and overall the proof is not much shorter, we forgo the details.
§3. C o m p a r i s o n L e m m a s 3.1. R e d u c t i o n t o quasisplit groups Recall from 1.6 that 0 ( T ^ , 7 G ; 1H,7G)
= A(7if,7G; 1H,JG)/^eilH^o
]
IHHG)
where JH^JH ^tre images of JG^JG relative to (H^ , G^ ) . We consider the limit of 0 as JH^JH approach e^ through fixed Cartan subgroups denoted respectively TH,TH , and JG^JG approach e^ . Theorem 1.6.A, which we have to prove, states that this limit is 1. Fix embeddings TH ^ T,TH -^ f for (He^^G*). Recall that we assume G*,ip to be quasisplit data for Ge . Write 7,7 for the images of JH^JH 5 as usual. If we factor 0 as we factor A, A^ then only 0 i = 0iiii depends on JG,JG rather than on 7,7 alone. The next lemma allows us to replace G by G*,€^ by e and JG^JG by 7 , 7 .
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R. LANGLANDS and D. SHELSTAD
L e m m a 3.1.A. © I ( W , T G ; 7H,JG)
= 1•
Proof. We use the notation of [I, 3.4], putting a subscript e on those objects attached to G^ . We may take u{a) to be the image in Gl^ of the element U^{(T) in (G*)sc ? and h and h to be the images of h^ and h^, Then V{(T) and v{a) are the images of Ve(cr) and Veio") • The cochains v^ and v^ have the same coboundary and it takes values in the inverse image W (a finite group) of Z*^ in {G*)sc • So we have a cocycle i with values in where we use a notational principle that admits several variants: the subscript e — sc indicates inverse image in (G*)sc • The classes inv(7/^,7G ; 7 ^ , 7 G ) and inve{jH, JG ] JH^JG) are the images of the class of i under the homomorphisms induced by
(3.1.1)
Dual to (3.1.1) we have (3.1.2)
V
To prove the lemma it is sufficient to show that the images of su and su^ in V are the same. The X-data for G* (see 1.4) provide T - ^ f and T - ^ T . We have two commutative diagrams. V
Pe-ad X Te_ad
V
y
U
^ ^ad X ^ad
>
U,
-^e—ad X i £_ad
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503
Using them we extend (3.1.2) to a commutative diagram
^ V
^
X
u *—
T^ X T^
^
-«sc ^ •'sc
I ^v
Ue
I f
- ^
^—
Te_sc X T , _ , , *—^ ^ Te.- S C
X
if—J
We may write ? G T^c as the product of the image of J^ in 7^_sc with an element of the identity component 1Z of the subgroup {x G T^c • Oi^{x) = l , a ^ G i2(Ge,T)}. Embed 11 diagonally in T^c ^ ^ c without change in notation. Then we have just to show that the image of 7^ in V^ is trivial. But IZ is connected and V —^ Te-ad x ^e_ad ^^^ finite kernel because Tc_sc X fe-sc —^ V does. Thus it is enough to show that n
^V -^
fe_ad X te-ad
is trivial. That, however, is immediate, and the lemma is proved. 3.2. Remarks and notation We assume from now on that G = G* , 6^ = 6, JQ = j and JQ = j . Then 6 ( 7 i f , 7 ; 7/f,T) is a quotient 0 ( 7 i f , 7 ) / 6 ( 7 f f , 7 ) . So also is each of ©I, ©11, ©2 and 0IV . The embeddings TH --^T, fn^f being fixed, we may delete 7 and 7 from the notation. We may further fix Borel subgroups BH D Be^ DTH,BD B,D T, BR and so on, for which TH -^T.TH -^f are the attached embeddings. There is then a canonical identification of the roots of T in G with those of f . It carries the B -positive roots to the JB-positive ones, the (positive) roots of T in Ge to the (positive) roots of T in Ge, and the roots from H to the roots from H . Thus we use the simpler notation R = R{G)^Rc = R{Ge)jRs = B,{H) , and so on, for root systems. Also R{G) — R{Ge) will be abbreviated as R{G/Ge) and R(G)-{R{Ge)yjR{H)) as R{G/G,,H). In this notation ©iv(7fl^) is written as
n
KT)-ip/2
a6fi(G/G„/f)
and so lim 0iv(7i/)=
n
l«W-ir^'
aeR(G/G^,H)
=
hm
©iv(7/f) •
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R. LANGLANDS and D. SHELSTAD
T h u s 0IV contributes 1 t o the Umit and it remains to examine
T h e First L e m m a of Comparison will allow us t o examine QI{JH)/QI{7H) and the Second to examine 02(7if ) / 0 2 ( 7 f f ) • For the remaining t e r m observe for now t h a t 7 H hm
eu{7H) =
llx4^^^^^
where the product is over representatives for the orbits in R(G/Gej der t h e Galois action for T . There is a similar formula for lim
H) un0ii(7/f)
involving the Tf -orbits. To compute 6 i we need also to fix F - s p l i t t i n g s (B,T,{Xa}) of G and ( B e , T e , { y ^ } ) of Ge, but then we can use B , B e along with B,Be t o identify T, T^ with T , and again identify all roots as roots of T . Recall t h a t we write T for G a l ( F / F ) , TT for {CTT : cr G T} , T for {a = (TT : cr G r } and so on. In [I, Section 2] we identified TT as a subgroup of TQ{G,T)x r . Now we find it convenient to work with Q{G,T)>^TT and t o identify Tf as a. subgroup of Q C X F T , where fie = Q{Ge,T) . T h u s for cr G r we may write af as u; x CTT , where cj G fie • We shall often write a for Cji and 0 a~^a
where the product is over a G R{G/Ge)
.
.
DESCENT FOR TRANSFER FACTORS
505
On the other hand, for 9 E: ^e»^T we have n(0) € Norm(G,T) on regarding Q C X F T as a subgroup of fi>4 T [I, 2.1] and ne{0) G Norin(Ge,T'e) if we regard Q^XTT as a subgroup of Q^x T ^ • Then n{0i)ni02) = n,(0i)n,{02)
t{0i,02)nieie2) =U0i,02)11,(0102)
and if Ti01,02) =
ht{0u02)h-^Ki,{0u02)-^K^
then
r{0u02)=
n
(-l)"^
a>0
with the product again over a G R{G/Ge) (see [I, Lemma 2.1.A]). The restriction of r to F T is also the coboundary of y [I, 2.2.A]. Since T = r~^ we can write the more convenient: (3.3.1)
dy = T-'
(see also [I, 2.3]). For u e^€ we define 6(u;) G T by hn{(jj)h~^ =: h{(jj)hen^{uj)h'^^ . Finally, suppose ^{a) represents the cohomology class / i . Then for ujxa ^ fieXFy we set z{(jj xa) = b{uj)uj{y(a)~^fi{(T))T{u,a)-^ . Note that LJ = 1 yields fi{a) = y{(T)z{a) ,
cr G FT .
Lemma 3.3.A. (First Lemima of Comparison) fi{a) = y{a)z{a)
,
aeTf
.
Proof. Write & as u; x a. Then we calculate //(o-) as hx{d')n{uja)a(h^^ hc)nc{ua)'~^ Xe{d')~^ hj^ = y(^)r(a;, (T)"^hn(Lj)n(a)(T{h~^he)nc{o')~^nc{u))~^h~^ = y{a)T{u), a)"^ hn(u})h~^y{a)~^ fi(a)h^n^{uj)~^ h~^
= J/(^)^(^,<^)~^^(y(<^)"V(<^))K^) = y(^)^(^).
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R. LANGLANDS and D. SHELSTAD
and the lemma is proved. Most of the time we will assume a condition that is satisfied, for example, if TH is maximally split in H^ . It is: (3.3.2)
r ^ C fioXiTT, where fio is the Weyl group for a root system RQ C R{HC ) that has a base UQ stable under TT •
The assumption will be harmless because of the transitivity property in Lemma 4.LA of [I]. Lemma 3.3.B. Given (3.3.2) we may choose the cocycle /jL{a) representing fi such that uj{fi{a)) = fi{a) , a; G fio , cr G F T . Proof. Multiplying h or h^ by an appropriate element of T replaces /L/((7) by an arbitrary element in its class. So we need only verify that the class of fi lies in the image of H^(T^^) -> H^{T), where T^° is the centralizer of fio in T . If EQ = {ai, • • • ,a,.} then t —» (ai(/), • • • ,ar{t)) yields an exact sequence 1 -^ T^° -^ T -^ S —^ I with S an induced torus. Since H^{S) = 1, the lemma follows. We suppose now that /i(<7) is fixed by QQ , inflate it to QO'>^TT and then restrict to Tf . Suppose the cocycle so obtained is i/. Then
Lemma 3.3.C. Given (3.3.2) we have
Proof. We identify T as T with Galois action Tf and thus T x T as T X T with action T^^xT • ^^^ element Sf of 7ro(T^) is then identified with ST and so we drop subscripts. Working in T x T we have {fJ'^ST)/{l^ySf)
= {(/i,i/),(s,S-^)) .
If we define the F -torus A by X,iA)=
{(Ai, A2) G X . ( r X f ) : Ai - A2 € (S^)} ,
with Galois action induced by TrpxT ^^^^ we have Galois homomorphisms
A—^T
xf
, f xT—^A.
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507
Since the image of (s,s~^) under the map induced by TxT -^ A is trivial we have only to show that (/i, i/) lies in the image of H^(A) —^H^{TxT). The diagonal embedding of T in TxT factors through A but the induced homomorphism T —^ A is not defined over F . However, by (3.3.2), if we retrict to T^^ then we do get a map over F . Since /i takes values in T^° and has image (//,z/) under H^iT^'^) -^ H\A) -^ H^{T x T) the lemma is proved. Finally, continuing to assume (3.3.2), we set v{u} X a) = T{uj,(T)(jj{y{a))b{uj)-^y{&)~^ . Then ?; is a 1-cocycle of F^^ in T = T. By the Lemma of Comparison it coincides with i/p,~^ and we conclude: Lemma 3.3.D. Under the condition (3.3.2) we have QI{IH)IQI{IH)
= {v,Sf) .
3.4. The Second Lemma of Comparison The term A2{JH) was defined in [I, 3.5] using the construction of [I, 2.6]. For once and for all, we assume 7i is an Z -group, as we may without loss of generahty. The choices of 3.2 provide f -^ T , TH-^T. As in [I, 2.6] we extend these to ^T^^G, ^TH ^ ^H by defining m{w) = r{w)'n{a) x w , ms(w) = r,(w;)n,(cr) x w . Here w; —»
and a G
= A2(7i/)/A2(7^)
= {a,7){a,7)-^ . The definition of the cochain r appears in [I, 2.5]. Set c{w) = r(w)rs(w)~^
,
c{w) — r{w)fs{w)~^ .
H^{W,f).
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R. LANGLANDS and D. SHELSTAD
The coboundaries of the cochains n , n, defined on Q(H)>\TT are denoted t, ts (see [I, 2.1]). Set r = ttj^ . Finally for u; G Q{H) we define 6(0;) G T n(u;) = 6(a;)n,(a;) ,
a; G ^{H) .
We will freely transport objects among T,T notation.
and T without change in
Lemma 3.4.A. (Second Lemma of Comparison). Let w E W map to (T £TT o.nd to uj X a £ Tf . Then a[w)~^ = c{w)b{Lo)u{c(w)~^a{w)~^)T{uj,a)~^
.
Proof. fh{w) = f{w)n(Lja) x w = r{w)t(Lij,a)~^n(uj)n(a) =
x w
r{w)t(uj,a)~^b{uj)ns(u))r{w)~^m{w)
= f{w)t{u, a)~^ b{Lj)uj{r(w)~^ a(w)~^)ns{u))C{fT^s{'^)) and C{fhs{w)) = =
fs(w)ts{uj,a)~'^ns{u;)ns{(T) rs{w)ts{uj,(T)~^iv{rs{w))ns{u;)^{ms{w))
so that a(^w)~^ = c{w)T{uj^(T)~^b{u;)uj{c(w)~^a{w)~^) , and the lemma is proved. Let zi{ujw)
= b{Lj)uj{c{w)~^a{w)~^)T{uj,a)~^ .
Then a{w)~^ = c{w)zi(l,w)
,
and a{w)~^ = c(w)zi{u;jw) . Note the similarity to the First Lemma. The Second Lemma will be applied a little differently however. We shall give an example of the technique in the next section.
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509
Note that t —>• uj{t)t~^ lifts to a homomorphism a^^ : T -^ Tsc - Here, as elsewhere, Tie denotes the inverse image of T in the simply-connected covering Gsc of the derived group of G . Thus if we multiply zi (a;, w) by a(w) we obtain ^{cjjw) = b{Lj)uj{c{w)~^)T{u;, a)~'^u;{a{w))~^a{w) in the image of Tsc in T. Gsc • Then
More precisely, 6, c, r may be constructed in
2sc(u;, w) = 6(u;)a;(c(w;)-^)r(u;, cr)"^a^(a(ti;)-^) has image ?(u;, it;) under Gsc —^ G . To calculate ©2 we also need A^ which is attached to (G^, H^ ) - For this we pass, if needed, to an admissible extension H^ of H^^ (recall 1.3). We then have exact sequences 1 —^ f — y i i — ^ Z i —^ 1 , and
^ 1 — ^ T — y T —>Zi —^ 1 .
