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London Mathematical Society Lecture Note Series. 235
Number Theory Seminaire de Theorie des Nombres de Paris
1993-4
Edited by
Sinnou David Universite Pierre et Marie Curie, Paris
CAMBRIDGE UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press The Edinburgh Building, Cambridge C132 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521585491
© Cambridge University Press 1996
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Number Theory Paris 1993-94
Table des Matieres S. BOCHERER and R. SCHULZE-PILLOT On the Central Critical Value of the Triple Product L F unction .................... I I.B. FESENKO
Abelian extensions of complete discrete valuation fields ...........................47 D. HARARI
Obstructions de Manin transcendantes ....................................................75 H. HIDA
On Selmer Groups of Adjoint Modular Galois Representations ..................89 B. LEMAIRE
Algebres de Hecke et corps locaux proches (wie preuve de la conjecture de Howe pour GL(N) en caracteristique > 0) ........................................133 A. NITAJ Aspects experimentaux de In conjecture abc ...........................................145 W.M. SCHMIDT
Heights of points on subvarieties of G a ................................................. 157 T.N. SHOREY
Some applications of Diophantine approximations to Diophantine equations ..........................................................................189 U. ZANNIER
Fields containing values of algebraic functions and related questions ..... 199
Les textes qui suivent sont pour la plupart des versions ecrites de conferences donnees pendant l'annee 1993-94 au Seminaire de Theorie des Nombres de Paris. Ce seminaire est financierement soutenu par le C.N.R.S. et regroupe des arithmeticiens de plusieurs universites et est dotee d'un conseil scientifique et editorial. Ont ete aussi adjoints certains textes
dont la mise a la disposition d'un large public nous a paru interessante. Les articles presentes ici exposent soit des resultats nouveaux, soit des syntheses originales de questions recentes; ils ont en particulier tous fait l'objet d'un rapport. Ce recueil doit bien sur beaucoup a tous les participants du seminaire et a ceux qui ont accept& d'en reviser les textes. Il doit surtout a Monique Le Bronnec qui s'est chargee du secretariat et de la mise au point definitive du manuscrit; son efficacite et sa tres agreable collaboration ont ete cruciales dans l'elaboration de ce livre.
Pour le Conseil editorial et scientifique S. DAVID
Number Theory Paris 1993-94
Liste des conferenciers 4 octobre : B. PERRIN-RIOU et D. BERNARDI. - Fonctions L p-adiques des courbes elliptiques 11 octobre : S. LOUBOUTIN. - Problemes de nombres de classes pour les corps a multiplication complexe 18 octobre : L. MEREL. - Operateurs de Hecke, symboles modulaires et courbes de Well
25 octobre : A. CORTELLA. - Le principe de Hasse pour les similarites de formes bilineaires 8 novembre : Sir P. SWINNERTON-DYER. - Rational points on certain intersections of two quadrics 15 novembre : U. ZANNIER. - Fields containing values of algebraic functions
22 novembre : J. COUGNARD. - Anneaux d'entiers stablement libres et non libres
29 novembre : M. HARRISON. - On Bloch-Kato conjecture for Hecke characters over Q(i)
6 decembre : R. GREENBERG. - Two-variable Iwasawa theory 13 decembre : P.G. BECKER. - Transcendental values of the DouadyHubbard function
3 janvier : A. NITAJ. - Consequences et aspects experimentaux de la conjecture abc
10 janvier : C. BACHOC. - Classification et construction de reseaux unimodulaires
17 janvier : C.-G. SCHMIDT. - Generalized Kummer congruences for Siegel modular forms
24 janvier : J. MARTINET. - Classification des reseaux eutactiques 31 janvier : A. ABBES. - Theoreme de Hilbert-Samuel arithmetique
7 fevrier : M. KANEKO. - Atkin's polynomials on supersingular jinvariants and hypergeometric series
28 fevrier : W. McCALLUM. - Dualite en Theorie d'Iwasawa a plusieurs variables
7 mars : B. LEMAIRE. - Conjecture de R. Howe pour GLN sur un corps local de caracteristique positive 14 mars : E. URBAN. - Congruences de formes modulaires et Theorie
d'Iwasawa 21 mars : D. HARARI. - Obstructions de Manin transcendantes 25 avril : F. BEUKERS. - Units in quaternion algebras as monodromy groups
25 avril : L. MEREL. - Bornes pour in torsion des courbes elliptiques sur les corps de nombres
2 mai : J. ASSIM. - Sur les corps de nombres (p, i)-reguliers 9 mai : P. COLMEZ. - Fonctions zeta p-adiques en s = 0 16 mai : T. NGUYEN QUANG DO. - Sur les coryectures de Lichtenbaurn
30 mai : R. SCHULZE-PILLOT. - Theta series and L -functions
13 juin : J. NEKOVAR. - p-adic regulators 20 juin : L. FAINSILBER. - Formes hermitiennes sur les algebres padiques 27 juin : S. FERMIGIER. - Annulation de la cohomologie cuspidale de sous-groupes de congruence de GL,, (Z)
Number Theory Paris 1993-94
On the Central Critical Value of the Triple Product
L-Function S. Bocherer and R. Schulze-Pillot
Introduction Starting from the work of Garrett and of Piatetskii-Shapiro and Rallis on integral representations of the triple product L-function associated to three elliptic cusp forms the critical values of these L-functions have been studied in recent years from different points of view. From the classical point of view there are the works of Garrett [9], Satoh [22], Orloff [21], from an adelic point of view the problem has been treated by Garrett and Harris [101, Harris and Kudla [ 121 and Gross and Kudla [ 11]. Of course the central critical value is of particular interest. Harris and Kudla used the Siegel-Weil theorem to show that the central critical value is a square up to certain factors (Petersson norms and factors arising at the bad and the archimedean primes) ; the delicate question of the computation of the factors for the bad primes was left open. In the special situation that all three cusp forms are newforms of weight 2 and for the group Fo(N) with square free level N > 1, Gross and Kudla gave for the first time a completely explicit treatment of this L-function including Euler factors for the bad places; they proved the functional equation and showed that the central critical value is a square up to elementary factors (that are explicitly given). We reconsider the central critical value from a classical point of view,
dealing with the situation of three cusp forms fl, f2, f3 of weights ki (i = 1, ... 3) that are newforms for groups Fo(Ni) with N = lcm(Ni) a squarefree integer 1. The weights ki are subject to the restriction ki < k2 + k3 where k1 > max(k2, k3) ; the distinction whether this inequality holds or Both authors were supported by MSRI, Berkeley (NSF-grant DMS-9022140). R. Schulze-Pillot was also supported by the Deutsche Forschungsgemeinschaft and by the Max-Planck-Institut fur Mathematik, Bonn.
2
S. BOCHERER and R. SCHULZE-PILLOT
not played an important role in [11] and [ 121 too. We start from the simplest possible Eisenstein series E of weight 2 for ]po3) (N) on the Siegel space H3 ("summation over C = 0 mod N"). After applying a suitable differential operator (depending on the weights ki) to ]E we proceed in a way similar to Garrett's original approach : we restrict the differentiated Eisenstein series in a first step to N1 X H2 and integrate against fi, the resulting function on H2 is then restricted to the diagonal and integrated against f2, f3. The necessary modifications to Garrett's coset decompositions (that were for level 1) are not difficult (for the first step they have already been carried out in 12]). The actual computation of the integral is elementary and needs only standard results from the theory of newforms. It yields a Dirichlet series (2.41) whose Euler product decomposition is then computed in Section 3. The cases that p divides one, two or all three of the levels Ni or is coprime to N must all be treated separately, which makes the discussion somewhat lengthy. However, the actual computation in each of these cases is again fairly straightforward. In Section 4 we show that the Euler factors defined in Section 3 are the "right ones" by proving the functional equation. In order to exhibit the central critical value as a square (up to elementary factors) we follow a similar strategy as 1121: the Eisenstein series B at s = 0 is expressed as a linear combination of genus theta series of quaternary positive definite integral quadratic forms. At most one of these genera (depending on the levels Ni and the eigenvalues of the fi under the Atkin-Lehner involutions) contributes to the integral. Eichler's correspondence between cusp forms for Fo(N) and automorphic forms on definite quaternion algebras allows then to express this contribution as an (explicitly computable) square of an element of the coefficient field of the fi ; this element arises as a value of a trilinear form on a space of automorphic forms on the quaternion algebra and may be interpreted as the value of a height pairing similar to [I I]. It may be of interest to compare the advantages of the different methods applied to this problem. Although the adelic method makes it easier to obtain general results, the explicit computations needed here appear to become somewhat simpler in the classical context. In particular, by making use of the theory of newforms and of orthogonality relations for the theta series involved from [2] we can use the same Eisenstein series B independent of the fi. This is of advantage since the pullback formalism is especially simple for this type of Eisenstein series and leads to the remarkably simple computations in Sections 2 and 3. Most of this article was written while both authors were guests of the MSRI in Berkeley during its special year on automorphic forms. We wish to thank the MSRI for its hospitality and financial support. R. SchulzePillot was also supported by Deutsche Forschungsgemeinschaft during a visit of one month at MSRI and was a guest of the Max-Planck-Institut fur
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
3
Mathematik in Bonn in the final stage of the preparation of this manuscript.