In place of a^^oLe we have ^^^GLC that take values in T , T . These two cocycles have the same projection on Zi [I, 4.4]. The Second Lemma of Comparison becomes Sf(t/;)""^ = c^{w)zi^^{uj,w) and 'a^{w)zi^^{u)^w) is equal to he{uj)uj{c^{w))~^Te{(jJ,ay^U)i^^{w)~^)Ze{w)
which takes values in T (or Tsc or Te_sc )• The terms h^^c^^Tc are h^c^r for the group Ge. 3.5. A n application The following was stated in [I] as Lemma 4.4.A but not proved in general. Lemma 3.5.A. There is a character A on the center Z{F) of G{F) such that ^{zjH. no) = A(z)A(7^, 7G) , ze Z{F) for all jH^lG '
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R. LANGLANDS and D. SHELSTAD
This applies to arbitrary G but we reduce immediately to the quasisplit case for it is only A2 that is affected by replacing ^H ? TG by z^u, Z^Q • Proof. All we need to show is that (a, z) — (a, z) or, in T xf
, that
((a,a),(z,z-i)> = l . Define a torus S over F by X*(5) = {(Ai, A2) £ X*(T) X X*{f)
: Ai - A2 G
X*{T^)}.
Here again we identify f with T over F and thus X*{f) with X*(T) . We have Galois homomorphisms T x T —^ S and S ^^ T x T. Since (z,^-^) lies in the kernel of T{F) x f{F) -^ S{F) it is enough to show that (a, a) lies in the image of H\W,S) -^ H\W,f x f). The torus S is isomorphic to T x Tsc • We obtain the factor T by factoring the diagonal embedding T-^TxT = TxT through S ; so clearly T ^ 5 is not compatible with the Galois action. On the other hand Tsc ^ 5 is obtained from X*{f^) -^ {0} x X*{f^^) C X*(5) , and so does respect Galois action. By the Second Lemma of Comparison the cocycle w -^ (a(w),a(w)) with values in T x T is the image of the cochain w —>
{a{w),c{w)~^%c{^,'w)~^)
with values in S = Tx Tsc • We have to show that this cochain is a cocycle. It may be written as
{a{w),l){hc(w)-'){lXc{^,w)-'). The coboundary of the first term is {wi,W2) —^ (l,aa;i(cri(a(ti;2)))) if Wi —^ LOi X (Ti E Tf ; the coboundary of the second is (l,f{wi,W2)) . Note that f{wijW2) = T(ai^a2) = f{uji x cri,L02 x 0-2). Thus it remains to show that the coboundary of ?sc is {wi,W2) —>T{UJI X ai,U2 X (T2)a^^,{o'i{a{w2))). This is Lemma 4.2.A.
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§4. Analysis of 6,6 and a reduction 4.1. Galois action By definition, 6(a;) = hn{(jj)h~^hen^{(jj)~^h~^ ,
cj G ^e ,
and in the dual setting, 6(a;) = n(cj)n,(a;)-^ ,
u G fi(if) = fi, •
We have immediately, 6(cji)a;i(6(cc;2))6(cc;iu;2)"^ = r(a;i,a;2) ^ ^ ^
(4.1.1)
h{uJi)ljJi{h{(jJ2))K'^l^2)
=T{(JJI,UJ2)
'
Recall that these objects may be constructed in Gsc and Gsc ? the simplyconnected forms for G and G. For G^ and H = Gg we may work in the simply-connected forms and project into Gsc or Gsc • This will be the rule throughout, although in notation we may identify an element with its image in G or G . To describe the effect of Fy on 6 and 6 we consider the two cases together but keep in mind that in the former we have a genuine Galois action and in the latter an algebraic action. First define f{u;,a) =
ht{a,Lj)t(au>,(T-^)a{u;){t{(T,a-^)-'^)h-^
for a; X (7 E ^e XI F T and similarly f{ij0^a) for u; x cr 6 ^ ^ X F T . Then we see easily that hn{a)n(uj)n{a)~^ h~^ =
f(uf,o')hn(a(u;))h~^
where n(cr) = n(cr) x cr G Gsc>^r (recall from 3.3 that we have changed slightly the notation of [I]). In the dual case we have (n(a) X w)n{u;)(n(a) x w)~^ = where w -^ a under I ^ ^^ F . Set e = ff~^ (3.3.2) for the first part of the next lemma.
f{uj,a)n(a{Lj)) and e^ = ff~^ . We assume
L e m m a 4.1.A. (a) e{uj,a)b{(T{ij)) = a{uj){y{a))y{ay^(T{b{(j)) and
for LJ x a e ^e>^^T ,
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R. LANGLANDS and D. SHELSTAD
(b) 'e{u},a)b{a{u)) = a^(^)(c(ti;)a(i<;))cr(6(a;)) for w —^ a and Lo X a e fiaXlTT . Proof. This is a straightforward calculation. For (a) we suppress h,he from the equation in order to make the calculation more transparent. Then n((T{uj)) = b{(T{uj))ne{(T{u;)) implies n(a)n(uj)n{a)~^ = e(u,a)b(a{u;))nc{a')ne{uj)n^{o')~^ and so it is enough to show that n{a)n(uj)n{(T)~^ = a{uj){y{a))y{a)~'^a{b{uj))ne{or)ne{u;)ne{(T)~'^ . But n{a)n(Lo)n(a)~^ = n(cr)6(u;)ne(a;)n(cr)~^ = a{b{u;))n{a)ne{uj)n{(7)~^ , and all we need is that n{(T)n,{u;)n{a)-^ = a{u)(y{a))y{a)-'^n,{a)n,{uj)ne{(T)-'^
.
Suppressing /i, h^ we deduce from the equation hx{a)n(a)a(h)~^
= fi{a)h^Xc(a)n^{a)a{h^)~
that n(a-) acts on nc{uj) as fi{(T)xei(T)x{a)~^n,{or) = fi{a)y{a)-'^n,{a)
.
Since cr(u;) fixes A*(
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4.2. Calculation of a coboundary Recall the cochain ? on QgXWT : ?{uj,w) = b{u;)uj{c{w)~^)T(u;,a)~^ai^{a{w)~^)
.
Lemma 4.2.A. The coboundary of 7 is Ta : {uJiWi]a;2W2) —> r{u;iO'iyUJ2(T2)a^:,{cri(a(w2))) .
Proof. Because r^ is a 2-cocycle it defines an extension of fi^xPVr by Tsc • This extension is generated by T^c and elements 5(7;), 77 G QS>\WT , with s{r])ts{r))-' = r){t) and
s{m)s{m) = TairjumMrjim) • We prove the lemma by showing that T)—>?{r))
h{7))=zsi{r})
splits the extension. Let w-^(T ETT ^ Then SI{LJW) = b(u;)''^u;{c{w))r{u y (T)au;{a{w))s{ujw) =
b{u;)~^Uj(c(w))Ta{LOyW)s(LOw)
= b(Lo)~^ s(u))c{w)s{w) = si{uj)si(w) . Moreover, ^1(^1^2) = b{u;iu;2)~^ s(ujiu}2) = u}i{b{u2))~'^b{u}i)~^T{u;i,u;2)Ta(LOi,L02)~^s{u;i)s(uj2) = u;i(6(cc;2))~^6(u;i)"^s(<x;i)s(a;2) = si{ui)si(uj2)
.
Finally, Si{wiW2)
= c{WiW2)
^s{wiW2)
=
c{wiy'^Wi{c{w2))~'^T{wi,W2)Ta{wi,W2)~'^s{wi)s{w2)
= Si{wi)si{w2)
.
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R. LANGLANDS and D. SHELSTAD
To show that SI{UJIWI)SI{LJ2W2)
=
Si{uJiai{uJ2)wiW2)
and thus complete the proof of the lemma we need only verify that (4.2.1)
5 i ( ^ H ) = s^{w)si{uj)si{wy^
.
The right side is z'{w)~'^ ^{^(LJ))'^
a{u})(z{w))s{w)s{uj)s{w)~^
which equals a{uj){c{w)~^)c{w)a{b{w))~^s{w)s{uj)s(w)~^ and s{w)s(u})s{w)''^
is 5(cr(a;)) times
Ta{w,Uj)Ta{wu;,LJ~^)a{uj){ra{w,W~'^)~'^)
.
It follows readily from the definitions that this last product is equal to
or, since a{w) is a cocycle, to e(u;,(T)a^(^)(a(t/;)-^) , Thus the right side of (4.2.1) equals a(b{u;))-^aa(u;){c{w)a{w))-^€{u;,a)s{a{u;))
.
Since the left side is b{(T{uj))^^s{cr{uj)) we need only appeal to part (b) of Lemma 4.1.A to finish the proof. Recall the definition of the cochain z on ^^XTT Z{UJ,(T)
- h{ijj)(^{y{a)-^)T{uj,(Ty^
.
A similar, but simpler, argument establishes the next lemma. Lemma 4.2.B. z has cohoundary r . 4.3. Explicit form In the setting of [I, 2.1] suppose that /? is a positive root. Let /3 = z//?o where iy E: Q, and /?o is simple. Write LOQ for (jp^ .
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515
L e m m a 4.3.A. n{uj^) = 6j,{/3)n(i/)n{u;o)n{u)~^ where and the product is over roots a for which a >0,
LOpa < 0
and
i/""^ct;^a > 0 .
Proof. n{ujp) = n{uLOoJ^~^), and this is the product of t{i/u;o,i'''^)~^t{u,uo)~^(jiJp{t{iy,i/~^)) and n(i/)n{ujo)n{i/)~^ . Moreover t{i/^ujo) = 1 since a > 0, u^^a < 0 and ujQ^i/~^a > 0 impUes that i/~^a = —/3o and hence that a = —^^ contradicting a,/? > 0 . It remains then to show that 8^{p) ^tivujo.u-^y^Lj^itiu.u-^))
.
The right side is n ( ~ l ) ^ ? *^^ product being taken over those a for which a > 0, z/~^a;^a < 0, ujpa > 0 and those for which ujf3a > 0, i/^^upa < 0 . This coincides with the product over a < 0 , u~^ujpa < 0 , LJpa > 0 and thus with St,(/3) ; so the lemma is proved. Let Rf3 = {a E R : a > 0 , upa < 0 , a ^ ^ } . Then a root a hes in R/s if and only if —a;^a does and then the two elements are distinct. We can therefore choose a subset Rt of Rp such that Rj3 is the disjoint union of Rt and —ujp{Rt) . Then
6am = n (-1)"' aen^ is well determined up to a factor (db 1)^ because —ivpa"^ = —a^ + {a"^,/3)l3^ . Notice that we may take Rt = {a E Rp : u'^u^a > 0} since i/~^Lj(3(—u)pa) = —i/~^a and i/~^ujpa = ^^^{ly'^a) have opposite signs. Thus (4.3.1)
n(u-^) = SGm±
lf\iiy)niL,o)n{i^)-'
.
Clearly this also holds when /? is negative. We return to the setting of 4.1, working in Ggc and Gsc • Since (3.3.2) is in force, there exists RQ C R{H^ ) with Fy -stable base EQ . Then F^ C Q O X T T , fio being the Weyl group generated by So . We are interested in
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R. LANGLANDS and D. SHELSTAD
6(u;) , b((jj) for uj E QQ . By (4.1.1) we need only consider b{ujp) , b{u}^) with /? G Do . Set 6{I3) =
6G{P)8GXP)~'
and
Lemma 4.3.A There exist bp G F^ , bp G C^ such that
and
Proof. The first relation is immediate from (4.3.1) once we recall that n{u)f3) = b{ujp)n^{ijjfi), and note that both n = n{i/)n(uo)n{u)~^ and n^ = ^e('^6)^e(^o,e)^£(^e)~^ lie in the image of SL(2) in G determined by /?, so that n = a^ ric, with a E F^ . The second relation is proved in the same way. Note that ujp -^ b'i has a natural extension to a 1-coboundary 6^; of Qo in Tsc as follows. Choose t G T^c such that j3{i) = bp , /? G Do • Then clearly uj -^ tuj{i)~^ has the property that LJ(3 ^*- b^ . It is independent of the choice of t, provided we take i in the image of F ^ (g) (DQ } . Define S{u;) , a; G ^0 J by 6(cj) = bi^S{u;) and the dual S{uj) similarly. Note that dS = r , dS = T . 4.4. Root types To prove Theorem 1.6.A we use R{H) and R{Ge) to partition the roots. The notation will be as follows. type(a) R{H) n R(G,) = R{H,^) = i?(^) type(6) R{H) - R{G,) = R{H/H,^)
= R^^^
type(c) R{G,) - R{H) = R{GJH,,)
= R(<^^
type(d) R - {R{H) U R{G,)) = R{G/H, G,) = i?(^). This also gives a classification of the TT -orbits O and the Tf -orbits O in R. Observe that
e..(™)= OCR(^) n ^.(2^).
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517
with a similar formula for QII{JH) • In the next section we so arrange some choices that roots of type (b) or (c) contribute nothing to 0i(7if,7iy). 4.5. Analysis of 0 i , 0 2 Recall that if v{&) =
T{u;,a)u;{y{(T))b{uj)-^y{a)-^,
where a = UJ x a , then we evaluate QiijH^jH) by pairing the class of the cocycle v w i t h s j . . By definition, r(a;,cr) = I K " ! ) ^ 5 where the product is over roots a for which a > 0 , uj~^a < 0, a~^uj~^a > 0 and a G R{G/Ge) . Such a are of type (b) or of type (d) and we consider the products r^*^ , r^^^ over roots of each type to define r = r^^V^^) . The same can be done with y and y, Then to write v as v^^\^^^ it remains to factor h as b^^H^^^ . To describe the contribution from roots of type (b) we recall that these are the roots of H outside H^fj . But 6 was attached to the pair (G^Gc). Now we use the fact that EQ C R{Hcff) in the assumption (3.3.2) to observe that we can attach b^ to {H^H^j^) in the same manner. From (4.3) we have b^{uj) — b{j6^{w) , a; G fio ? along with b(uj) = bi^6(ij). Recall that there is some freedom of choice in 6^ , 6 . For /? G Do we may arrange that 6^{ujp) is the same as the contibution S(^\u;^) to S{u;^) from roots of type (b). Then we define 6(^\UJ) = 8^{uj) for all u; G fio and 6^^^ = 8/8^^) . Note that 38^^"^ = r^ = r^^) and QS(d) = r^d) Set 6if) = 6^ and b[^^ = b^/bi'^ . Finally, 6(^) = bi'h(^) has coboundary r^^^, while fcO = bi, ^S^''^ coincides with b^ . This yields a factoring v = t;(*^v'^''), and t)(*),t;('') are cocycles. Thus
because 5 central in Ti implies that
Observe that if there are no roots of type (d) then v^^^ is trivial and We shall use a similar but dual argument for ©2 . Here we have roots of types (c) and (d) to deal with. The term A2(7/f)/A2(7jy) is obtained by pairing the cocycle {a{w),a{w)) = =
{a{w),a{w)c{w)~^b{oj)~^T{uj,a)ij{c{w))auj{a{w))) {a{w)ja{w)c(w)^^'z(uj,w)~^),
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R. LANGLANDS and D. SHELSTAD
of 1^ in f X T with the element (7,7"^) of T{F) x f{F) . We may thicken T to T as at the end of Section 3.4 in order to compare directly with A2,€(7^)7^2,c(7^). In notation we will not distinguish between T and T , or T and T . cocycle
Thus for 02(7if)/©2(7/f) we have to pair the
{a{w),a(w)){ae(w)jac{w)~^)
= {a{w)ae{w)~^
ja{w)ac{w)^^)
with ( 7 , 7 " ^ ) . Now b is attached to (G, H). Similarly we have b^ attached to {Ge,He^). Factor 6(u;) as t'\uj)b(^\uj) = b,{u;)U'^\u;), as we did 6(0;). The factoring of c, r and c is immediate, again as before. Set a^^\w) = ae{w) and a^^\w) = a{w)la^{w). Then clearly a{w) factors as a^^\w)a^^^{w) where af<^\w) = a^{w) and 02(7/f)7ff) equals {{a^'\w),a^'\w)),
(7,r')>,
or, more explicitly,
{{a^'\w),a(''\w)^''\w)-'^<'\w,w)),
(7,r')>,
with ?<*^)(a;,it;) = 6(^)(a;)r(^)(u;,cr)-^u;(c(^)(ti;)-^)aa;(a(^)(«;)-^). Observe that in the case that there are no roots of type (d) our proof of Theorem 1.6.A is complete.
§5. Final Reductions 5.1. Introduction To complete the proof of Theorem 1.6.A we have to show that
and satisfy 5.1.1 The limit as Jff,jff
approach ejj of
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519
is equal to
n ^.(^) n ^^)Here Orb^^^ denotes the collection of orbits in R^^^ , and Orb has a similar meaning. It is moreover understood that a is a representative of O , and a of O . In addition, for an asymmetric orbit O we may choose, when convenient, the x - d a t a and the a - d a t a trivial, but then a^ = —1 for a € — O . We propose to verify (5.1.1) in part by induction on the dimension of Gder • Suppose i^o satisfies the condition (3.3.2). For example, RQ could be R{Hc) itself. We choose fi so that the condition of Lemma 3.3.B is satisfied. Suppose moreover that RQ C Ri C R ^ where Ri is also TT invariant. Then the dual system R'l may be taken as {a^:a G i2i} , and we may construct a_ quasi-split group Gi containing Cartan subgroups identical to T and T (and identified with them), and with Ri as its root system. The group Gi need not be isomorphic to a subgroup of G but it will have an endoscopic group attached to ST in T or to ST in T. Denote this endoscopic group by Hi . The groups T and T have images in Hi. Let D T = Z2 X F T . Taking the non-trivial element of Z2 to act as —1 we obtain an action of E T and of QQ X E T on R and on i?^i = R — Ri . Let A be a set on which Ey acts, and denote the image of A E A under the non-trivial element of Z2 by — A . Suppose that —A is never equal to A . Finally, extend the action on A to QQ x Sy by letting QQ act trivially. The critical lemma for the reduction is the following. Lemma 5.1. A. If there is a mapping from R^i to A compatible with the action of QQ X FT' then Assertion 5.1.1 is true with respect to G , T, T, € if it is true with respect to Gi , T , T , e. Before proving this lemma it is as well to remind ourselves what it means. We have endoscopic data for four groups G , G^ , Gi , Ge^i all of which share tori T , T. However the factors A2 are defined on covering groups T , T , Te, T^, Ti , Tj , Te,i , f^^ defined by central extensions of the four groups. For G itself we have of course made the necessary extensions at the very beginning, so that for G the tori T, T are covered by themselves. We must also choose e in T^, mapping to e and therefore common to T^ , as well as ei G Ti fl T^ , ei G T^^i fl T^ ^ with similar properties. Thus the Assertion 5.1.1 is to be understood as applying not literally to Gi,e but to the covering group of Gi and to ei in it. Of course, only the limit of ©2 could possibly be affected by the various choices, and it is not.
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R. LANGLANDS and D. SHELSTAD
5.2. Beginning of the proof of critical lemma We may as well assume that the mapping from R^i -^ A is surjective. Observe that the classification of roots in Ri into types is the same in Ri as in i ? . The choice of a - d a t a and x - d a t a remains at our disposal. It is clear that, copying the definitions of §2.1 and §2.5 of [I], we may define fields Fx , F±x , A G A , and, associated to A , collections of a-data, {ax}, and X""data, { X A } - Since the mapping from i?^i is compatible with the actions of TT and Tf we choose a^ = «« == ^A and Xa = Xx o Nmp^fpy^ , Xa = XA o ^ ^ F a / F x if a —>^ A . Inside Ri we choose the same a -data and x -data for G as for Gi . If OrbJ:^]^ , Orb^j denote the orbits outside Ri it is easy to show that with these choices of a -data and of x -data the second expression in (5.1.1) divided by the analogous expression for Gi is equal to
To prove Lemma 5.1.A we show that these same choices lead to (5.2.2)
e\'\jHnH] = e['\jH.7H)i,
(5.2.3)
lim 0^/^(7if,Ti/) = lim 6^/^(7/7,7if)i •
The subscript 1 on the right indicates that we are calculating with respect to Gi . If Ta, Tx consist of the elements in the Galois group TT fixing a , A respectively, then
«.(^) = «4n
ax
P /P <^"(f) -
Thus the first product on the left side of (5.2.1) is the product over a set of representatives for the orbits of the image of R^i in A of
"An *A
«A
a(e)
The same calculation barred yields the identical result, so that (5.2.1) is 1. The left side of (5.2.2) is obtained by pairing the cocycle v^^^ with Sf . We shall factor v^^^ as v^ , the cocycle attached to Gi times v^l and then show that v^^l is a coboundary. The relation (5.2.2) follows immediately. Since v^^^ is a product.
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521
we easily factor it by factoring each term. The first two and the last are given as products over the roots in R^^^ , which we may decompose as products over R[^^ and i?(^) - R[^^ = R^^l . Finally we define b^^l{uj) by the equation Since Ri and i?^i are invariant under QQ , we may define
%
<'
,.W
a>0
on all offio X ^T • The a - d a t a have been so chosen that y^l is obtained by restricting 2/L1 ^^ Tf . Moreover, it follows from Lemma 2.2.A of [I] that the coboundary ofy^^l is the inverse of ri'? . Thus
and (5.2.2) is a consequence of the next lemma. Lemma 5.2.A. We may so chooseb^^\u)
and b\ ^(cj) thai
In addition to b['^ , we may define and 61 , and then Moreover b^iiu) = b^^liuj)b^^liu;). Lemma 5.2.A will follow from the next lemma, applied first to the pair G, Gi and then to the pair H, Hi , the group Hi being defined by Ri n R{H). Lemma 5.2.B. We may so choose 6(0;) and bi(uj) that
Proof. Both sides of this equation have the same coboundary, so that it is sufficient to verify that it can be satisfied for uj = up , /? G So . Recall from Lemma 4.3.A that (5.2.4)
b(cjff) =
A similar equation is valid in Gi'.
bl"s{p).
522 (5.2.5)
R. LANGLANDS and D. SHELSTAD 6i(u;0) = 6f>i(/?).
The factors S{/3) and 6i{/3) are defined by sets Rt and Rf^ . We may suppose that Ri n Ri = R'^Q . To define Rt O (Rf^i) = R'ti Q choose a set A"*" of representatives for the orbits of Z2 in A and agree that a G Rt n (iZ^i) if and only if a -^ A G A"*" . Observe that if a —> A then —a;/?a —» —A . From (5.2.4) and (5.2.5) we obtain a factorization
with
The expression on the right in Lemma 5.2.B is (5.2.6)
n
«""' '
a>0,a'~^a<0
when (jj = Ljp . Putting a and —Loa together, and noting that
(-!)-«" = (_i)""(_i)<-".W^ , we see that (5.2.6) is equal to
s^m
n
«:!'''•'"''•
a>0,w~'a<0
To show that
(5.2.7)
n
«=i"''''
a>0,a^~^a<0
is a possible choice for 6^1/? we apply the next lemma. To state it, set e^i{u),(T) = e(a;,cr)ei(a;,(7)-^ with e(a;,(7) = r((7,u;)r(cra;, (T-^)(7(a;)(r((7, cr"^))"^ as in (4.1).
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523
Lemma 5.2.C. Suppose that for each /? G Do w;e are given c{/3) G F^ . We may choose 6^1,/? = c(/?) for all /3 if and only if the following equations are valid for all cr £ TT '
= y^iHtr(«)(y^i((T)-i)e^i(a;, <7)c(a^)''^"6^i((T/?).
If we define c(/?) by (5.2.7) this equation is simply (7(2/^1(0;/?)"^) = yr^i{(T)a{ujp){y^i{(Ty^)e^i{wp,(T)y^i{u)a(3)~^, for in the lemma LJ is O;^ . This equation is better written as
Inserting the factors of 2/-i(^W/?)2/-i(
which is true by definition. Lemma 5.2.C is verified by applying the definitions and the following lemma for Gi as well as for G . Lemma 5.2.D. Suppose that for each /? G Do we are given c(/3) C F^ , We may choose bp = c(/3) for all /3 if and only if the following equations are valid for all a E TT : <Tic(p)rP\iS{/3))
=
y(a)<Tic){y(^r')e(u;,a)ci
Proof. The necessity, namely the equation (7(6(0;)) = yi)), is the first part of Lemma 4.LA. To prove the sufficiency we observe that what the conditions of the lemma determine are the quotients {(T(c{/3))c{a/3)~^Y^'^ ' , where A is any weight. Thus it permits multiplication of any given collection by a collection {d{/3)} satisfying {
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R. LANGLANDS and D. SHELSTAD
On the other hand, we do not destroy the condition of Lemma 3.3.B if we multiply h by an element t such that /3{ta'{t~^))^ = 1 for all f3 £ Ho . Replacing /? by
Since the change in h replaces b{u;p) by/?(t)6(a;/?) the equality (5.2.2) is proved. 5.3. The term 022 • The relation (5.2.3) causes the most difficulty. For the calculations we must pass explicitly to a, z -extension G[ of Gi . The associated quantities will be denoted with a prime. All tori T , T , T^ = Ti , T ^ = T^ , as well as T^ , T^ , f,,i , T^ ^ were identified. Thus we have embeddings T ^-^ T{ , T, <^-> T/ ^ and so on. We transfer cocycles with values on one group to another group in which it is imbedded without change in notation. Suppose e' in T' maps to e. The left side of (5.2.3) is obtained by pairing {a{w)a^{w)~^ ^ a{w)a^(w)''^) with (e,6""^) or, passing to T" , with (£',e'~^) . The right side is obtained by pairing (ai(ti;)ae^i(w;)~^ , ai{w)ac^i{w)~^) with (e',e'~^). Thus it suffices to show that (a(w)ai{w)~^ , a{w)ai(w)~^) yields 1 upon pairing with (e',^'"^) and that {ae{w)ac^i{w)~^ , de{w)ae^i{w)~^) also yields 1. We shall prove the first assertion which is a statement about G and Gi . The second follows from it upon substitution of G^ for G, and G^ i for Gi . Set a2{w) = a{w)ai{w)''^ , a2{w) = a(w)ai{w)~^ . We want to show that (5.3.1)
((a2,a2), (e'.e'-i)) = 1.