Notations We use some standard notations from the theory of modular forms, in particular, we denote by 1H,,, Siegel's upper half space of degree /n (for n = 1, the subscript will be omitted); for functions f on N and
g = I a d I we use (f Ik g)(z) = det(g) 2 (cz + d)-k f (g < z >) and similarly for the/action of double cosets (Hecke operators). The operators T (P) and U(p) however will be used in their standard normalisation. The space of cusp forms of weight k for ro (N) _
I E Sl2 (Z) c
will be denoted by [I'o(N), k]o.
0 mod N }
///
JJJ
1. - Differential Operators We have to deal with two types of embeddings of products of upper half spaces into 1H13 namely
HXI2 -` t12
(z, Z)
z 4
1
H3 0
0 Z
and
I (zl, z2, z3)
H3
H
(zi Z2
z3
Without any danger of confusion we may denote by the same symbols the corresponding "diagonal" embeddings of groups : t12 : S12 x Sp(2) -* Sp(3)
and
till
: S12 -+ Sp(3).
One might try to apply Ibukiyama-type differential operators [16] in the integral representation of the triple L-functions (equivariant for S12 X S12 X S12 - Sp(3)). However in the actual computation of the integral, it is more convenient to have equivariance for S12 x Sp(2) y Sp(3). Therefore we use Maa13-type operators (see [201) and the holomorphic differential operators
introduced in [61; we describe these operators here only for Sp(3), but of course they also make sense for Sp(n). We start from a natural number r and three (even) weights kl, k2, k3 with kl = max{ki} and satisfying the condition (1.1)
k2 + k3 - kl > r.
S. BOCHERER and R SCHULZE-PILLOT
4
Then we define nonnegative integers a, b, 1,21 113 by
r+a = k2+k3-kl kl = r+a+b k2 = r+a+v2 k3 = r+a+v3.
(1.2)
Then we have
b = v2+v3.
(1.3)
We use two types of differential operators on 1H13. The first one is the Maa6 operator
Nl« = det(Z 3
E3 (a)
=
(1.4)
Eµ
Z)2-« det(er) det(Z
tr ((z - Z)[µ[
-
Z)«-i
(a.))
µ_p
= 63(a) + -
+ det(Z - Z) det(aOj)
where (following [20])
Ep(a) =
1
a.(a-2)...(a-'
p=0 µ>0
and for a matrix A of size n we denote by A[IL] the matrix of p x p-minors. We put
Nl[«1=M«+i-to...oM«+1oMa. We recall from [20] that (1.5)
for all g = (a
.A4[« `I (.f Ja,p g)
_ A4[/'If )
I a+µ,/3-1i g
d) E Sp(3, R) with (f I«,p g) (Z) =
det(cZ+d)-«det(cZ+
d)-1 f (g < Z >). Here a and ,0 are arbitrary complex numbers, but it would be sufficient for us to take a = r + s, ,3 = s with s E C.
The second type of differential operators was introduced in [6]
:
it maps
scalar-valued functions on 1H13 to vector-valued functions, more precisely to C[X2, X3]6 -valued functions on ]El[ x ]fl(2 H3 where C[X2, X3]b denotes the
space of homogeneous polynomials of degree b; we realize the symmetric
5
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
tensor representation Ob of Gl (2, C) on this space in the usual way. The operator L,() as defined in (6) satisfies IW+b,Q 91 =
(f I.,# t1,2(g1, 14)) (t1,2(w, Z))
(II' (b) f) I a,l3,ab 92 =
(f 1.,0 /1,2(12,92)) (t1,2(w, Z))
(1.6)
for all gl E S12 (k) and all g2 E Sp(2, Ik), where the upper indices Z and w indicate which variable is relevant at the moment and ( \L (b)f) I a,/3,,,, 92) (t1,2(w, Z)) =
det(cZ + d) -a det(cZ + d)-aab(cZ + d)-1 (La) (t12(w, 92 < Z >)) This differential operator can be described explicitly as follows : (1.7) La(b)
1 (DTDj)" (D - DT = 1 a! !c* v!(b - 2v)r !(2 - a - b)! ! 0<2"
Dj)b-2v
with t* denoting the restriction to 1H[ x H2 -# 13,
Dl = all D1 = E 0,3XtiX3 2
D-D1 -Dj =2(012X1+013X3) and a!"]
_ F(a+v) I(a)
1
1a(a+1)...(a+v-1)
v=0 v>0.
We should remark here that !L.) has coefficients, which are rational functions of a with no poles for R(s) > 0. We shall use the operators E)(a'b)
with 2a' = a and
:_ )L(b+ai ..A4[.']
defined by
Da(a'b)f =
f) tlll(zl, z2, z3)
6
S. BOCHERER and R SCHULZE-PILLOT
Denoting by Da(a'"2 v3) the operator which picks out of Da(a'b) its x2 X33component, we get a decomposition v*(a,b) f _
(1.0)
(D (a,v2.v3)J f) x22 !1 x13 a V2+V3=b
with *(a,V2.V3)
Da
f k,/3 011(91)92793) Z3
Z2
Zl
Dk
)f) la+a'+b,13-a' 91 la+a'+v2,P-a' 92 Ia+a'+v3,/3-a' 93
for all (91,92,93) E S12(R)3.
If f is a holomorphic function on IEI(3, then (yly2y3)-a
.
Da(a,v2,v3) (f)
is a nearly holomorphic function (in the sense of Shimura) of all three variables z1, z2, z3 E IEL.
To apply Shimura's results on nearly holomorphic functions, it is more convenient to use his differential operators Sam, which differ (in the onedimensional case i.e. on IE11) from the Maal3 operators only by a factor constant x y/` : ba
+a
27ri \ 2iy
say=ba+2µ-2 p... Ob.-
By elementary considerations about the degree of nearly holomorphic functions (as polynomials in y-1) Shimura observed that nearly holomorphic functions on H are linear combinations of functions obtained from holo-
morphic functions by applying the operators 61' (at least if a is not in a certain finite set, for details see [24, lemma 7]. By the same kind of reasoning we get an identity (yly2y3)-a'
,
D+(a,v2.V3) a
_
fail
6/i2
a+a+b-2µ1 a+a+v2-/i2Ua+a+v3-2 i3
(1.9)
0
-
We understand that in (1.9) the operator 5'
acts with respect to zti,
i = 1, 2, 3; moreover ID , (...) is a holomorphic differential operator mapping
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
7
functions on H3 to functions on H x ]H[ x H. Following again the same line of reasoning as in lemma 7 [loc.cit], adapted to our situation, we easily get that the IIDa(...) satisfy Da(a,v2,v3,171,A2,1-13) (f I.,# 1111(91,92,93)) (1.10)
= IIDa(a, v2, V3,,41, P2, µ3)(f) Ia+a+b-2µ1,P 91 z3
Z2
ICY+a+v2-2µ2,A 92 a+a+v3-2µ3,0 93
for all (gl, g2, g3) E Sl (2, R) 3 and all holomorphic functions on 1H13 (and
hence also for all C°°-functions). The upper indices zi on the right hand side of (1.10) indicate, on which variable gti operates. We have to remark here that Shimura's condition "k > 2r" in Lemma 7 [loc.cit] is satisfied in our situation as long as a is non-real or a + a + b > a
a + a + v2 > a
a+a+v3 > a. However the coefficients of the aij on both sides of (1.9) are easily seen to be rational functions of a, therefore (1.9) (and subsequent equations) are true for all a E C as rational functions of a.
It is crucial for us to see that in the identity (1.10) the "holomorphic part", i.e. IlDa (a, v2, v3, 0, 0, 0)
is different from zero. For this purpose we consider as a polynomial in 012, 013, 023. It is easy to see that this is a polynomial of total degree 3a + b, the component of degree 3a + b being given by (J1y2y3)-a
1
(a12a13a23)a x
1 .
1
.
1
2b
(J1J2J3)-a
.
Da(a,b)
-
as a polynomial in 812, 013, 023. It is easy to see that this is a polynomial of total degree 3a + b, the component of degree 3a + b being given by 1
(19120131923 )a x 1 .
1 .
2b
1
- (012X2 + 013X3)b
In particular, this component is free of yi 1 and atii, so it can only come from the IIDa (a, v2, v3, 0, 0, 0) with v2 + v3 = b.
S. BOCHERER and R. SCHULZE-PILLOT
8
Now we define a polynomial Qa of the matrix variable S = St = (si, )1
lDa(a, b, v2i V3, 0, 0, O)etrace(SZ) = Qa(S)esl1Z11+s22Z22+333 Z33
By the same kind of reasoning as in 11, Satz 151 we see that by
Ixixl xix2 xix3 (1.13)
(x1, x2,
x3)'_4 Pr(xl, x2, x3)
x2x1
Qr
x3 X1
x2 x2 x2 x3 xix2 xgx3
we get a polynomial function of (XI, X2, x3) E (C2r) 3 which in each variable is a harmonic form of degree a+b, a+ v2 , a+ v3 respectively; more precisely, Pr defines a non-zero element of (1.14)
(7(a+b(2r) ®Na+12 (2r) ®7-la+V3 (2r)
)0(2r)
For our investigation of the functional equation of triple L-functions we have to modify Da(a, b, v2i v3, 0, 0, 0) still further (we switch notation now from a to r + s). We consider the operator 0 = Or,,, (a, V2, v3) given by (1.15)
F' - > (y1J2J3)sDr+s(a, v2i v3, 0, 0, 0) (F x det(Y)-s) .