Let Y be the span over Z of i^o , and define a torus 5 by the relation X*(5) = {(A,/i)|AGX*(f'), /xGX*(TO,
A^/iGY}.
Notice that X^{T^) and X*(T') are, if desired, identified, so that the locations of A and /i are specified only to make the Galois action clear. The inclusion X^{S) -^ X^{V) 0 X^{V) defines S -^ V x T and T' xf -^S, Under the latter, (e',e'-i) -^ 1. Thus to establish (5.3.1) it suffices to show that (a2)«2) is the image of a cocycle with values in 5 . The decomposition X*(5) = X . ( f O e y : ( A , / i ) - (A, A - / / ) yields an isomorphism S c±T' x R /\{ R is defined by X^{R) = Y . It is not an isomorphism of Galois modules. Nonetheless if ai^ is a cocycle with
DESCENT F O R TRANSFER FACTORS
525
values in T ' then w —^ ci2{w) x 1 = as(w) does take values in S. are two points t o verify: (5.3.2)
T h e cocycle ai^ can be so chosen t h a t as is a cocycle.
(5.3.3)
It can at t h e same time be so chosen t h a t t h e image of as
There
is in t h e same class as (02,02). This will take some effort. T h e tori f , T , f^ = f , are identified in a fixed way with T . T h e normalizer of T in ^ G projects modulo T itself t o CI{G)XTT - Denote t h e inverse image of QQXTT by ^M . Since fio is contained in Q ( G i ) , Q{M) and Q{Mi) we m a y define ^M{ , ^ M , , ^ M , ' ^ in t h e same way, t h e kernel of ^ M i - ^ QQXTT
or of ^Ms^i
-^ Q O X T T being V .
T h e cocycles used t o define 02 are defined by means of homomorphisms attached t o t h e X - d a t a : e:^T--^^MC^G
;
^[^V
—^ ^M[ C^G[
and t o imbeddings:
T h e cocycle 0,2 is defined in a similar manner by ^ , ^J , ^5 , ^^ 1 . We shall construct homomorphisms (5.3.4)
V^,:^M, — . ^ M , '
with the following properties: (a) They are compatible with t h e projections t o QQ^^TT • (b) Let TT , TT be t h e n a t u r a l homomorphisms
defined by t h e imbedding f ^
f',
f
^
V . Then
(c) T h e r e is a < E T ' such t h a t ip o T] = a d t o r/J o <^3 , on Ms , t h e inverse image of QQ in ^ M , .
;
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R. LANGLANDS and D. SHELSTAD
These conditions axe clarified by a diagram LT
-L^
Jt^
LM
LM[
I'
h
For technical reasons we need variants of y>, (fg . Let T be the inverse image of Tg in the simply-connected covering G of the derived group of G with the action of the Galois group defined by T . Define T in a similar fashion with Tf replacing TT . Let H be the connected centralizer of 5 in G , and let Gi be the connected S -group associated to the group with root system J?i and Cartan subgroups dual to T. Let Hi be the connected centralizer of 5 in Gi . We have imbeddings
Tj'.H^G
,
rjiHi^Gi
.
Let M be the inverse image of QQ in the normalizer of T in G , and define Mg , Mi , Mg^i in a similar fashion. There are obvious maps
We shall also construct (p: M —> Ml ,(ps: M, —^ M,,i so that the diagrams M
(5.3.5)
-^
1 LM
—.
Ml
M,
i
i
^M(
^M,
^
M,,i
i —^
^Mi.
,
are commutative. Observe that ^ M acts on M , ^M{ on Mi , and so on. Moreover there will be a lifting of ^ to < in T such that ^ o ^ = ad t o ^1 o ^ , . Finally, if m G ^ M , m, € ^Mg , m G Mi , Ms £ Mg , then (5.3.6)
(p{m{fh)) = (f{m){ip{ffi)),
^s{ms{ms))
= <^5(m5)(^5(m5)) .
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527
Granting the constructing of <^ , ^ , , ^ , ^5 we verify (5.3.2) and (5.3.3). Set V^i = 9? o 77 , V'2 = ad t o T/j o <^, , V'l =<porj ,rp2 = a^dToTJiOifs and let if m G ^Ms projects to w. It is of course condition (c) that guarantees that 02 is a function of this projection alone. Since a{w)^{m) = rjo ^5(771) ,
m G ^T, m —)• ti; ,
and a i ( ^ ) 6 ( » ^ ) = ^1 o 6 , i ( ^ ) 5
me^T,m
—^ it; ,
we conclude from condition (b) that a2(w) = a(w)a^^{w)t
,a^,{(T(a'2){w2)))~^{a'2iwiW2)~'^,1),
which is simply au;i(<Ti(03(1^^2)))"^ or ^i(^<7"Va; )(^2(^2))~"^) • Thus we have to show that for all a; G fio and all w; in 1^ (5.3.7)
a^{a'2(w)) = 1 .
Since a^^io^sC ^ ) = a;i o aa^2( ^ V i ( 0 ^^ suffices /? G Do • Then if Rf^ is defined by X^{Rp) = obviously factors through V ^ T -^ Rf3 . Since homomorphism Xp: Rp —> T . If it were injective establishing the relation ^(«2(^)) = «2(^) y
to verify this for LO =ujp ^ Z/?, the homomorphism Z/? C X*(T), there is a we could verify (5.3.7) by
528
R. LANGLANDS and D. SHELSTAD
for the composition \p o a^^ takes i to (jj{i)t~^ . Unfortunately the homomorphism Xp is not always injective. If, however, we replace T by T then it becomes so. Of course a2{w) is not defined in T , but let ai^iw) lie in T and have an image in T, that is congruent to ai^iw) modulo the center of G[ . Then a^^{a2{w)) is the image of af^(a2{w)) and a^^Ca^iw)) = uj{a2(w))a2{w)~'^ . Choose m G Ms mapping to a~^(uj), where w —^ a ETT- Then ipi{w){jpi{m)) = xlii{w{m)) = ^p2{w{m)) . On the other hand, il^i{w)(4>i{m)) = a2{w){xl;2{w)(4>2{m)))a2{w)-^ -a'2{w)xl)2{w{m))a'2{w)~^ . We conclude that 1 = a'2{w)4)2{w{m))a\w)-^\i)2{w{rri))~^
- a2(u^)u;(a2(w;))"^ .
5.4. Construction of ^ and ^5 We begin with a general system 7^ , as in §2 of [I], and assume further that we have a surjective map 7^ —^ A, A having the properties of (5.1). We suppose that X -data are given on A and that if a —»• A then X « = X A o N'^Fc^/Fx '
Let /> be a given gauge on 7^ . On A we can construct a gauge pA by choosing in each orbit a representative A , as well as coset representatives CTi,... ,(Tr for r ± / r and then defining PK{(^T^^) *^ ^^ ^' Define, in the notation of §2.5 of [I], r^^ by n
1=1
where
Set P\Q) = pA{X{a)), and define a gauge Po on TZ in the same way as PA was defined on A . Lemma 5.4.A. The cochains w —>• Sp^pi(w)rp^{w) and ^ ~^ ^p/Po('^)''^p' (^) ^^ff^''^ ^y ^ coboundary. It follows readily from §2.5 of [I] that the quotient of those two cochains is a cocycle. It is enough to prove this lemma when the module X has
DESCENT FOR TRANSFER FACTORS
529
{a G 7^ I p\c^) = 1} as a basis over Z , 1Z is a single orbit under the group D introduced in §2.5 of [I], and for each A E A ,
To prove the lemma we have to prove that the quotient factors through a Galois group, and that if a G 7^ then the projection on the one-parameter subgroup Ra associated to Za of the restriction of the cocycle to W±a is a coboundary. Fix one a G 7i , and let A = A (a). Take the Weil group W to be WL/F where L is a large, finite, Galois extension of F. We first observe that for x E L^ C W^fp , (5.4.1)
rp^{x) = rp>^{x) .
The right side has been computed in §2.5[I] as
n x^iNm^F.cTxr~'^= n ^4n^w the equality following from the relation Xap{(Tx)=
Xf3{x) .
Since X _^ = X 7^ , the gauge pg can be replaced by p ' . If a -^ A then in the same way the left side is
n >^.(n-w) PA(/X)=I
Ver^
/
which is equal to
n n^. n n^-) or
n n>^^(n-w' Thus (5.4.1) is clear. Since the restrictions of Sp/pi and Sp/p' to W±a always have trivial projections to Ra , it is only the quotient rp^(w)rp'(w)~^ that need be
530
R. LANGLANDS and D. SHELSTAD
considered. Taking the projection of the restriction also allows us to suppose that A = {±A} , r = T±x , with PA(A) = 1. Choose ?;o = 1 ^ W±x , and if A is symmetric vi = v £ W±x — W^x . If w; G W±x let w = x{w)vi, x{w) G W^x • Then X = X x may be regarded as a character of W^x = ^ F + A / F = ^ » ^^^
Projecting on Ra we obtain X (x(w))^ . Since r_p/ = Vpt , we may assume that pg = 1. It is convenient to consider three cases separately, although in all of them the conclusion will be that r^/ (w) projects to X {x(w))^ . (i) a asymmetric, A asymmetric. We calculate with the notation of §2.5 of [I], noting that with our choice of X a ? X ai'iJ'ii'w)) = X{WiWW~^) .
Taking, as we may, w;i = 1 , so that wi> is also 1 for w E. W^a == W±a , we see that the projection of
on Ro, is X(wiwwY^y= X (w;)" . (ii) a asymmetric, A symmetric. Let v -^ r E T and let ( T I , . . . ,(7r be a set of representatives for 74.c^\r+A • Then {cri,... , cr^ , crir,... , o-^r} is a set of representatives for r ± a \ r . Lift <7,- to Wi in W , and set x(w) = vx(w)v~^ . We take (ji = 1, wi = 1. If w = x{w) then ^piH = n
^ {xix{w)wr^)^^
i
Y[x{wix{w)w-,^y~'''^
,
i
where j ' = j'ii^w) is defined by(7,o' = pi{d')(Tj/ if ^(ly) —» ^ . If w^ G r ± a then the projection of the right hand side on Ra is X {x{w)) because a is asymmetric. If w were x{w)v and a the image of t/; in r then era would be a and crA would be A . This is impossible, (iii) a symmetric, A symmetric. We take the representatives of r ± c r \ r to lie in Tx and we take v in r ± a \ r + a . If w^ = x{w) lies in W^a then rpf(w) is calculated as in (i). If «; = x(w)v lies in W±a\Wa then the projection of Vpi on R^ is again X {x{w))^ . Returning to our special situation we choose p\ on A, PQ on R^i , and PQ on Ri . Together pj ^^^ Po define po on i ? . We have natural factorizations ^P/Poi^)
= ^p/Po,iH^P/PoM^)
rp{w) = rp^i{w)rp^^i{w) .
=
^p/pli^H/p'oi^)
DESCENT FOR TRANSFER FACTORS
531
T h e r e are similar factorizations for the barred quantities. Recall from §2.6 of [I] t h a t ^{w) = rp{w)'n{cjT((T))
Xw
= ^p/po.lW^Po,!W«P/PO,-I(^)^PO,-IH^(^T(
X W .
In view of the lemma, we are free to replace «p/po,'^i^po,'^i ^y ^P/P'^PA J obtaining a homomorphism t h a t we still denote by ^ ,
We modify ^{w) in the exactly the same fashion. T h e homomorphism (p is defined on T as the imbedding of T into T" . To define it on all of ^M , we must define (p(n{ojujT{o'))>^u)), CJ G fio 5 w EW , w -^ a . Since Spfp/ is defined on all of Qo>^r7' we may set (p{n{ijJujT{(T))>^w) = s~J^,{uj(T)r~l{w)ni{ujLJT{(T))^w
.
There are three conditions to verify in order to show t h a t <^ is a homomorphism. For simplicity, write n{ujT{(^))>^'^ ^ ^G as n{w). (i) ip{n{ijji))ip{n{uj2)) = i{^i,(^2)^{n{LU\uj2)) , wi,<^2 G ^ 0 (ii) <^(n(t/;i))v?(n(if;2)) = t((Ti,cr2)<^(n(u;it/;2)) , Wi EWT , Wi-^ (TI (iii) (p{n{w))(p{n{ijj))(f{n{w))~^ = 6^(0;, (7)v?(n((7(a;)) , CJ G fio , <^ ^ WT , W —^ a . Here (5.4.2)
et{u;,a)
= t{uj,a)t{au;,(T-^)(T(Lj){t{a,a-'^))-^
.
T h e relation (i) a m o u n t s to (5.4.3)
5^|^;(a;i)a;i(5^lj^,(a;2))^i(a;i,a;2)sp|p/(a;ia;2) = ^(^1,^2) •
T h e p t h a t occurs here is really the restriction of p on i? to ii^^i , and in this sense tt]^^ = tp . Since tp'{(jj) = 1 for a; G ^ 0 because p' is invariant under QQ t h e relation (5.4.3) is clear from Lemma 2.4.A of [I]. T h e relation (ii) is valid for a similar reason. One has only to observe that rpl{^i)^i{rp^{^2))rp^{wiW2)=tp^{(Ti,a2) .
T h e relation (iii) amounts to the equality of
and
532
R. LANGLANDS and D. SHELSTAD
if e^i is defined by the analogue of (5.4.2), Gi replacing G and ti therefore replacing t. If e^i is defined as in (5.2.8), the element LJ^ being replaced by to then e^^i = e^e^/ • On the other hand, the boundary of Spfpf being tp/ipi we have
Clearly Moreover tp>{uj,((T,a-'^) = cr(a;)(fp/((j,
(T"^))
,
because p' is invariant under QQ . Thus
(The inelegant appeal to the fact that we are dealing with cochains of order two is entailed by an infelicitous definition of ^ in §2.6 of [I]). The homomorphism (ps is defined in a similar fashion. Condition (a) is of course manifest, and condition (b) follows easily from the definitions. To verify (c) we have to prove the existence of t ET such that (5.4.4)
ip{r){ns{uJi3))) = adt(77i(^,(n,(a;;,)))) ,
/? G So .
Suppose we can prove the existence of fi G T and for each /? G EQ of A/5 G C^ such that (5.4.5)
(p(7j(ns{ujf3))) = X^^^dti{r)i((fs{ns{u^/3))))
Then we choose ^2 such that /?^(^2) = ^p for all /? G So , and (5.4.4) follows with t = ti<2 • Observe first of all that if to = co^ then riiusiuj)) = b-'^{u;)n{u;) , so that ^i^{^s{(-^))) =^~H^)Splp,{^)ni{Lj)
.
On the other hand, mi^sirisiu;)))
= 5-^^/p, (u;)77i(n,,i(a;)) = s-^f^,^{uj)b^\u;)ni{u)
.
DESCENT FOR TRANSFER FACTORS
533
Thus the pertinent factor is
Since p' is invariant under QQ )
a>0,cc;"^a<0,p'(a)=:l
with Rs = R{H).
Observing that p\—u~^a)
= —p'(a) , we write this as
aeR-^-(Rt^Rt) with
t,=
n
(-1)' •
a£R+-R+^ p'(a)=-l
Equation (5.4.5) now follows from part (b) of Lemma 4.3.A. The coroots a that appear in the definition of Spfpi and r^^ are also coweights of T , so that we may interpret the expression defining them as giving functions Sp/^/ and Fp^ with values in T. Then (p is defined by ipiniuuTia))
x w) = s'^^^,{u;(T)rpl{w)ni{ujUT{(T)) x w ,
and even on a group containing M that we could call ^M . That it is a homomorphism is proved in exactly the same way as for (p . The homomorphism (ps is defined in a similar fashion, and the diagrams (5.3.5) are clearly commutative. That the element t can be also lifted to t so that (p o rj =: a.dt o rji o ifg is also clear. Finally (5.3.6) is valid because we can define ip on ^M and (ps on ^ M , . 5.5. Reducing the dimension of Gder Since are arguing by induction, we can exclude from consideration any group G with subgroups G^,... , C such that dim G^^j. < dim Gder for all i and such that the truth of the statement (5.1.1) for all of the G* implies its truth for G . Since that statement is clearly invariant under z extensions [K i ] and since any two z -extensions are covered by a common z -extension, we can immediately suppose that Gder is simple over F , and indeed that G is obtained by restriction of scalars from a group over a larger field. Then a simple argument that we prefer to omit allows us to assume that Gder is absolutely simple. There are two obvious ways of constructing a root system Ri between RQ rz R(He) and R invariant under F T and with a map to A satisfying
534
R. LANGLANDS and D. SHELSTAD
t h e conditions of the critical Lemma 5.1.A. T h e first is to take Ri = {a | a(£:)^ = 1} 5 A = {Qf(e)} C F^ and the m a p a -^ a(e) . T h e second is to take Ri = {a \ a ^ ( s ) 2 = 1) , A = {^^(5)} C C^ with F T acting trivially and the m a p a —^ a^{s). Since we can always use the transitivity of L e m m a 4.1.A of [I] t o choose T = T^^ , we can also suppose t h a t (3.3.2) is satisfied. T h u s we are reduced to the case t h a t a(e) = ±1 and a'^{s) = ± 1 for all a. In general let us call (G^e^s^T^T) (i) (ii) (iii) (iv)
primitive if
T h e element e is not central in G, and s is not central in G, Gder is absolutely simple, T h e set R^^^ is not empty, T h e r e is no system Ri C R satisfying t h e conditions of t h e critical lemma, so t h a t in particular e^ is central in G and s^ central in G.
We now assume t h a t (G^e^s^T^T) is primitive, and see what this implies, although we shall see t h a t the critical lemma allows us to impose even further constraints on the problem. On p . 708 of [L 2 ] a diagram, either a Dynkin diagram or an extended Dynkin diagram, together with an action of the Galois group was attached t o the pair ( T , s) (there denoted ( ^ T Q , /c)). In the present case t h e diagram D ^ (there denoted D) is the union of XQ^X^ (there denoted 3to, 3ti ). t h e same construction can be applied to T , 6 yielding D , 3to 5 3Ci . L e m m a 5 . 5 . A . / / (G, 6, s, T, T ) is primitive then both diagrams D, D^ are extended Dynkin diagrams, and Xi,X^ both consist of a single simple root, whose coefficient in the expansion of the largest root is 2. It suffices t o t r e a t the diagram D . T h e n XQ is just a set of simple roots for Gc . If it were a subset of a set of simple roots for G then G^ would be t h e Levi factor of a parabolic subgroup of G . This parabolic subgroup need not be defined over F. Take Ri = R{Ge) and let A(a) be the restriction of a to the center of Ge . It is clear t h a t A(a) ^ A~^(a) for a G R{G/Gi). T h u s a - ^ A(Qf) satisfies the condition of L e m m a 5.1.A, contradicting the assumption of primitivity. We conclude in particular t h a t D is an extended diagram, and t h a t XQ contains the negative of the largest root a^ . Let a i , . . . , a^ be the simple roots. By construction every root a such t h a t a(e) = — 1 is the s u m of a single element of Xy and an integral linear combination with non-negative coefficients of t h e elements of XQ . If aj G Xi then —otj assumes the value —1 at e, T h u s for some otk ,
(5.5.1)
-OCj = <^A: +
^
CiOti - CoOCo
DESCENT FOR TRANSFER FACTORS
535
with Cj > 0 , Ci = 0 if Qfj ^ XQ . Since t
(5.5.2)
ao = y^6fQfj
with 6,- > 0 for all i, the equation (5.5.1) is only possible if j 9^ AT and Xi = {aj,afc} , Co = 6j = 6A; = 1 or j = fc and Xi = {aj} , cbj = 2 . If Xi — {ttj} and 6j = 1 then (5.5.2) implies that ao(e) = — 1, which is out of the question. Thus to prove the lemma, we need only exclude the possibility that Xi = {aj,ak} . The action of Ge on the quotient g/g^ is a direct sum of distinct irreducible representations p. Thus to each a G R(G/Gs) we can associate the p = p(a) in which it appears. Clearly F acts on the set of these p and cr{p(a)) = p{(Toc) while p{—Oi) = p , the contragredient of p. We extend the action of F = F T to E = E T by demanding that the non-trivial element of Z2 C E send p to p. We claim that 01 = 06 + y ^ 0p , Qp being the space of /?, is a subalgebra. The root system of Gi is obviously a possible Ri , for we can take A = {p | p Cii p} . Each p has a minimal weight and by the definition of XQ , Xi this weight lies in Xi . Let it be aj = aj (p) . Then Qp ~ 0 - if and only if —aj = aj 4- 2_^ ^i^i ~ ^oOLo ,
c, > 0 .
a,63to
Comparing this equation with equation (5.5.1) we see that g^ = g if Xi consists of a single element, and that gi = g^ if it consists of two elements. We infer first that it is in both cases a subalgebra, and then that Xi consists of a single element, that we denote aj . The connected components of XQ decompose it into the disjoint union of connected diagrams permuted amongst themselves by Fy and, in the same way, Tf , for we could as well define XQ , Xi starting with T rather than T. The result is the same. Suppose that XQ is the disjoint union of two non-empty sub diagrams X'Q , XQ , each invariant under F T and each the union of connected components of XQ itself. We shall use the transitivity of Lemma 4.1.A of [I] and the Critical Lemma to show that this case too may be reduced to that of a group of lower dimension, so that we may impose one further restriction, (v) XQ is not the disjoint union of two non-empty subdiagrams X'Q , XQ each invariant under F T and each the union of connected components of XQ itself. Moreover X^ satisfies the same condition.
536
R. L A N G L A N D S and D. SHELSTAD
Supposing t h a t such a decomposition exists we let XQ be the subset containing t h e negative of the largest root. Since XQ is the Dynkin d i a g r a m of Qe this leads to a direct s u m decompositon, (5.5.3)
0e = 0' e g" .
To b e definite, we p u t the center of ge in g ' . It is pertinent to observe t h a t 0e has the same rank as g because Xi consists of a single element. We have seen, moreover, t h a t the representation /> of 0 on g/g^ is irreducible. T h e decomposition (5.5.3) implies a tensor-product decomposition p c^ p ' 0 /?" . Since p C:^ p we have p' '::±'p' ^ p " ^ 'p" . Moreover every element OL of R{G/Ge) may be represented as a = ( a ' , a " ) , where ( a ' , a " ) are t h e weights of p' and p" . L e m m a 5 . 5 . B (i) a' is never zero; (ii) There is an a for which ot" is not zero, moreover a" is zero if and only if a is a rational linear combination of roots in XQ . Every root of XQ is orthogonal to every root of XQ . T h u s XQ , XQ span mutually orthogonal subspaces of X*(Tder)0R whose s u m is X * ( T d e r ) 0 R • T h e components a ' , a " of a may be identified with its components in t h e two s u m m a n d s . T h u s a " is zero if and only if a is a (necessarily rational) linear combination of roots in XQ and a ' if and only if it is a linear combination of roots in XQ . Since XQ is contained in t h e set of roots simple with respect t o a suitable order, a root t h a t is a rational linear combination of the elements of XQ is necessarily an integral linear combination. T h e first assertion of the lemma follows. If a " were 0 for all a then p" would be trivial and t h e roots in XQ orthogonal to aj and t o all the roots in XQ . This contradicts the assumed absolute simplicity. Take i^o = R{He) , and let RQ he the set of a £ RQ t h a t are roots in g' and RQ the set of a e RQ t h a t are roots in g" . If fi', Q " are the Weyl groups of g ' , g" then fio = ^ 0 ^ 0 with fi'o = fio H fi', fi'J = fio H Q" . In t h e same way He factors as H[ • H'^ . If Vf and
if UJ{(T) = UJ\(T)U"{(T)
= {ijj{(T)(T I O - E T T }
, (JJ'{(T) G %
, UJ'\(T)
G ^O
then
TT' - {a;'(cr)(7 I cr G F T } is a subgroup of ^Q^^YT
and
r ^ = {J\a)a'
I a' G F T / } .
As the notation implies, YT' gi^es the Galois action for a group T' t h a t could b e substituted either for T or for T . By Lemma 4.1.A of [I], it
DESCENT FOR TRANSFER FACTORS
537
suffices to prove Theorem 5.LA for the pairs {T,T') and {T,f) . To deal with the pair {T^T') we apply the critical lemma, taking the RQ that appears there to be RQ and Ri to be the union of R{Ge) and {a e R{G/Ge)
I a" = 0}
and introducing yet a third Cartan subgroup, whose projection on H'J is stably conjugate to the projections of T or T ' , but whose projection on H'^ is maximal split. Then A is defined on R{G/Gi) by A(a) = a" . For the pair T ' , T we take the RQ of the Critical Lemma to be RQ . The set Ri is R{Ge) and A is defined on R{G/G\) by A (a) = a ' . It is again necessary to introduce a third supplementary torus and to use transitivity. L e m m a 5.5.C. Suppose thai (G^e^s^T^T) is primitive and that in addition the condition (v) is satisfied. Then all roots in R{G) are of the same length. We examine the possibilities for groups with roots of unequal length, using the appendices of [B]. (i) Bi: The diagram is
> 0C2
o • • -
o ai-i
>
o OLt
The dual diagram is O
>*
O
O
.
.
•
O
O
V—^'^^ O
and ao = a i + 2a2 + . . . + 2a^ , /?o = 2/?i + . . . 2/?^_i + ^t • Condition (v) and Lemma 5.5.A imply that X\ — {oct] , that i — 2k is even, that X\ = {Pk} and that F T acts nontrivially on the diagram. We conclude that a{€) = 1 if a is long and that a(e) = 1 if a is short. On the other hand, a^(s) = — 1 only if a^ is short, although a"^ (s) can be 1 for some short roots. Since R^^^ = {a I a{e) = - 1 , a'^is) = - 1 } , it is empty, and groups G of type Bi are excluded. (ii) Ci: By the symmetry of the conditions, groups of type Ci are also excluded.
538
R. LANGLANDS and D. SHELSTAD
(iii) F4: The diagram is
and ao = 2ai + 3a2 + 4a3 + 2a4 . Then Xi = {^4} . Thus, with the notation of Table VIII of [B], a{€) = ~ 1 if and only if a = ^(±^1 ± 62 it 63 ± 64) , and therefore, in particular, only if a is short. By duality a^(s) = —1 only if a is long, so that R^^^ is empty. (iv) G2'' The diagram is
ai
—ao
Oil
and ao = 3ai -f 2a2 . Thus 3Ci = {a2} . This however, is incompatible with condition (v).