This operator (acting on functions on T113) is easily seen to satisfy
(1.16)
A (F 1, 1,111(g1,g2,93)) = 0(F) 1 r+ r+a+b 91
r+2
a+V2 92 r+ 1
93
for all 91,92,93 E S12 (R). By the same kind of argument about nearly holomorphic functions as above we get CIi1
Ar,s (a, V2, V3) =
0 < µ1 < [] o
(1.17)
0 X
113 <
/2
r+a+b-2µi 6r+a+v2
[n,}V]
6113
r+ a+V3-2µ3pr,3(a, V2, V3, /11, ,02, /13)
with holomorphic differential operators Or,s (a, v2, v3, /t1, 1'2, µ3) mapping functions on 1H3 to functions on lEll x IE)( x H. We should mention here that
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
9
the differential operators coming up in (1.17) do not have poles as long as r is positive and s is non-real or Re(s) > 0. Again the "holomorphic part" A,,,, (a, v2i v3, 0, 0, 0) defines (as in (1. 12),(1.13) an element of (7-la+b(2r) ®fla+v2 (2r)
This space is known to be one-dimensional : by a result of Littelmann ((191, p. 145) the decomposition of (2r) is multiplicity (2r) free and contains 7-la+b(2r)), hence there is a unique invariant line in the threefold tensor product. There exists therefore a function c = cr(s) such
that (1.18)
A,,, (a, v2, v3, 0, 0, 0) = c, (s) Dr (a, v2, v3, 0, 0, 0).
By comparing coefficients of (012013023)a a12a13 on both sides of (1.17) we get
(1.19)
cr(s) _
(r + a')<<'1
(r+s+a')[b]
_ r(r + a' + b)
r(r+a')
r(r + a' + s)
r(r+a'+s+b)
2. - Unfolding the integral For a squarefree number N > 0 and three cuspforms af(n)e2"" E [ro(N),ki]o
f_ _
ao(n)e2"" E [ro(N), k2]o ap(n)e2""z E
[ro(N),k3]o
with k1, k2, k3 as in section 1 we want to compute the threefold integral A(f, 0,,0, s), defined by
(2.1)
3 (D*(a,v2,v3) (G'
J (1'o(N)\H)3 ff(z1)(z2)(z3)
kl+s-a' k2+s-a' k3+s-a'
X (t111(z1,z2,z3)) J1
J2
Y3
s))
3
dxidyi
ri
Y2
i=1
a
10
S. BOCHERER and R. SCHULZE-PILLOT
where G3 s is the Eisenstein series on 1H13 defined by G r' s
E
_
1 Ir+$,s M
MEr,\ro(N) (2.2)
D)-r-s det(CZ
det(CZ + C
D
+ D)
-8 .
MEr3_ \ro(N)
In the applications we shall need modified versions of the integral (2.1) ; it is appropriate to describe these here : we use the well-known fact (see e.g. [95, equation (2.28)1) that holomorphic cusp forms are orthogonal to (C°O)automorphic forms in the image of the differential operators b, therefore we k1+s-a' k2+s-a 2 k3+s-a 2 may replace (D*(a,v2,v3) (G3r,s)) (G111(z1, z 2, z 3)) J1 J2 J3 in the integrand of (2.1) by (D,+,, (a, v2, v3, 0, 0, 0) (G ,s)) (t111(zl, z2, z3)) Jl i+sJ22+sJ3s+s
or by (2.3)
Cr(s)Dr(a, 112, v3, 0, 0, 0) (Fr,s) (t111(z1, z2, z3)J11J22J3
3
where E3",,(Z)
= det(Y)3 Gr,9 =
det(Y)s I r M.
>2
MEr3_ \r03(N)
The actual computation of A(f, ¢, V), s) is however most conveniently done using the integral in the version of (2. 1).
2.1. - The first integration To understand the integration with respect to z1, it is better to consider first the integral (2.4)
I(s) =
f
dx1dJ1
.f(z1) (D(a,1)G3,s)
(t1,2(z1,Z))yi1+s
x
det(Y)s-a
y1
ro(N)\lkl
with Z = X + iY E 1H12. We recall from [2, Thm. 1.1) that the double cosets
F",)\r3
2
(ro(N) x ro(N))
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
11
can be parametrized by the following set of representatives m E N U {0} , m= 0 mod N}
(2.5)
where 13
0m0 UM
m 0
0 0
0 0
03 13
We split the integral (2.4) into the contributions of the double cosets (2.5) :
I(s)
(2.6)
Im(s)
.
It is easy to see that the double cosec with m = 0 decomposes into left cosets as follows
-r E r00\ro(N),S E D*(a,n)Gs+s Therefore its contribution to s is just {t1,2(-Y,6)
Dr+s,s
I
1 Ir+s,s L1,2(7, s) ry,b
-
b)
Dr+9 l
(1) Ir+s+a'+b,s-a' 'Y Ir+s+ai s-a'
0b
t5.
It is obvious that D*(a,b) (1)
r+9
_ f0
b> 0
E3(r+s)E3(r+s+1)...E3(r+s+a'-1) b=0.
Unfolding the integral defining Is(s) we easily get (by the cuspidality of f)
that
b(s)=0
(we omit the standard calculation).
For fixed m > 0, m - 0 mod N the left cosets are given by (see (2, Thm. 1.2]) (2.7)
{g7 Ll,2(7, l(h)g I 'Y E ro(N), h E r[m]\ro(N),g E C2,1(N)\ro(N)}
where
r[m] = I'o(N) n
(
0
01) ro(N) (m
0-1)
S. BOCHERER and R. SCHULZE-PILLOT
12
l = ill and C2,1 is the standard maximal parabolic subgroup of Sp(2) given by
with C2,1(Z) n r2 (N) The summation over y unfolds the integral for Im(s) to C2,1(N)
Ira(s) = *(a,b)
f(z1) (2.8)
H
Dr+s
Z (1 I r+s,s 9 ne) I r+s+a',s-a',ab
h,g
x l(h)g yl'+s
dxidp1 2
x
det(Y)s-a
.
By lemma 4.2 of [6] and (1.4) we have E)*(a'b)
(2.9)
(1 Ir+s,s 9m)
= 2-a(2r + 2s - 2)[a](r + s)[a 10r+s+ai (1 Ir+s+a m2z1Z*)-r-s-a'-b
= A(r + s, b) - (1 -
,s
9rn) m2z1Z*)-s+a'. (mX2)b
(1 -
where for Z E TEl2 we denote by Z* the entry in the upper left corner of Z and A(s b) _ (_,)b 2-a(2s-2)[a](,)[a'](2s+a-2)[b]
2-7rib!(s+a' - 1)[b]
This implies
Im(s)=A(r+s,b)E (2.10)
(f
g,h .f(z1)(1-m2z1Z*)-r-s-a
2
H
x (MX2)b) +s+a',s-a',Qb i(h)g x
det(Y)s-a
.
The integral in (2.10) is exactly of the same type as in [2, (1.4), (1.5) ]. Using
the same notation
s) _ (-1)
k
7r
k+s-1
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
13
as in [2) we get by the same reasoning as there Im (s) = A(r + s, b) x
p(r + a + b, s - a') .-r-2s
.fP Ir+b+a ro(N)
(2.11)
(0M
) r°(N)
0
k+a
x
''> (ab ®det
< Z >) s_a
det Im
(j(g,Z)))_1
X2 ( Im(g(< Z
>`))
((a )
with j , z) = cz + d. This is (essentially) a vector-valued Klingentype Eisenstein series attached to the modular form
.fP Ik, ro(N) m
-MI ro(N)
where
fP(z) = f(-2). From now on we assume that f is a normalized newform (eigenform) of level Nf I N; we write N = Nf Nf whenever it is convenient. The Fourier coefficients of f are then totally real and we have an Euler product expansion of type (2.12)
afnn = H af(71)f-s rIN (1 2()_3
_.,
1
1
ptNI
(1 - app') (1 - a'Pp 3)
Moreover for any prime q dividing Nf we have of (q)2 = qkl
_2
and
f
.flkiV = flkiVq 1 = -af(q)g1 where V denotes the "Atkin-Lehner-involution" given by NN q
J)
- (xN q
with xq - Ny = q and qI x (for details we refer to 118]).
Actually we have to work not with the newform f itself, but with
Df 0
f I k'
0
1
where Df is a fixed divisor of Nf
14
S. BOCHERER and R SCHULZE-PILLOT
To simplify (2.11) further (for f Ik1 (2.13)
0
we have to study
1
1
(0f
E f Ik, m=0(N) =N-g
Df
0 ) ro(N)
01) Ik, ro(N) (m f
0o
E
0) Iklr°(N) (o
MI=1
m 2N)
r0(N)
ol)m
Ikl
Now we are essentially in a "local" situation, because we may decompose the "Fricke involution" into Atkin-Lehner involutions :
(N Ol) --yo
q 9I N
with -y E r0(N).