§6. T h e Order-Two Case 6.1. Introduction According to the last section it remains only to prove (5.1.1) under the following assumptions: (6.1.1)
a(e) = ibl,a^(5) = ± l ,
a^R',
and (6.1.2)
all roots in R are the same length.
In this case the roots of type (d) are those ex ^ R for which a(e) = —1 = a ^ ( s ) . We begin by describing formulas for ©^ ^ and lim ©2 . Let T(5) be the F-torus with dual T(5) = T^^/{l,Sj.} endowed with the action induced by Yf . Then dual to
we have 1 — ^ — 7(,) ^
T,, -^
1
DESCENT FOR TRANSFER FACTORS
539
where A has order exactly two. Each of the terms in v^^^ , and hence also v^^^ itself, can be constructed in T^^ . We do so without change in notation. A lifting v of v^^^ to a cochain in T(,)(F) will be described in (6.2). The coboundary dv takes values in A and so defines an element ej of H^{T,A) = {±1}, Then (6.1.3)
e^'^(7.,TH) = e i .
To check this we note that the inclusion {l,Sf}
"^-^ T^^ yields
We have a commutative diagram
I'
I'
(see [LI] or [M]) and since Q\ KIH^IH) ^^ given by evaluating v^^"^ , as an element of H-^{X^{f^^)), on s , (6.1.3) follows. The term Um ©2 is handled similarly. Following (4.5) it is given as
the pairing being that for H^{W, fxT) and T{F) x f(F). We shall move to the torus S = T x T^^ from (3.5). It has the Galois action (6.1.4)
^:{tuh)
—^ {cr{h),a^(cT(h))ija{t2))
,
where & = u x c , We also have TxT -^ S over F and dually S (see (3.5) for definitions). Recall that w -^
^^TxT
(a(^)(^),c(^)(w;)-iz<^)(a;,ti;)-i)
is a cocycle in S with image w — . (a(^)(t/;),a(^)(t/;)c(^)(tx;)-^S<^)(u;,w;)-^) under
^
H\W,S)
—y H\W,fx
f) .
More precisely, this was shown in (3.5) for cochains without the superscript (d) , but a factoring argument as in (4.5) allows us to consider just the contributions of type (d). We may therefore compute lim &2{7HJ7H) ^ ((a(^)(tx;), ^^\wr'z(^\LJ,w)-'),
cs)
540
R. LANGLANDS and D. SHELSTAD
where e^ is the image of (e, 6"^) under T{F) x f{F) ^_S{F) . Note that es = (l,ead) where ead = e'J^ is the image of 6 under T —>• T^^ . Next define a torus T(g) over F by
-^n^)
1 —^ {l,fad!}—T^ Then we: have 1- -
^ B
•^ ^ e ) -
^T
—^ 1 ,
with B of order 2, and 1 —* 1
fx%)- - ^ f X
X B—>
L
1,
or, more simply, 1 —•*B
—
-fx%)
—^S-
1.
In TxT(^f) the Galois action is given by (6.1.4) with a^ now taking values in T(£), as is possible because the roots of EQ lie in Xt{T(^e)) • Dual to this is
l-^{l,es)^S-^Tx
T(,) - ^ 1 .
In the next section we will define a lifting of to T(e) and thus a lifting of the cocycle
C^^\W)~^Z^^\UJ,W)~^
E T^^
u('^\w) = (a(^)(tx;),c(^)(i/;)-iz<^)(a;,«;)-!) to a cochain u(w) with values in T x T(g) . The coboundary du takes values in B and factors through W -^ T . It then defines an element €2 of if2(r,J5) = { ± l } and (6.1.5)
lim02(7H.7H) = ^2 •
This follows from the commutativity of the diagram below (see the observations at the end of Section 6.5).
I' {TxT^,)){Fy
I' —
S{Fy
V ^
{1,65}*
We observe that Ac± B c^ ^2 and that the homomorphism
DESCENT FOR TRANSFER FACTORS
541
is injective. This is essential in all that follows. 6.2. Liftings Fix i = ip G F^ such that P = —I, We write v^^\a) y^^\&)~^v^{u; X a) , where
as
V,{LJ xa) = r(^)(a;,(7)a;(2/(^)((7))6(^)(u;)-i with a = LJ X a , and lift term by term. First
y^''\^)= n yoi^) where
Voi^) = n <"" • aed a>0
We shall define y^(^) E T(^s) ^^^ then set y
(a) = fl^oC^) • a To fix a-data for all asymmetric orbits choose one orbit, say O , from each pair ±0 and set a^ = — 1 for a £ O ^ a^ = 1 for a E —O . Define
la(^)= n *""' a60 a>0
and On the other hand, resentatives (7i = 1 , for each j . Thus O / o ^ = cr. ^^/a^. We
if O is symmetric fix a > 0 in O and then rep(72,... ,(T„ for r ± a \ r such that aj — ^~^^OL > 0 = { ± a j : I < j < n} . Fix y ^ G F ^ and define have
mi^) = n
^«i •
Set
5d(^) = n (v^)^""' It remains to lift the terms in v^{uj x cr). We define 5^^^(cr) in the same way we did y
(o-) and then lift Lj{y^^\o'))
as uj{y^^\a)).
The
542
R. LANGLANDS and D. SHELSTAD
term T^^\UJJ(T) is a product of elements (—1)^ over certain a of type (d). Lift T^^\U;J(T) to the corresponding product T^^\LJJ(T) of elements P^"" . Recall from (4.5) that 6(^)(a;) = bl^h(^\u;). The factor 6(^\ij) is a product of terms (—1)^ each of which we lift to P^ . Recall from (4.3) that 6L ^ is of the form JJ ^k ' where Xk £ F^ , /?ib G So • Such an element k
is naturally lifted to T(,) (by the same formula) since On the dual side we have to lift the cocycle {a^^\w),c^^\wy^z^^\u,w)-^)
SQ C X * ( T ( , ) )
= {l,c^^\wy^)u^{u;
.
x w)
with u^Lo xw) =
(a^'^\w),z^^\uj,w)-^)
= (a(^)(w;),f(^)(cc;,^)a;(c(^)(ii;))aa.(a^^^(ti^))6(^^(a;)-^) . Now i = ic will denote a square root of —1 in C . We start with c^^\w),
lifting it to f^\w)
in f(e). Then {l,c{w)-^)
is to be lifted
to {l,f'^\w)-^). The term ^^\w) is a product n ^ ± d ( ^ ) ^^^^ pairs ±0 of orbits of type (d). We proceed term by term in this product. If O is asymmetric then we take x~data for ±0 to be trivial. There is still a nontrivial contribution to r^.^ , namely the term Spfg of [I, 2.4]. Here p is the gauge on ±0 defined by the fixed order on the roots and q is given by q{a) = 1 if and only if a G ^ . Thus ([I, 2.4])
v.(^)- n (~ir a>0
and r^^{w)
= 5p/g(cr) if w; ^^ cr under W ^^ T . We define r^o(w)
to
be the product P^ over the same roots. If O is symmetric the contribution Sp^g is trivial because we have arranged that jp = q . On the other hand, the x~^2ita {xa} now are nontrivial. Following [I, 2.5] we write n
n
i=i
i=i
Fix some square root of the complex number s{uj (w)) , denoting it by y/s(uj{w)) , and then set
DESCENT FOR TRANSFER FACTORS
543
It remains to lift ti* {uj x w). Again we proceed term by term. We lift cW(tr) as we did c(^)(t/;) , r(^)(a;,o-) as we did r(^)(ct;,(7) , and U'^\oj) as we did b^^\Lj). We shall regard a^j as taking values in T(e). Then S,(a; xw) = (a^^)(w;),?(^)(a;,(7)u;(c<^)(ti;))a^(a('^)(ti;))?^)(a;)-^) . 6.3. Some coboundaries For a of type (d) the element (—1)^^ of A is nontrivial and all such elements coincide. Similarly B = {(±1)^^; a type (d)} . We identify both A and B with ^2 = { i l } • At the same time we identify ic and ip , and then the subgroup 5 ' of Tu) generated by {P^:a type (d)} with V
the subgroup A^ of T^,) generated by { r " : a type (d)} (recall that all roots have the same length); this of course does not respect the action of F T or Tf . According to (6.1.3) and (6.1.5) we have to compute the 2cocycle dv du with values in fi2 • This coincides with dv/du which is more convenient for calculations. Where needed, we inflate cocycles of F to W without mentioning it in notation. In this section we investigate the contributions to dv/du from iT* and u^ . First, v^ and v^ are well defined on S^o>^r7'. From Lemma 4.2.B we have dv^{(jJlP,(jJ2
T^^\(J0IP,(JJ2O-)~^
and so if we lift r^^) to r^^) by replacing each term (—1)^ by v^^ then we conclude that dv^f^^^ takes values in A — pi2 - Similarly u^ andS* are well defined on Q.Q>\W and Lemma 4.2.A shows that du^r^^^ takes values in B = A = fi2 - Here a; x cr G Q>o>iW acts on T x T(f) as in (6.1.4). Hence dv^ , du^ take values in B' = A' ^ and dv^/du^ takes values in A since f^^^ is identified with f^^^ . The cochain dv^/du^ is not in general a cocycle (since the operator d in the numerator is that for the Galois action on T^,) and the operator in the denominator is for the dual algebraic action). We calculate its coboundary as the coboundary of dv^T^^yOu^T^^^ and thus as {u}lp,UJ2Cr,UJ3T)
^
pT^'^\u;2(r,UJ3T)/pT^'^\(jJ20',U)3T)
which equals
n
2a"
(?)'
When A is identified as fi2 this becomes N(UJ2<^,^3T)
{LJip,u;2(T,u}3r)
(f
544
R. LANGLANDS and D. SHELSTAD
where N(U;2O;UJ3T) is t h e number of roots a appearing in t h e product above. We embed //2 in fi^ = ^iA{F), t h e group of fourth roots of unity in F . T h e group Qo>^M^ acts on ^4 through F . Fix ^ G F ^ such t h a t ^^ = i and consider M{iJip,U2(T)
= ( "7J
where N{uj2^) is t h e number of a of type (d) for which a > 0 , cr~^u;^^a < 0 . T h e cochain M. takes values in ^4 . Observing t h a t N{U)I(T)
we find t h a t dM have proved:
+ N{UJ2T)
- N{(jJi(7ijj2r) =
2N{u)ia,uj2T)
is t h e inverse of (6.3.1). Let 0 = Mdv*/du*
L e m m a 6.3.A. 6 is a 2'Cocycle O/QOXTT
. T h e n we
with values in fi4 .
L e m m a 6.3.B. e{uji X p,u}2X
a) = e{p,a)
.
P r o o f . Because ^ is a 2-cocycle it defines an extension of QQXTT by //4 . T h e extension is generated by fi^ and elements fj, 77 6 r2o>JFT , with ffxfi-^ We have t o show t h a t {uJip)^{uj20'T
= r]{x),
f}ifJ2 = 0{r}i,'q2){'nimT
= 0{p,a){cjipuj2(r)^
'
•
For this it is sufficient t o verify (6.3.2)
Q1Q2 = {(^1(^2)^ ,
(6.3.3)
Qa = (ioa)^
and (6.3.4)
aQa-^ = (aua-^)''
.
Moreover (6.3.4) need only be verified for a; = a;^ , /? G So • Indeed, assume t h a t (6.3.4) holds for ui and (JJ2 . T h e n ?(a;iu;2)^?~
= ?a;ia;2^~^ =
{ai0iuj20'~^)^
so t h a t (6.3.4) is also valid for UJIUJ2 •
DESCENT FOR TRANSFER FACTORS
545
If all these conditions are satisfied then (WI/))^(W2
= QipQ2^ = pip~^^ip)^<^2^
or p?(
e{p,a){pcr)^{(p(T)-Y{^iP^2p-Y{p
which is e{p,(T){u)ipu)2(r)^ and the lemma follows. The three conditions may be rewritten as (6.3.5)
e{u>i,U2) = l
(6.3.6)
e{w,(T) = l
(6.3.7)
e((T,u;)(T(u))(e(
First observe that M(ui,u}2) = M(U,
/ ^ \ ^M
But N(u;) is twice the number of roots of type (d) in R^ ; so we may rewrite this expression as (A-) ^^ . Turning now to dv^ and du^ , each taking values in A^, and C = dv^/du^ taking values in A = //2 , we have C{u)i,UJ2)
= db{ljJi,UJ2)/dh{(jJi,LJ2)
and C(uj,a) is the quotient of Z(u;)-'u{y{a))^u;,a)-'u;{y{a))-'b(u^) by
and this is
_
= 1
546
R. LANGLANDS and D. SHELSTAD
Here, and below, the superscript (d) has been omitted from the notation. Similarly we find £(
wher
Es = S{cT(uj))a(S(uj))-'e{uj, a)
E2 = c(w)(T{uj){c{w))~^au;{a{w)~^) and E3 = 6{(T{uj))a(S{Lj))-^^{Lj,a) . Recall that uj = ujp . To simplify this expression for £((7, a;) we need some preparation. L e m m a 6.3.C. There exists rjp = ±1 such thai
and
DESCENT FOR TRANSFER FACTORS
547
Proof. This is a long calculation in which we repeatedly use the fact that
First, e{u>, 0 , uj~^ 0 and that over era;" V - ^ a > 0 , oj'^a^^a > 0 of (—1)"^ . When multiplied together they yield the product over a < 0 , a ; ~ ^ c r " ^ a < 0 , cru;~^cr"^a> 0 . The first factor is the product over a > 0, (T"^a < 0, u;~^cr~^a > 0 of (—1)^ . For e(a;,<7) we then have the products over (1)
a > 0, Lj-^a^^a
> 0, cr'^a > 0, (rcj^^a'^a
<0
and (2)
a > 0, c j - V " ^ a > 0, a'^a
< 0, aLj-^cr'^a > 0
of ( - 1 ) - ^ . The contributions of o'(6(c<;^)) , S{
(T-^a > 0, LJ-^or-^a < 0, (r'^a E R'^
and (4)
a > 0, (TUJ-^a-^a < 0, a G i?^
of the same term. Here uj =• Uf^ . The left side of the second equation is an exactly analogous product, roots replacing coroots in the exponents. Consider the contribution of {ifcQr,±a'} , where a' = —(TUj~^(T~^a . If a and a' have opposite signs then these can be contributions only to (2) and (3). Taking a > 0 and supposing that a;~^cr"^a and (T~^a have opposite signs, we see immediately that these contributions are: (T-^a > 0, a-^a G Rp.
cT-^oc < 0, -(7-^a G a-^Oi < 0, -a-^a
^ Rp,
{-iy'\-ir'']
R-^:{-iy'^{-iy^', (-1)''^{-1^'^.