We use the following formal identities :
pIN:
jf I ro(N) (o °l) ro(N)X` _
(2.14)
l=0
(1 - X)(1 - p2X2) (1 - pX)(1 - aP2p-k,+2X)(1 - c ,2p k,+2X)
Proof. Standard (2.15)
pI Nf 00
t=0
_
f I ro (N)
(10 P2lo 1) ro(N) o VP . Xl =
pl-
f (p
l
, II k1VNI
-1
1-X'f
Proof. Standard, using af(p) = af(p)- and a1(p2) = Pki-2 pINf 00
.f ro(N) (2.16)
+1) r0 (N) o dP x XI
0
1
(0
P2
l=0
=p
k 2
(fIU(p)-P2'flk1(o °) .X) (1 - a2ki+2X)(1 - a, 2p k,+2X) PP P
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTYON
15
Proof. Standard. Using
0) 1
and
flk,
(o
)Ikiv=f
,
we get for (2.16) :
(p O1
_1-s
irv_
(1 -
al2,p-kl+2X)(1
(2.17)
11
,..V\ r
I
- a'P2pki+2X)
In quite the same way we get (still for the case pl Nf) :
Eflk, (o °) I ro(N) (p2° 1) ro(N)Ikivp Xl = l= 0 p(1+pX).f-af(p)pz+2X.f
Ik1
(0
(1 - a2p ki+2X)(1 - a' 2p kl+2X)
k1VN I
r
P
P
p(l+PX)flkl
°)
( °) -af(p)p2-+f.X
(1 - a2p-ki+2X)(1 - a'P2p-kl+2X)
The usual procedure (X H p-') yields for (2.13) (2.19)
Df
I Iki
O) Iki ro(N) (m
M=O(N)
= N-3 H (1 - p-3)(1 1
f2-2s)
-p-s+1
PCN
x
II PINT
-1 (1 -
)-k-s)
o-1) r0(N) o (N -l)-m-8
MNI
(1 - aP2p-s-kj+2)(1 -
x -s
with (2.20)
fs = E a(d, D, S)f lk, djNf
(0 0 )
ap2p-s-kj+2)
S. BOCHERER and R. SCHULZE-PILLOT
16
where
a(d, D, s) = fj a(dp, DP, s)
(2.21)
vINI
is a multiplicative function given by (2.14)-(2.18). Here we denote by t p the
p-part of a positive rational number t. In the sequel we write cv (d, D, s) instead of ca(dp, DP, s).
2.2. - Second Unfolding To continue the computation of the integral (2.1) we first need to find a good parametrization of C2,1(N)\r02(N); we shall follow 1221 (with the modifications necessary for level N > 1). We first remark that two elements of Sp2(7G) are equivalent modulo C2,1 (Z) if and only if their last rows are equal up to sign (the same is true for ro(N) and C2,1(N)). For C2,1(Z)\Sp2(Z) the parametrization given in (22] is as follows : (2.22)
{11,1(12, h) I h E S12 (Z),,. \Sl2(Z)}
(2.23)
{d(Jf)it,l(h,12) I h E S12(z)I 00 \S12(z)}
,
(2.24)
U
{d(M) o cl,l(h, h') It E S112(Z),,\Sl2(Z), h' E Sl2(Z)+,,\Sl2(Z)} I
au,vEN
u,v coprime
where
G12 (T) -f
S1p2 (IR)
0 Jf
= (1
A)
1) and M is an element of Sl2 (7G) with M
= (u
v) . Among
(2.22), (2.23),(2.24) precisely the following elements have their last row congruent to (0, 0, *, *) modulo N : (2.22')
(2.23')
{t1,1(12, h) I
It E r.\ro(N)}
{d(J)t1,1(h,12) 1 It E F 00 \Fo(N)}
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNC ION
17
h E Sl2(Z),\Sl2(Z), It' E Sl2(Z)+\Sl2(7G) (2.24')
d(M) o L1,1(h, h')
I
M= (u v)
E S12(Z)+\Sl2(Z)
U, v E ICY, uc
0(N), vc' - 0(N)
Here c (and c') denote the lower left entry of h (and h').
At this point we should emphasize that (2.22')-(2.24') do not give represen2 tatives of C2,1 (N) \r20 (N), since these elements are in general not in ro(N), but they are equivalent modulo C2,1 (Z) to such representatives (by using a suitable transformation we shall finally transport them into ro(N)). To describe (2.24') more appropriately we fix two decompositions
and
and consider for the moment only those h =
(c
) and h' =
(C,
d, )
with gcd(c, N) = N1 and gcd(c', N) = N1'. Then the data
71, V, (c d) (c, d,) describe an element of (2.24') if and only if N21u and NNIv. These data exist only if NIN1 N1', because we require u, v to be coprime. It is a
standard procedure to translate these considerations into more group theoretic terms : we denote by TN, = TNi an element of S12 (Z) with TN, _ (a Q) and N2Ia. This implies in particular that (TN,)2 E ro(N). N1 N2 Then (2.24') can also be described by (2.24")
h E (TN,ro(N)TNi
d(M)Li,l(h, h')
U
N1N2=N
N1N2 = N NINiNj'
I
\TNiro(N)
It E (TN'ro(N)T;, ) + \TNiro(N) 1
it, v positive, coprime, N2 u, N2'1 v
By a routine matrix calculation, we see that (with h, h', M, N1, Ni as in (2.24")) (2.24.,,)
G1,1(T(N1,N`i), 12) o d(M) o ti,l(h, h')
is indeed in ro(N), if we require (as we are allowed to do!) that M is of type M = r s with vir. 71
v
18
S. BOCHERER and R. SCHUIZE-PILLOT
Now we denote by
(z2 i z3,
s) the X22 X33 -component of the
(C[X2, X3],,-valued function I,n(s), restricted to (z2, z3) E )El X IH( -* H2With f, 0, 0 as before (f now again an arbitrary element in [ro(N), k1]o) we consider the double integral
(2.25)
f f
V2,V3
O(z2) 4'(z3)Xf
k2 k3 dx2dy2 dx3dy3 (z2, z3, S)y2 y3 y2
(ro(N)\})2
y2
where
1Cf(Z,s) _ f(9*) (Qb ® deck+a(i (9, Z))) 9EC2,i(N)\r2(N)
det Im(9) *
X2' Im(g )
is the same Klingen-type Eisenstein series as in (2.11), but with the Hecke operator removed; again /1.12,13 denotes theX2X'3-component of )C f, restricted to IHI x H. We Split into three parts according to the three types (2.22),(2.23),(2.24) of left cosets : K12,'3
(2.26)
IC V2, V3
f
i=1
It is again easy to see that JC121V3 and Kf 2''3 do not contribute to the integral (2.25). Using (2.24") we further split 10123"3 as JCV2,V3
(2.27)
f,3
/ f,N1,N,'
=
YV2,V3
Nl ,Ni
We can express 1Cf2,1 Ni more explicitely as follows 1.KV2,V3
f,N1,Ni
=2 (2.28)
E f 1k, T(N1iN,) (112h(< Z2 >) + u2h'(< V2
h h' ,u,v ,
X V12Uv3j(hv Z2)k-a-V2j
z3)-k-a-V3
t (h',
Im(h < z2 >) . Im(h' < z3 >)
x ( v2Im(h)+112Im(h')
ON TFIE CENTRAL CRITICAL VALUE OFT HE TRIPLE PRODUCT L-FUNCTION
19
u, v is given by (2.24") with Ni and Ni
where the summation over It, fixed.
It is well known how to unfold integrals like
1(f, 0, ,N1,Ni,s)
f
(2.29)
1
O(z2)0(z3)IC
dX2dy2 dx3dy3
12
1iN,1 (z2, z3, s)y22y33
y2
y3
(ro(N)\H)2
by applying TNi and TNi : The result is (2.30)
Z(.f,0,0,N1,Ni,s) _ I k2T, i) (z2). (Y' I k3TN;) (z3) (TN11o(N)TNi)_\ro(N)(TN,ro(N)TNi) 1
1
\Fo(N) 00
(V2
X >.f Ik1 T(N1,Ni)z2+U2z3VL22LL3 U 'V
k2
2
y2y32
$
v y2 + 2L y3
k3 dx2dy2 dx3dy3
X y2 y3 y2
y3
We do not want to work with Fourier expansions at several cusps, therefore we assume from now on that f, 0, V) are normalized newforms (eigenforms of all Hecke operators) of levels N f, No and N1. (all dividing N). We decompose N1 and N2 as
Ni =Ni,f'Ni,N2 =N2,f - N2
(2.31)
(and the same for Nl, NN and also for 0 and 0). We mention here the following facts, which we shall use in the sequel : N
R yOT(RNl)
with ,y E Fo (R) .
For any divisor d of Nf we have d
(0
I
O)OTNN
1=
with 'y E ro(Nf).
°
(0
1
Nf
°T(NI,N,) =ry°TNi,! 0
(d, Ni) 0
0
20
S. BOCHERER and R. SCHULZE-PILLOT
N TNl
=yo
N
H
1 Nz
o
q
0) 1
0
gIN2
with y E ro (N) and (using the same notation as in [18] X
VN
\N qY
q
with xq - Ny = q and qI x. For any newform g of level N and weight k we have (see [ 18, Theorem 3]) _ k
91k Vq = -a9(q)gl 7 . g.