Thus we are done with the case of opposite signs. Observe that explicit calculations were not really necessary. It suffices to observe that the contribution to each of (2) and (3) is (-1)^'' , 7 G {±a, ± a ' } . If a, a' have the same sign, then { i b a , ± a ' } can contribute to (1), (3) or (4). There must be a contribution from exactly one root to (4). There is
548
R. LANGLANDS and D. SHELSTAD
a contribution from at most one root to (3). It occurs if and only if there is no contribution to (1). Because we may identify each root with its coroot the same argument applies on the dual side to yield the second equation. The lemma is thus proved. We now recall the equations in Lemma 4.1.A. The first, e(u;,<7)6(
= [^(
because uj = uj^ ^ and
y(a)a(a;)(2/( 0 , cr""^a<0. Thus the equation may be rewritten as ^P"
or, since we are in Gsc , as
Now we pass to T(^,) , obtaining
K,cr{h,)-'J[a'>f'P)
ap^
in this torus. The left side is E1E2 and so
E2E2 = rff Similarly on the dual side, we find ^1^2 = r)f . But rf'p'' rj^^ is rjp. We therefore cancel E1E2 with E1E2 IS identified laentmed with witn rjp and conclude that C{a,uj) = E3/E3 . This quotient is simply
^^A^^^^
or
T\
(-h:)
^ .
C * ) " " ? ™ . Hence £(
In /i2 this is
DESCENT FOR TRANSFER FACTORS
549
cancels with and the proof of Lemma 6.3.B is complete. We now return to dv/du and write it as a product of cocycles {Md^/dc)-^
• (Mdv^/du^)
.
Lemma 6.3.B says that we may calculate the second cocycle as Mdy/dc using the action of F T rather than Tf . Factor M as Ho Mo where
M.,.)=(^)'
Mc
No{cr) being the number of a in O for which a > 0 and a~^a < 0. Then we have Mdy/dc= JJA±C? ±o where X±o =
MoM-odyody-o/dr±o
if O is asymmetric. By now familiar arguments show that dyody-o/dr±o ( O symmetric) and dyo/dro {O symmetric) take values in ^2 and that A±o is a 2-cocycle with values in ^4 . Similarly,
Md'yjd'c-
Y[^±o ±0
where now
Moih-^)^ (^)^a(^:) . We conclude: Theorem 6.3.C.
\±o
J
\±o
6.4. Remaining steps To complete the proof of (5.1.1) we will show: (6.4.1)
\±o is trivial for ± O asymmetric,
(6.4.2)
invi^Ao = Xa(aa) for O symmetric.
550
R. LANGLANDS and D. SHELSTAD
and (6.4.3)
n
Xa(-2) = n
Xa(-2)
Here invi^ denotes the isomorphism H^{T^^^) = Z4 . Since we have chosen i G C^ and i G F^ we may identify this Z4 with fi4{C), In this section we prove (6.4.1) and (6.4.3). Suppose that O is asymmetric. Then
a>0 <7~^a<0
and so
w/>,-)= n ^ n f<7~^a<0 a£OU-0
<7"^a<0 aeO
Let C/PC = e . Then e^ = 1, p(2)/i = 6"^ and \
(
n
^±o{p,
a>0 \
Of
€0
e 'e
-2
n
a>0 cr-^a<0
= (e-'Y /
where Ni is the number of terms in the first product and N2 is the number in the second. We observe that Ni = N2 and thus prove (6.4.1). Let <^ be the group of all automorphisms ^ of R^^^ such that ^ (—a) = —
det p2i
i-lf^'^K
The group QO^^TT is imbedded in <^ and it is easily seen that N{u;) is even for cj G fio • Thus if a in F T corresponds to a in Tf then (6.4.4)
det p2{cr)/det />i(cr) = det p2{^)/det
pi{a) .
DESCENT FOR TRANSFER FACTORS
551
L e m m a 6.4.A. Suppose a* G F T corresponds to x £ F* , Then Y[ X^{x) = det p2i(T)/dei PI{(T) . o Since the lemma will be valid for T just as well as for T , it together with (6.4.4) implies (6.4.3). Proof. We note that pi is the direct sum over a set of representatives ^ for the double cosets ^±a\^/^T of the representations
Pr = i
=r±^
and ^2°ad V is the character Xp regarded as a character of GaJ(F/F±^)
= r±^. If (7i,... , (Jr is a set of representatives for r ± / ? \ r and Cjcr = PJ{(T)(TJI then
det pf (
X;,(/?,(
,
3
and, by local class-field theory, this is X p{x). Since det p2i(r)/det pi(a) = J J d e t p^ (
the lemma follows. 6.5. S y m m e t r i c orbits Throughout this section O will be a symmetric orbit. Recall that O = {iaj-: 1 < i < n} , where aj = o'J'^a > 0 and 0-1 = 1, cr2,... , cr„ are representatives for T±a . Define aj(/9) , ajf(cr) G F^c^ by ajp = aj(p)aji and (Tj/(T n ajf(a)ajti . For cr € F±ct define <^(^) = 0 if aa = a and (5(cr) = 1 if era = —a . L e m m a 6.5.A. <5(a,/(<7)y
dyo{p.(T) = Y[(Tj^
aj{p)y/a^
2ar
552
R. LANGLANDS and D. SHELSTAD
Proof. Let 6 be the character on r±a given by 6{
yoiP) = U^^ Kyo(.))=n-i
[ ^
if^M>)) = - i /
(.,(,)(V^/(M.))
if^(a,,(.)) = - l |
and
Thus
yo{P.^) = ll^i'Ap where Aj is given by the following table: e{aj{p))
e{ajia)) 1
1
-1
1
1
-1 -1
-1
Aj
1
»j(p)\/«r/\/ar 1 y/a^/aj{p)y/a^
Since ocj{p)y/aZ = :ty/a^ if aj(p)a = a , that is, if 0(aj{p)) = 1 , the lemma follows. Recall that to define VQ and ro we have chosen Wj mapping to aj under W —^ T , writing WjW = Uj{w)wj' , W± = W^VQ U W+vi , VQU = vo{u)vk with Ar = 0 or 1 and s{u) = X a(^o(w)), u G W± . Then
J wro{w') =
Y[y/l{ujiw')Y^''^^'^^^''^
where w ^>- p under VF ^ F, and 2a,
Hence L e m m a 6.5.B.
d?o{w,w') = Y[B]'''
DESCENT FOR TRANSFER FACTORS where ^
y/s{Uj{w)Uj'{w'))
From these two lemmas we conclude that
M»,»')=n(-i^)(^) Let
and define A' by
v=(n.r(^))v. L e m m a 6.5.C. A' is a cohoundary. Proof. Because aJ^Aj/Aj
=
we may write A' as
(T~^A-
/ fi \ ^(«j'(<^))
From the definition of Aj we see that
<^j'Ai _l<^j'{i)X^'''^''^^'^"''^''^'^ _
where T}J = cj'^{^)/^
2«(a,(^))H«,'(
. We also find that
coincides with {pVi'/ruY'^"''''^''^^ • Thus J On the other hand, the cohoundary of Yi'Hj j
is
553
554
R. LANGLANDS and D. SHELSTAD
Because H^jip))
+ ^iM
= 26(a,(p))6(a,-(<7)) ,
this coboundary coincides with A', and the lemma is proved. We discard the term A' from XQ leaving
J
/Vs{ujiw)uj.iw')))-']. Consider the cocycle A^^ of W±a in //4 given by (w, w') -^
(^ y/^/pi^
V^)Y^''\
V^{w){ ^/^{w')<'^''^/V~s{ww'))-'
where w^w' -^ pjC under W±a -^ T±a • Then Xo is the image of A^ under the corestriction homomorphism from i/^(H^±cr,/i4) to H^{Wjfi4). Since invi^^Ao = invF^^Aa , to prove (6.4.2) it is sufficient to prove that (6.5.1)
invAa = Xa(aa) •
If i^i —> R2 is an isogeny of tori over a local field F with kernel D then local class field theory ([M], Chap. 1) yields two sequences in duality (6.5.2) (6.5.3)
D{F) -^ H^(F,D)
R^{F) -^
^ - H\W,Ri)
^-
R2(F) -^
H\F,
H\W,R2)
^—
D) H\F,D)
.
The pairing is in all cases to C^ or a subgroup of it, and D = Eom{D,Hoo{F) 0 Aioo(C)) . We use in two different ways the compatibility of the two sequences with the pairing, for the commutativity of diagrams (6.6.1) and (6.6.2) are special cases of it. Since we use the compatibility at both ends, we need to pay attention to signs. Moreover we know of no reference for the compatibility, although it follows from standard results. So we include here some very brief remarks, based on the constructions in [L3]. First of all, when proving the compatibility, one can confine attention to elements of H\F,Ri) and thus of H^{K/F,Ri) for some large K (notice the proof that A^ factors through T±a in the next section). Then
DESCENT FOR TRANSFER FACTORS
555
a G H^(K/FjRi) when paired with the cup product /3 U j of /3 E H-^{K/F,X^{Ri)) and the fundamental class of K/F yields (a,/?} = (/?,a} . The pairing between a ' G H'(F,D) and /?' G H^{F,D) is given by /?' U a' G i y ' ( F , i ) (g) 5 ) —^ /f2(^^ ^ ^ ( ^ ) ^ ^^(C)) - Ain(C) . Here n is sufficiently large and /?' U a':/?, tr -^ /?'(/>) 0 poc'{o-) • Choosing n so that nX*(i22) C X*(/Zi) C X*(i22) we see that it is enough to treat the case that nX^{R2) = X*(i^i) . Then 1 —yD-^Ri —^R2 —^ 1 is obtained by tensoring (6.5.4)
1 - ^ ^n{F)
—^F*-^F*—^l
with X^{Ri). Suppose a ' G H^{F/D) has image oc and P' is the image of /? U 7 . Then by Proposition 5 of [Se], Chap. VIII, Sect. 3, p' = puSj, 6 being the map H\F,F*) -^ H^{F,Hn{F)) attached to (6.5.4). Thus /?' U a ' = /? U (57 U a ' = - / ? U a ' U ^7 .
Choose n divisible by [K:F].
Then
H\K/F,pir,{K)) -^ H\K/F,K*) , is an isomorphism as is H\K/F,^t„iK)) — H\K/F,Hm{K)) if n\m . The product /? U a ' = 71 i s a class in H~^{K/F,fin{C)). Lifting 7,71 to 7,71 with values in fin^(K) , //„2(C) we obtain a product of chains 7i U 7 that projects to 71 U 7 . Since 5(71 U 7) = dji U 7 4- 71 U 57 must be trivial in H^{K/F^iin{K)^^n{C)) , and dy = 6^ , dji = (/?, a) , we conclude that / ? ' U a ' = {/?,a)7. This is one of the compatibilities. For the other, take a G H\K/F,5), pUjU6a
= pu6aUj
and a' = 6a , /3' = pUj. = S{PUa)Uj
,
Then
R. LANGLANDS and D. SHELSTAD
556
and S{/3 U a ) = {/?, a) . 6.6. Final Calculations First we observe that B = ^/2{w){^/i{w')Y^^\^/^{ww'))-\w,w'
G W± ,
is given by the following table. The elements t,f
tvi
w' t' t'v t'
tvi
t'vi
w
t t
lie in W^ .