At this point we introduce - as we already did for Nf in the previous
subsection- divisors DO and D'' of NO and NO; as usual we further factorize them (for given Nl and Ni) as DO = Do D2 and DO = D110 D'2 Using these facts we get the following Fourier expansions (2.32)
D I
Ik2
0O
l
l
0D
1) I k2TNi) (z2) = 0I k2 C
I k2TNi (z2)
O
O/
0 _ OI k2TNi m Ik2 ( o
D2
II V.
J
D
NO)
0Ik2
l (Z2)
Ik2
N- 4 O )(z2) 0
9IN2,0
D2
11 -a(q)q 2 +1
=
-.
2 D2 x N22 ,0 DT
= D1
2 D2
2k
' ( D2DN2,, 1
2
-ak(q)q-k2+1
gIN2,m Dq'
x>aO(n)e
z2)
D2N2,0
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
DV' 3
0
(2.33)
k
0
Tl
)
(z3) = 4(Ik
J 2>rti
Eap(n")e
V,(4')9-ka+1
11
21
D''''
-ax3 D2 N2,y n x3
vIN2,,
and for a divisor d of Nf : (2.34) CdO1
01
N NI OT (NJ, N,)(Z) _ .f I k1T(N,,Ni)f
fIk1TNf
(Ni, ((d1(N11NDI) 0 o
(d,(N
(d, Ni, Ni
(Ni f Ni,1)
0
= d- 2 . (d, N , Nlf )kI
0
/
0
d,N
k
11
qjIcm(N2,f,N2,f) (4)4-ki+l)
(-af We use here the simple fact that
B
(2.35)
N, NNi f
of
= lcm(N2, f, NZ f) and
1
(d,N1,NJ, f) 2
lcm N2 f , N 2,f) 2,f)
d
Now we are ready to plug these Fourier expansions into the expression (2.30) for
Z
(d
1)
0
1J
(DI 0) 0
(DI
1J
0 7
Nl, 1>
17
s.)
By integration over x2 mod N2 and x3 mod NN we see that only those terms a f(n)a,6 (n') ap (n") give non-zero contributions, for which (2.36)
Bnv2 =
n' N2,,
Do1 D2
n"
D'i
N2,0
D'2
and
(2.37)
Bnu2 =
S. BOCHERER and R. SCHULZE-PILLOT
22
Using
00 00
k2-2 k3-2 y2dy3 Y Y2
Y2Y3
G2 Y2
f f 0
0
(2.38)
+ U
r(s+k2+k3 - 2)r(s+k2 -1)r(s+k3 -1) r(2s + k2 + k3 - 2) x
(41rBn)-s-k2-k3+2
v-2s-2k2+2
2s-2k3+2
.
we obtain (2.39)
Z
Cd 01
lki
(
0
2( b ) x
1
0) 0 k2 (DI B-s-k2-k3+2
Zoo (s) N2 N21
d-
k3
1
.
(DI 0) N 1N11,1 s)) = 0 1
Do!?Do 1 2
'D'10 2
1
. (d Ni Nif )ki 11
11
(-af(4)fq-ki+l)
eIN2,0
vllcm(N2,f,Nzf)
eIN2",p
u,v,n
(-ap(9)4-k3+1).
1
D'0
0
af(n)a.(nv2BN2,0 L2)ap(nua2BN2,,,
x
H
(-ao(9)9-k2+i)
'
_.. ) DIV' 1
x ,v-2s-2k2+2+v2 u-2s-2k3+2+v3 n-s-k2-k3+2
with (2.40)
Z°°(s) =
/
s-k2-k3+2 r(s + k2 + k3 - 2)r(s + k2 - 1)r(s + k3 - 1)
r(2s+ k2 + k3 - 2)
To finish this section, we collect all the information obtained so far; we must take care of the fact that we worked in section (2.2) with the Eisenstein series 1Cf(Z, s) rather than with 1Cf(Z, s - a') as is required by (2.11). The
23
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
value of the threefold integral A as defined by (2.1) is (2.41)
(
0f
1
1)1
2("1/2 l (1_P21_P- -r
(F pJN
(DI0O Ik2
r)
l J
/'
,Y
(DI 1 1k3 OJ
S) =
1
Pp (PINE (1_.2-*- 1 )(1-a'P P- .-
r7
11
1
)
-1
PI Nf
E a(d, Df, 2s - r) d# (d, Ni , N'i )k' (Di)
(Dz )-
dINf
k
4
(Dik)
4
N2 . N2 . B-s+a'-k2-k3+2
2
Ni N2=N,NjN2=N,NIN1Ni
II
(-af (p)p k1+1)) ( T7 11 (-a,,(p)p k2+1)) Pl1cm(N2,f,N2,f) PIN2,0 TT
(
(PI
f
(-aV,(p)p k3+1))
2,.1
x E a f(n)a0(nv2BN2,4 J)a,'(nu2BN2I n,u,v
,,j,D'0)
XU_2s+a-2k3+2+v3V-2s+a-2k2+2+v2n s+a'-k2-k3+2
We remind the reader that the summation over u and v is subject to the condition N21u and N2Iv.
3. - The Euler factors Using the multiplicativity properties of the Fourier coefficients of f, 0, and that the conditions of summation are of multiplicative type we may now write (2.41) as (3.1)
2(b) A(r+s,b)-y(r+a+b,s-a').I \v2/
TP(s) P
To save notation we write (3.2)
TP= 7 P-0 'TP
where Ti denotes the last four lines of (2.41). In all cases to be considered, the pair [(N1)P, N1P] can take the values [p, p], [p, 1] and [1, p] ; therefore we
S. BOCHERER and R SCHULzE-PILLOT
T1 =CPP+Cri+Cir. If dp can take both values 1 and p, we further decompose C.. as
C.. = C'. + C;
(3.4)
according to the cases d,, = 1 and dp = p.
Part I : Df = DO = DP = 1 p
IA : The case pINf, pINt, pjN0 (This is the case also considered by Gross/Kudla[ 111).
The conditions imply d,, = N = NO = NO = 1, ap(d, Df, s) = 1 and for the case
1 1
B'' =
otherwise.
P
Cpp(s) =
(3.5) 00
00
of (r)1)aO(T)1 (E 1=0 t=0 00
t) ao (p1)
(pt)) -s+.'- k, -k3+2 (7)t)) -2s+a-2k2+2+v2 +
00
+ E E a f (p1)aO()1)a0
(1)1+2t)(I)1))-s+a'-k2-k3+2
(pt))-2s+a-2k3+2+V3)
1=0 t=1
This equals (we use the fact that ao(p2) and a,,(p2) are powers of p) 1
1-
af(7))aO(p)a0(p)p-s+a'-k2-k3+2
1
(Il
q 1-aO(p2)p 23+a-2k2+2+V2 1
1
1- a
I)
2s+a-k2+V2 +
+ 1 - p-2s-r 1
f(p)ao(p)a,p(p)p-3-3a'-2r-6+2
+ 1 - aV, (P)P 1
1
1- of (1))a,6(1))aV,(2))p s+a'-k2-k3+2
_
aV,(t)2)p-2s+a-2k3+2+V3
r
p-2s-r
2s+a-k3+2+V3
p-2s+a-k3+V3 -p-2s+a-k3+V3
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
(3.6)
x
ps-a'+k2+k3-2
Ci,P = p ,
00
>af(p1)ao(p1 1=0 t=1
-
(p)1-k,+1)
_ (-af
00
p)aV,(pt+2t
.
. (-aop k2+1) x
)(pt)-2s+a-2k3+2+L3 (p1)-s+a -k2-k3+2
-
af(p)a,(p)a,,(p)p3-s-3a'-b-2r
1
1 - af(p)ak(p)ao(p)p s+a'-k2-k3+2
1 - a,(p2)P 2s+a-2k3+2+v3
1
af(p)aj(p)ap(p)p3-s-3a'-b-2r
l
1 - p 2s-r
s-3a'-2r-b+2
1 -
25
Quite the same computation shows that CI,,1 =C1,r
(3.7)
Hence
T1 =
1
1-
(3.8)
af(p)aO(p)a.V,(p)p-s-3a'-2r-b+2
1 + 2a1 (p)a,(p)ap(p)p s-3a'-b-2r+3 + p 2s-r
1 - p-2s-r
Now we write the numerator as 1+
2af(p)ak(p)ao(p)p-s-3a'-b-2r+3
_
+P-2s-r
af(p)aO(p)aV,(71)7-s+3-3a'-b-2r)2
(1 +
J
(3.9)
(1 -
2s-r)2
(1 -
of(p)aO(p)a0(p)p-s+3-3a'-b-2r)2
Therefore we obtain (3.10)
T, (s) _
-1 1 -
1 af(p)ao(p)a,,(p)p-s-3a'-2r-b+2
(1
-
of(p)ao(p)aO(p)p-s+3-3a'-b-2r)2
IB : The case pINf, pINO, pIN,, These conditions imply
=DO=D, =1
and
dpE{1,p}
26
S. BOCHERER and R SCHUIZE-PILLOT
cep(d, D f , s)
if dp=1
-(1 +P 1 -8)
_
pl-4
if dp = p
- of (p)
and
dp=(N1)p=Nip=p
if
otherwise.
dp
What we get is this : C,,,(s) =
(3.11)
-(1
+pl-2s-r)(1
(p)p-s-3(i'-2r-h+2)(1
(1 - apab(7p)at]
-
28-r)
+
apaO(p)aV,(Tt)T-s-3a'-2r-b+2)(1 - p 2s-r)
(similar calculation as in (3.5)) ; Ci1,(s) _
(3.12)
_ _(1 +p 1-2,-r) - p'
(-ao(p)p-k2+1)x
00
00
Y, E of Vil a, (p' . p) ao
(l )-s+a -k2-k3+2
(P1+2t)(Pt)-2s+a-2k3+2+v3
1=0 t=1
J
(1 1 + p1-2s-r) .
p
2s-r ,(p))p-s-3a'-2r-b+2)(1 - p-2,,-r)
(1 - cEpa4,(}))a11,(p)p s-3a'-2r-b+2)(1 - a,a4,(p)a
- lp
C1PI _ C1
(3.13)
C77 (s)
(3.14)
=
P1- 4
of (P)7'
of
4
= pkl p-s+a -k2 -k3+2 x (pl)ak(pl+2tp)aV,(pt
p)(pt)-2s+a-2k2+2+v2
(pl)-s+a -k2-k3+2+
lI'
L=0 t=0 00
00
E Y, a f (pl)a4, (pl p)a,j,
(1+2L)
(pt)
-2s+a-2k3+2+V3 It''Ml) -s+a' -k2 -k3+2
l
1=0 t=1 (p)ai(p)aq,(p)ps-3a'-2r-b+3(1
af
(1 -
cxpa4(p)aV,(p)p-s-3a'-2r-b+2)(1
+ p
2s-r)
- ai,ao(p)aq,(p)P s-3a'-2r=b+2)(1 - p-2s-r)
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
(3.15)
C1,P
pl-4af(p)p4 00
27
. p . P,-.'+k,
00
I: a f (pI) aqs (pI 1=0 t=1
p
. p) ao
+k2-2(-ao(p)p k2+1)x (pt+2t 1)(P,)- 2s+a-k3+2+v3 W) -s+a'-k2-k3+2
l/'
(p)ps-2r-3a'-b+3
-af (p) ao (p)a,
(1 - apas(p)a,,(p)p s-3a'-2r-b+2)(1 - a' ak(p)a.,(p)p s-3a'-2r-b+2)(1 CP -_ CPlP.
(3.16)
PI
All the C;,, have the same denominator, the numerator of > C;* being equal to
1 - p 2s-r + pl-2s-r - afa ,a,i Jp3s-3r-a-b+3
- p s-a-2r-b+3}
This can be factorized as -(1 -
p2s-r)(1
(p)p-s-2r-3a'-b+3)
+ apa,5(p)a
a'Pao(p)a.,(p)ps-2r-3a -b+3) .
(1 +
Now we also write the denominator of TPO as product of linear factors in p-8
using (1
2pkl+2-2s-r) _ (1-a1,ama,ps-3a'-b-2r+3) (1_
(and the same with a'' instead of aP) ; therefore we get some cancellations and arrive at : (3.17)
T-
(1-a1ao(p)aV,(p)ps-3a'-b-2r+2)(1-a''
ao(p)a,,,(p)p-s-3a'-b-2r+2)
1
x (1-aPa4,(p)a ,(p)p s-3a'-b-2r+3)(1-a,a0(p)a,,(p)p-s-3a'-b-2r+3) IC : The case pIN f, pINO, pIN'O We have (B)P
NJ =Ni=p
1
if
1 p
otherwise,
p2s-r)
28
S. BOCHERER and R. SCHUIZE-PILLOT
and we get (3.18)
C1P =
=p,
ps-a'+k2+k3-2
00
, (-af(p)p ki+l)x
00
t)-2s+a-2k3+2+v3(pl)-3+a' -k2-k3+2 Y
l=1 t=1 00
00
=
- l'=0 EE af t=1
(p6')
a,5 (p
l')a
1'+2t)(pt)2s+a-2k3+2+v3(p1 )-s+a -k2-k3+2 + (p
(3.19) 00
00
=-1: 1: of (tr/l)a,6
Cpl
pI)(pt)-2s+a-2k2+2+v2 ,T`I)-s+a'-k2-k3+2
(p1+2t)ati,
V'
1=0 t=1
CPP(s) =
3.20) 00
00
E E a (p')ao L=o t=o 00
+
1+2t
/l
/l
00
E 1=0 t=1
af,(pl)a
(pL+2t)
(T/l)a
s+a'-k2-k3+2
t
2s+a-2k2+2+v2
(pL))-s+a -k2-k3+2 (pt)) -2s+a-2k3+2+v3
Summing up the C. we see that the summands in (3.20) with t
0
cancel against (3.18) and (3.19) and we get 00 af(p1)aj(p))aV,(d)(P1)-s-3a'-b-2r+2
IP(S) _ -(1 - p-2s-r)-l 1=0
-1 (1 x
af(p)I3PYPp-s-3a'-b-2r+2)(1
-
af(p)QP'YP,p-s-3a'-b-2r+2)
x
1
(1 - a f(p)f3 /' y
s-3a'-b-2r+2 )(1 - af(p)QP/
YPp/-s-3a'-b-2r+2 )
.
ID: The case p{N This case (which is in some sense the most difficult one) was previously considered in [9[ for the case of equal weights. It was already noticed in [21) and [22[ that the result from [9[ carries over to the case of arbitrary weights.
We just state the result here :
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-F JNC77ON
29
p-2s-r) (1 _ p2-4s-2r)Lp(f 0 0 ® &, s + 3a'+ 2r + b - 2).
(3.22) TP = (1 -
Part II: DP =NP ,DI =NN,DPI' =NP Only the cases B and C are to be investigated, the case B being the most complicated : II B : The case pINf. , pINO, pI NO
These conditions imply s1
(d D f s) _
B.=
{_af(P)P2__8
if
dp = 1
p(1 + pl-s)
if
dp = p
1
(3.23)
dp=(Nl)p=(Nl)p=p
if
P
otherwise.
CC1, = ap(1, Df , 2s + r) x
EE af(pt)ai1I11+2t)a,j (pi) rT
(pt)-2s+a-2k2+2+,,2
(M1)-s+a -k2-k3+2 +
1=0 t=0
00
00
1: 1: aj
()1)a,6(p1)aG(11+2t)(J)t)-2s+a-2k3+2+V3(PI)-s+a -k2-k3+2
1=0 t=1
ap(1, Df , s) (1 + p 2s-r)
(1 -
apao(p)a.p(p)p-s-3a'-2r-b+2)(1
-
a7,ao(p)ap(p)p-s-3a'-2r-b+2)(1 -
P 2s-r)
(computation similar to (3.5)). (3.24) C17, = cap(1, Df, s) p (-ak(p)p k2+1) x 00
CID
-k2-k3+2
1: 1: 1=0 t=1 -p-2s-rap(1, Df,,, 2s +/ r) (1 - aPa0(P)aV,(P)p s-3a'-2r-b+2)(1 -
(3.25)
1
aa4,(p)a+,(p)p-s-3a'-2r-b+2)(1 - p-2s-r)
1
CP 1 = C1,P ,
30
S. BOCHERER and R SCHULZE-PILLOT
and
(3.26)
Cs
-_
-af(p)p2-4-s
/
(p)aO(p)p-s-3a'-2r-b+2
(1-ara,k (p)a+o(p)p s-3a'-2r-b+2) (1-apa
(3.27) CPr(s) =
= ar(p Df , s + 2r) p a 00
pk'ps+a'-k2-k3+2 x
00 (pl+2tp)
E a f (pt) ac
a'G (p'p) (pt)
-2s+a-2k2+2+v2 (pt) -s+a' -k2 -k3+2+ lll''
1=0 t=0 ao
00
1: 1: a f (p1) a , (pt p)ap 1=0 t=1 _
(,,)1+2tp)
(pt)
-2s+a-2k3+2+v3 / 1) -s+a' -k2 -k3+2 1I
j,(p)p4-s-3a'-2r-b+2(1 +
ap(p, Df , 2s + r)a j(p)a
(1 -
apaO(p)a,,(p)p-s-3a'-2r-b+2)(1
p2s-r)
- apak(p)aV,(p)p-s-3a'-2r-b+2)(1 -p-2s-r)
(similar computation as in (3.5)).