B yfs{t)^s{t')yfs{U')-^ y/^{t)^it')y/^{tt')-'^ ^/J(<)^/J(*')"'^/^(«')~^ ^it)^(t')-^y/^{ti'vl)-^
The proof of (6.5.1) will be divided into the following cases: (I) i e F ± „ ; (II) i G F „ - F ± „ ; (III) i i F,. We shall delete the subscript a from notation. Thus Oa , X a , A^ , F±ot become a , X , A , F i ; we write F+ for Fa . To verify directly that A factors through r± , we choose an open subgroup U of finite index in F^ that does not contain —1 . Then V = U^ is also open and of finite index and u ^^ v = v? is a topological isomorphism between the two groups. Define a character // of F by ^{v) = X {u) . Then /i^ is equal to X on F . We may suppose U and V are invariant under V± . Then F is a normal subgroup of W± and we may so choose
y/s{t),
teW^,
that y/s{vi)
=
^(v)y/s{t)
Since fi(vv) = X (uu) = 1, t; G V^, it follows easily from the table that B factors through F \ W ± and thus through T± . We now prove (6.5.1). Case I. We may assume F nonarchimedean. Let L be the cyclic quartic extension F^{^y/a) of F± , r be the generator ^ y/a-^ i^y/a and fii be the character on Gal(Z//F±) given by //i(r) = f"^ . We pull //i back to a character on V± and observe that = A^i(/>),
pGr±.
Because X(—1) = X (i^) = 1 we may choose a character ^2 on F ^ such that //^ = X . Regard /i2 as a character on W4. and set y/s — ^2 •
DESCENT FOR TRANSFER FACTORS
557
As usual, let w^w' -^ p)<^ under W± —>• T± . Then in the table of values for B we obtain 1, l,/i2(^'^')~^ J A^2(^'^'VI)~^ J ^^ ^^^^ ^ ^^ given by {w,w') —>• [/i2 otrans(w')]''
^^^
where trans is the transfer homomorphism W^ -^ W^ . Since corresponds to restriction of /i2 to F^ we may write this as
fi2otrans
on identifying this restriction as a character on T± . Observe next that on F^ or T± we have //J = / i | = X so that 0 = fi2fJ'i^ is of order two. Also (/?,2{(^)~^^^^ is cohomologous to (/9,
We interpret this as the cup-product of 6 in /f^(F±,;/2(C)) and S in ifl(f±,//2(F)). In general the following diagram,
\\
labelled (6.6.1) and defined by x -^ x^ , F^ —> F^ , is commutative. Thus invA = 0(a^) = fi2(a^)f^T^{a^) = X(a)//f^(a^) . However, the norm of ^ y/a is i . ( _ l ) . ( _ i ) . ^ V = a2, SO that /ii(a^) = 1. Thus the relation (6.5.1) is valid in this case. Observe that slight variations of the preceding arguments allow one to verify readily in all cases that if (6.5.1) is valid for one choice of a and X then it is valid for all. Case 11. We may choose a = —i, y/a = ^"^ . Then A is given by B~^ . The diagram (6.6.1) is now to be replaced by the analogous diagram for the group R with the twisted action p:x —^ p{xy^^^ . Then C^ is replaced by R . We take n = 4 , and then the kernel of x —^ x"^, R -^ R is, because
558
R. LANGLANDS and D. SHELSTAD
of the twisted action and because i £ F^ — F± , isomorphic to ^^{F±) . The analogue of (6.6.1) is labelled (6.6.2) and is 1\
^
R*
^
R*
^
H\F±,l4y I
H^F^,,i4iF±))
^
H\W±,R)
^
H\W±,R)
^
^i(^±,M^±))
It is again commutative. The element B~^ lies in H^{F±, fj,4{F±)) , and comes as the boundary of a 1-cochain on W± with values in R, namely w —^ y/s{w)~^ . Taking the fourth power, we obtain the cocycle r:w-^ 5(1/;)"^ . The element invA is obtained, after our identification, by pairing B with a~^ = i in Z4 (or //4(F)) . Thus, by commutativity of the diagram, it is obtained as the value of the character 1/ associated to the cocycle r on a~^ . In general if z E R, thus if z E F^ and zz = 1 ^ then z = yy~^ and i/(z) = X (y)"^ because the function s^ restricted to W^ is X ^ . For z = i we have y z= I-\-i and X ( y ) - 2 = X ( 2 0 - ' = Xi-i)=
X(a),
because 2 is the norm of y. Case I I I . We have not been able in Case III to deal directly with fields of even residual characteristic. They can, however, be handled by a global argument. Suppose, for present purposes, that F^ is a quadratic extension of the number field F± , that a £ F^ and a = —a , and that X is an ideleclass character of IF_^ whose restriction to Ip^ is the character ^ F ^ / F ± associated to the quadratic extension. The construction of A in Section 6.5 can be carried out globally. At a place in F± that does not spUt in F+ , the global construction is compatible with the local. However, A can also be restricted to the local Weil group at a place v that splits in F^ . We claim that the relation (6.5.1) is valid at this local place. Thus invAt; = Xv{ci) = ^ To see this observe first that Xt;(—1) = 1 so that Xv is a square. Since 0{p) = I if /> lies in the decomposition group, the denominator of the expression defining A,; is 1. Since 6{a) = 0 if cr is in the decomposition group, the numerator is also 1. We conclude that if (6.5.1) is valid at all but one place then it is vahd at the remaining place. It is certainly valid at the archimedean places, for if they are not split they fall under Case II. Moreover, given local data at one place, we can extend these local data to global data and in such a way
DESCENT FOR TRANSFER FACTORS
559
that at any prescribed finite set of places not containing the original one the extension splits. Therefore it suffices to treat the case of odd residual characteristic. Since i ^ F+ we have a diagram of fields K F+ ramified
\
^ E
= F±{i)
/
unramified
F± All intermediate extensions are quadratic and [K: F±] — 4 . Let q be the number of elements in the residue field, so that ^ = 3 (mod 4) and 5^ — 1 = 0 (mod 8) . Let m be the largest power of 2 dividing q^ — I and let C be a primitive m *^ root of unity in E . We may suppose that
Since F^/F± is ramified, we may choose a so that a^ = zu is a uniformizing parameter in F± . Then K{^^a) and K{^) are linearly disjoint and Galois over F± . Thus we may enlarge our diagram of fields to
K{^)
K{^)
K{^)
There exists a r G r ± such that (6.6.3)
Ta = -a
r{^)
= £^^ ,
r(^)
= ij/C
We choose l , r as representatives for r ± \ r + . The group of units 0 | in 0± is a product {±1}C/, where U is the set of all a in O^ whose image in the residue field has odd order. We define X on F^ by the following conditions: X|C/=1;
X(-l) = - l ,
X(-a2) = l .
Then we extend it to F^ , obtaining a character X of order 4. This character defines a cyclic quartic extension of F^ that evidently contains
560
R. LANGLANDS and D. SHELSTAD
K because X ^ is unramified. Thus it is the quadratic extension of K associated to the character i/{x) = X (xx) . Since z/ is not trivial on units it is ramified, and thus of the form K{y/ja), where 7 is a unit. Then 1 = z/(—7a) = 1/(7) X (a^) = —^^(7), so that i/('y) = — 1. Consequently we may take 7 = C > stnd the field is K{y/^). Consider the element S of H^{F+, fi4{F^)) given by /> —» ^/p{^) and the element 6 of /f^(F^.,p4(C)) given by X . According to the diagram (6.6.1) the invariant of their cup product is X (a) . To complete the proof of (6.5.1) in Case III and thus of Theorem 1.6.A it remains to show that A is in the class of the corestriction of 6U6 . The cup product itself is given by
(pa) -^ i^/p(
= Xp,a{p(TJ.
We first examine the restriction of the cocycle /i to r_|. . There it is given by
/ip,. =
i^p{^)-T''^^^T-\^TpT-\^)-'r'^(^^^~'^
We claim that ord(ro'r~^) = —ord((j) . It is enough to verify this on an element a such that a{y/^) = i t f v ^ - Then, by (6.6.3), Tar-^iy/Ca)
= T(T(-iy/Ca) = T{±^/Ca) = T « V ^ = <^~^(\/^) •
Thus
/ip,. = i^p{<^)-'T-\^)-'pT-\^)r<'(<^^
= (MO-'r'^"^.
On the other hand, the numerator of the quotient defining A is trivial on r^. since S(a) = 0 for <7 G r + . The denominator is easily calculated since it is just the puUback to r + through X of the 2-cocycle of the extension 1 — . /i2(C) — . fis{C) -^
/i4(C) —. 1 .
If we choose Ci such that ord(cri) = 1 then for 0 < a,6 < 4 this is just
_J 1 ^<^?'-J=
_1
a-h6<4 a+ 6 > 4 .
DESCENT FOR TRANSFER FACTORS
561
Thus both cocycles are defined on G a l ( / ^ ( \ / ^ ) / F ^ . ) and if we take
?, = ?i, (
{<^i)=^Mo-\<^'ir
then p -a = \p^a{p<^) on F^. , because (T\{C) = ^^ = ^ • If a belongs to F^ then 1 • r = r , r • cr = T(TT~^ • r , r • r = r^ , so that ^,,,
= T - \ ^ T \ ^ y y ^ ' '
= ^-2ord(a) ^ .-ord(<7)
On the other hand,
A.,. = {Vx(l)r'Vx(^Vx{r
= v6tMx/x(r
and we have chosen i/x(
0 < a < 4,
A,,i = 1 . Then we set r = I3T , where /? is yet to be determined, and define (rcr) so that
This done, we have to verify that (6.6.4)
aT = X^^r{(TT) .
If this equation is valid then a '(rp) = a(Xr^p)~^aTp = (T{\r^p)~^K,T{<^r) p which equals ^{^r.pT
>^a,TX~j-\^jT{\T-iaT,p)K,r-^aTp{crTp)
- Xa,Tp{(^Tp) .
Moreover, if (6.6.4) is valid for p as well as a then {pa)r
= X-^^paT=:X-^^p{Xa,r)>
= Xpa,r{p(TT) .
Thus it is enough to verify it for ci . The left side is equal to (7i(/?)^<7j .^(crr)'^ ; the right side is
Since ord(r^) = 2 , 1 • r = r , rai =
while
{T(TIT~^)T
,
562
R. LANGLANDS and D. SHELSTAD
Thus we must have
Since o'i{i) = — 1 , we may take
It is easy to verify that (rp) a = )^Tp,a{Tp(^) if p^cr £T^ , and if (6.6.5)
r2 = A,,,(r2)-
it is easy to verify that {pT){aT) = XpT^aTipTcr) , p^a £T^ . Thus the proof of Theorem 1.6.A will be complete once (6.6.5) is established. It is clear that fij^r = 1 and that ord(r^) = 2 . Therefore
On the other hand,
and we are done. REFERENCES [I]
R. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann., 278, 219-271 (1987). [B] N. Bourbaki, Groupes et Algebres de Lie, Chs. 4, 5, 6, Hermann (1968). [C-D] L. Clozel and P. Delorme, Le theoreme de Paley-Wiener invariant pour les groupes de Lie reductifs, Inv. Math. 77, 427-453 (1984). [HC 1 ] Harish-Chandra, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc, 119, 457-508 (1965). [HC2] 5 (notes by G. van Dijk) Harmonic Analysis on Reductive p -adic Groups, Springer Lecture Notes, Vol. 162 (1970). [HC 3 ] , Harmonic analysis on real reductive groups I, J. Funct. Anal, 19, 104-204 (1975). [K 1 ] R. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J., 49, 785-806 (1982). [K 2 ] , Stable trace formula: elliptic singular terms. Math. Ann., 275, 365-399 (1986).
DESCENT FOR TRANSFER FACTORS [K3]
563
, Tamagawa numbers J Ann. of Math., 127, 629646 (1988).
[K 4 ] [K-S] [L 1 ] [L 2 ] [L3] [L-S] [M] [Se] [Si] [S 2 ]
[S3] [S 4 ] [S5]
[W]
, Sign changes in harmonic analysis on reductive groups, Trans. Amer. Math. Soc, 278, 289-297 (1983). and D. Shelstad, Twisted endoscopy, in preparation. R. Langlands, Les Debuts d'une Formule des Traces Stable, Publ. Math. Univ. Paris VII, Vol. 13 (1983). , Stable conjugacy: definitions and lemmas, Can. J.Math., 31, 700-725 (1979). , Representations of abelian algebraic groups, preprint (1968). and D. Shelstad, Orbital integrals on forms of SL(3), II, Can. J. Math, (in press). J.S. Milne, Arithmetic Duality Theorems, Academic Press (1986). J.-P. Serre, Corps Locaux, Hermann (1962). D. Shelstad, Characters and inner forms of a quasi-split group over R, Comp. Math., 39, 11-45 (1979). , Orbital integrals and a family of groups attached to a real reductive group, Ann. Sci. Ec. Norm. Sup., 12, 1-31 (1979). , Embeddings of L-groups, Can. J. Math., 33, 513-558 (1981). , L -indistinguishability for real groups, Math. Ann., 259, 385-430 (1982). , Orbital integrals, endoscopic groups and Lindistinguishability for real groups, Publ. Math. Univ. Paris VII, Vol. 15 (1984). G. Warner, Harmonic Analysis on Semisimple Lie Groups, Vol. II, Springer Verlag (1972).
School of Mathematics The Institute for Advanced Study Princeton, NJ 08540
and
Mathematics Department The University of Utah Salt Lake City, UT 84112 Mathematics Department Rutgers University Newark, NJ 07102