(3.28) CIPP(s) = ap(p, Df, 2s + r) . p-4 . P . 00
ps-a'+k2+k3-2 (-aop-k2+1) x
00 1)(pt)-2s+a-2k3+2+v3
1It1)a4,(p1-1
E E of 1=0 t=1 _
p)a - `,(t'1+2t
(m1)-s+a'-k2-k3+2 1I'
-ap(p, Df , 2s + r)aj(p)aq, (p)p- '-s-a'-r+2 (1 - arao(r))ap(p)p s-3a'-2r-b+2)(1 -
aya,6(T))aV,(p)p-s-3a'-2r-b+2)(1
- p-2s-r)
Cr rl_-Crlr'
G,r _
-aoao(1 + p1-2s-r)p --s-a'-r+3 (1--arao(p)a'p(p)p s-3a'-2r-b+2)(1-alpa,6(p)a.o(p)p-s-3a'-2r-b+2)
The numerator of E C;, is equal to
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
31
(3.31) -+r+3a'+b-1-s(1+a f(p)aO(p)aOp-s-2r+3-b-3a'+p1-2s-r)
_
a
We can therefore apply the same kind of trick as in case I B; denoting by ep(0) the eigenvalue of the Atkin-Lehner-involution VPN acting on qS (and similarly for V)) we end up with TP(s) = -EP(O)EP(W)p,
(1 -
apa4,(p)aO(p)p-s-3a'-b-2r+2)(1
-
1
x (1
-
aPa4,(p)aO(p)p-s-3a'-b-2r+3)(1
i-s '
a4,(p)aV,(p)p-s-3a'-b-2r+2)
-
apak(p)a,(p)p-s-3a'-b-2r+31 )
II C : The case pINf, pIN", pIN"
a
CPP(s)=p2
(3.33)
k3
p2 x
00
t af(pt)ao(pt+2t
I 1=1 t=0 00
1)(pt)-2s+a-2k3+2+V3(pl)-s+a'-k2-k3+2 l
1)ap(pl+2t
+ 00 1:
I
1=1 t=1
= P s+a'+2-k22k3 x 00
00
{ 1=0 t=0 CO
of (ltt+l) aO (pt+2t) aV, (pt) (pt)
-2s+a-2k2+2+V2 (pl) -s+a' -k2 -k3+2 +
00
of(j)'+') aj (pl)a+,( l+2t)(pt)-2s+a-2k3+2+V3(l-+ -k2-k3+2 E I 1=0 t=1
C1j = p-2+
(3.34) 00
2
p
00
a f (pt) a, (pt p) aV,1111+2t
p
1=0 t=1
-p-s+a +200
k
ps-a'+k2+k3-2
. (-(P)P-k,+1) x
1 1)(,)t)-2s+.-2k3+2+,,3 (1ll) -s+a' -k2 -k3+2 PP 1I
x
00 EEaf(pt+1)a,,(
1=0 t=0
1)a.
ll
1
lt,
32
S. BOCHERER and R SCHULZE-PILLOT
(3.35)
-p-s+a'+2- k
C1,1 =
00
x
00
of E 1=0 t-o
(pl+1)a,(p)+2t)a,(pt)(pt)-2s+a-2k2+2+V3 (pl)-s+a'-k2-k2+2
Hence in the sum of the C** only the "t = 0-part" of C,i survives and we obtain -EP(f)p1-s
TP
(1 - af(p))3P
YPp-s-3a'-b-2r+2)(1
-
3 af(p)0P1'Pp-s-3a'-b-2r+2)
1
x
(1 - a f(7/)QPYPp -s-3a'-b-2r+2 )(1 - af(p)pPY IPp-s I
/
-3a'-b-2r+2 )
.
Remark : Although our list of Euler factors is complete, the reader should be aware of the fact that in our integral representation (2.1) we are free to interchange the roles of ¢ and V (interchanging the roles of v2 and v3 at the same time), but f has to be the cusp form of largest weight. Therefore e.g. in case IB, IIB we should also consider the case where p does not divide the level of 0 or 0. It will be left to the reader to show by similar computations as above that in those cases the Euler factor will be the same (as should be more or less clear from an adelic point of view).
4. - The Functional equation The factors T,,. = T ,,(s) computed in the previous section are for p f N and for p I gcd(Nf, No, N,,) up to a shift in the argument and an elementary factor the known Euler factors of the triple product L-function L(f, ¢, z/), s) associated to f, 0, z&. We define therefore now : DEFINITION 4.1. - The triple product L function associated to f, 0, 0 is for S > k,+k22 k3-1 defined as L(f, 0, ',, s) _ fl1, LP(f, 0, 0, s) where the Euler factor LP(,f, 0, /i, s) is given by (4.1)
L1,(f,4>,V),s+3a'+2r+b-s) = T1, (S)
1(1 - p-2,,-r)-1(1 - p2-4s-2r)-17'(s)
if pIN if p { N
With (4.2)
L.(f, 4>,,O, s) = r'c(s)rc(s + 1 - k1)rc(s + 1 - k2)rc(s + 1 - k3)
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
33
(where I'c(s) = (2ir)`17 s) as usual) the completed triple product L -function is A(f, 0, 0, s) = L.(f, 0, 0, s)L(f, 0,,0, s).
With these notations we have for the integral A(f, 4i, 0, s) from 2.1
(with c2(s) as in (1.19), (7,(s) = (1 - p s)-1 for finite primes p and
( (s) = 7r-s/2r(s/2)) : THEOREM 4.2. -
A(f, 0,,0, s)
(4.3)
C2 (s)
(s+ 1)(2s + 1)
V2
s + k, + 22+k3 - 1) zeta,,,(4s + 2)( (2s + 2)
Loo
7r2+3ri
(b )ib+kl N-2s-223-2b-a.
L1,(s+ k1+22+k3 - 1) x
(1,(4s + 2)(p(2s + 2)
,s
riN(-Lp(f,
kl+k2+k3 2
- 1))
1
Proof. This follows from 2.41 and the results of Section 3. THEOREM 4.3. - The function A(f, 0,,0, s) has a meromorphic continuation to all of C and satisfies the functional equation (4.4)
A(f, 0,, s) =
-N-a(s+ kl+k2+k3 -1)(gcd(Nf,
k1+k2+k3 _j) 2
No,
(fl
NEp)A(ki+k2+k3-2-s)
where for p I N the number Ep is defined as the product of the eigenvalues under the p-Atkin-Lehner-involution wp of those forms among whose level is divisible by p.
Proof. We put r = 2 since for this choice the functional equation of the Eisenstein series is under s --> -s. It is then not difficult to read off the functional equation of E from the calculations in (1I]; actually things become somewhat simplified since we need the functional equation only up to oldforms. We need the following Lemma : LEMMA 4.4. - The Eisenstein series E2,3 satisfies thefunctional equation : (s + 1)(2s + 1)(,(4s + 2)(,,.(2s + 2) 11 (;7,(4s + 2)t p(2s + 2)E2,3(Z) pfN
_ -N-9s(s - 1)(2s - 1)(,,.(-4s + 2)(00(-2s + 2) 03 -13 x fl (,p(-4s + 2)(7,(-2s + 2)E2,-,l (Z) + )ES T4N
( N13
03
34
S. BOCHERER and R. SCHULZE-PILLOT
where ES is a linear combination of Eisenstein series for groups r3 (N') with N' strictly dividing N or conjugates of these by a matrix
M I (N/N').
03
(M13
-13 ) with 03
Proof. Let Fr,s(Z) be the Eisenstein series of degree 3 defined in the same way as Er,s in Section 2, but with the summation running over coprime symmetric pairs (C, D) with gcd (det C, N) = 1; one has 0
'r,slr N
(4.6)
_ N-(3r/2)-3sFjrs
1
0
The calculation of the local intertwining operators M(s) = Mp(s) in sections 5 and 6 of 1111 gives (in the notations of that article) Mp(s)4p(s) for p = 00 or p f N and allows for p f N to express Mp(s),DP(s) explicitly as a linear
combination of the sections P(-s), fir3,(-s), (Dp)K(-s), (4)p) K, (-s). An elementary calculation shows in this case that the coefficient at 4)P(-s) is zero and that the coefficient at 4) (-s) equals 7-6s-3
(p(2s - 1)(p(4s - 1)
((-4s + 2)(p(-2s + 2)'
We let Z = X + iY and g = (g.,1, ...) E Sp3(A) with g. _ 13 X) (Y '/2 y-1 1. Then it is well known that with C 03
13)
t13
4)0
_ doo X f 4)p X fl(4)p)K pIN
p(N
and analogously defined 4)3 one has E2,,,(2) _ (det Y)-'E(g, s, °) and 1F2,3(Z) = (det Y)-1E(g, s, 13) (see Proposition 7.5 of (111), and analogous formulae are true for all the global sections I for which the p-adic components fip are one of the 4 , fiP, (4) p) K, (4)p)K'. The assertion of the lemma follows upon using the duplication formula for the gamma-function for the contribution from the infinite place and then applying the functional equation of the Riemann zeta function. We can now finish the proof of Theorem 4.3. From Theorem 4.2, Lemma
4.4 and the fact that the integrand in A C2(s) contains by 1.18 a differential operator independent of s we see (using the results of Part II in Section 3) that 'O'V,'3
A(ss + k,+ 22+k3 (4.7)
- 1) = /7 7 ( 1TE
N,6, Nq,))
N ep)A(-s +
k'+ 22+ks
- 1)
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNC71ON
35
where for p I N the number eI, is defined as in the assertion of Theorem 4.3. Putting s' = s+ k1+k22+k3 -1 we obtain the desired functional equation
under
k1+k2+k3-2- s'.
5. - Computation of the central critical value Following the strategy of [12[ we now evaluate the integral 2.2 at the point s = 0 using a variant of Siegel's theorem, i. e. , expressing the value at s = 0 of the Eisenstein series G as a linear combination of theta series. The setting for this is basically the same as in [2, 4, 61. Let M1, M2, M3 be relatively prime square free integers such that M1 has an odd number of prime divisors. By D = D(Mi) we denote the quaternion algebra over Q ramified at oo and the primes dividing M1 and by R = R(M1, M2) an Eichler order of level M = M1M2 in D, i. e. the completion Ri, is a maximal order ,,
for p { M2 and conjugate to { I a
)
c-= 0 mod p} for p M2,
E M2 (Zp)
where we identify D ® Qr with M2 (Q) for p { M1. By gen(MI, M2, M3) we
denote the genus of 7L-lattices with quadratic form (quadratic lattices) of R(M1, M2) equipped with the norm form of the quaternion algebra scaled by M3. The genus theta series of degree n of gen(Mi, M2, M3) is then gen (n)
M1,M2,M3 (Z)
=
E {K}Egen(M1,M2,M3)
O(n) K, Z) if-)(W) I
where the summation is over a set of representatives of the classes in gen(MI, M2, M3), O(K) is the (finite) group of orthogonal units of the quadratic lattice K, Z is a variable in the Siegel upper half space IFIL and
exp(2iri tr(q(x)Z))
O(n) (K, Z) =
with q(x) = (2(M3tr(xtxj)))i,j We consider a double coset decomposition h=h(ML,M2)
DA =
U
DQyiRA
i=1
of the adelic multiplicative group of D with RAx = Dx x rjP#. Rp and representatives y2 with n(yt) = 1 and (yi). = 1. Then the lattices Iii _ yi,RyJ- (with the norm form scaled by M3) exhaust gen(Mi, M2, M3) (with
some classes possibly occurring more than once) and it is easily seen that
36
S. BOCHERER and R. SCHULZE-PILLOT
with Ri = Iii and ei = 1R2 I we have OH (Ij4, Z) = 2"OMn,M2,M3
(5.1)
eie4
i,j=1
(Z)
where w = w(Ml, M2) is the number of prime divisors of M1M2. With these notations we have from [2] (Theorem 3.2 and Corollary 3.2) and [6], (p.229) : LEMMA 5.1. - The value at s = 0 of G2,3 (Z) is
E
aMi,M2,M30gen
,
(3)2,M3
Mi,M
(Z)
M1M2M31N
with (-1)1+w(M1,M2)(MiM2)-3M3 68.7r4((N)(2)-2.
aM1,M2,M3 =
In order to compute the value at s = 0 of the differentiated Eisenstein series from Section 2 we have to compute (D*(a,V2,V3) 6 (3)
(5.2)
2
(K1 -)) (L111(z1, z2, z3))
for the individual theta series appearing in the sum in Lemma 5.1. We denote by U,, the space of homogenous harmonic polynomials of degree p in 4 variables and identify an element of UN, with a polynomial on D,,, by evaluating it at the component vector of an element of D,,. with respect to an orthonormal basis relative to the quaternion norm on D. Similarly, for v E N let U(,°) be the space of homogeneous harmonic polynomials of degree v on R3 and view P E U(°) as a polynomial on D(am) = {x E D,,.Itr(x) = 0}. The representations T of D'/1[8" of highest weight (v) on given by P(y-lxy) for v E N give all the isomorphism classes of irreducible rational representations of D ,/1[8". By we denote the invariant scalar product in the representation space by the invariant scalar product in the SO(Do, norm) =: H1 -
space U,,,. We notice that the invariant scalar products on the Uµ; can be normalized in such a way that they take rational values on the subspaces of polynomials with rational coefficients and that these subspaces generate the U, . Indeed, consider the Gegenbauer polynomial C(µi) (x, x') = obtained from [21
C('`) (t)
= 2'`i J:(-1 j=0
1
j!(Ni - 2j)!
(µi
-A 223
tµ:-2j
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
37
by
C(µ:)(xl,x2) = 2(µ`)(norm(xl)nornl(x2))1 /2C11`.)(
2
tr(x122) norm(xl)norm(x2)
and normalize the scalar product on U,, such that C(µi) is a reproducing kernel, i.e., ((C(µi)(x,x),Q(x)))µi = Q(x') for all Q E Uj;. Then the with rational x' are rational, generate U,,,, and the reproducing property implies that they have rational scalar products whith each other. The same argument applies to the It is well known that the group of proper similitudes of the quadratic space (D, norm) is isomorphic to (D" x D")/Z(D") via T
x) i--> Q
with a
(
= x i Y X2-
and that under this isomorphism SO(D, norm) is the image of
{(x1ix2) E D" x Dx I n(x1) = n(x2)}. is isomorphic to the H1 -space U2 and the isomorphism can be normalized in such a way that it preserves rationality and is compatible with the invariant scalar products on both spaces (which are assumed to be normalized as above). Denoting by S the Gram matrix of the quadratic lattice K we know from Section 1 that (5.2) is of the form Moreover, the SO(D00, norm) =: HH -space U(,°) ® U,..
P(S'xl, S7x2, SI x3) exp(7ri(S[xl]zl + S[x2]z2 + S[x3]z3)) a=(x1,x2,x3)E(Z4)3
where P E ®3 jUk _2 is a harmonic polynomial of degree µi = ki - 2 on R4 in each of the variables and is invariant under the (diagonal) action of Ha = O(D ® I(8, norm). Moreover, P is independent of S and has rational coefficients (up to a factor of 7r3r+21,) The HR-invariant trilinear form T on Uµ, 0 UI62 0 U,, defined by taking the scalar product with the invariant
polynomial 7r-3"-2hP (as remarked in Section 1 this is up to scalars the unique invariant trilinear form) is hence rational (i. e. takes rational values on tensor products of polynomials with rational coefficients). If all the pi are even (as is the case in our situation) then the decompofrom above gives furthermore that T sition of HH and of U,, as 0 factors as T (O) (9T(0), where the unique (up to scalars) D'-invariant trilinear form T(°) on U( Uf has the same rationality properties
38
S. BOCHERER and R SCHULZE-PILLOT
as T. Of course both T and To are just the ordinary multiplication if all the pt are 0. For any positive definite symmetric 4 x 4-matrix S we define the UN,ivalued theta series 6S(pi) by (5.3)
(z)(x')
_
C(µ:) (S1/2x, S1/2x') exp(7riS[x]z). zEZ4
We notice that if K is a quaternary quadratic lattice with Gram matrix S the right hand side of 5.3 does not depend on the choice of basis of K with respect to which the Gram matrix is computed (because of the invariance of C(µ:) under the (diagonal) action of the orthogonal group) ; we may therefore write it as O("i) (K) as well. We denote by O(Mi)M2 M3 the weighted average of the sµ') over the Gram matrices of representatives of the classes in the genus gen(Ml, M2, M3) as above. We find -))(t111(zl, Z2, Z3)) _ (5.4)
3a+2bT(15011)(S, z1) ® p(/t2)(S, z2) ® 15(P3) (S' Z3)) 7r
(where T is as above). For the value at s = 0 of (2.1) we obtain therefore 7r 3-+2b
EMiM2M3IN aMi,M2,M3T((?M1>M2,M3(zl),
P Z1#9
(5.5)
15(Mi)M2,M3(z2),O(z2)) ®(o MI, 2,M3(Z3),
(z3)))
where by (,) we denote the Petersson product. In order to evaluate this expression further we use Eichler's correspondence. We fix M1, M2 and an Eichler order R(M1, M2) C D(Mi) as above
and set M = M1M2. For an irreducible rational representation (V,,'-) (with it = T, as above) of DR /R' we denote by A(DA , RA, T) the space of functions co : D' --+ VT satisfying cp(yxu) = T(u;)w(x) for -y E D' and u = uoou f E RA". It has been discovered by Eichler that these functions are in correspondence with the elliptic modular forms of weight 2 + 2v and level M = M1M2. This correspondence can be described as follows (using the
D X y R' from above) : recall from double coset decomposition DA = i=i U Section 5 of [2] and Section 3 of [4] that for each p I M1M2 we have an involution wl, on the space A(Dx , Rx , given by right translation by a suitable element -7rl, E R, of norm p normalizing R. This space then splits into common eigenspaces of all these (pairwise commuting) involutions. On A(DA, RA, -r,,) we have furthermore for p { N Hecke operators T(p) whose
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
39
action on these functions is expressed by the Brandt-matrices (whose entries are endomorphisms of V,-j. They commute with the involutions 5;. On the space A(DA, RA, T,,) we have moreover the natural inner product defined by integration, it is explicitly given by h
ti_1
((co(yi), p(yi))),el eti
By abuse of language we call (in the case v = 0) forms cuspidal, if they are orthogonal to the constant functions with respect to this inner product. We denote for p dividing M2 the p-essential part by Ap,ess(DA, RA, r) consisting of functions cp that are orthogonal to all p E A(DA, (RA)", T) for orders R' D R for which the completion R11, strictly contains R. It is invariant under the T (p) for p f M1 M2 and the t , for p I M1 M2 and hence has_ a basis of common eigenfunctions of all the T(p) for p t N and all the involutions th1, for p I N. Moreover by the results of 17, 15, 17, 231 we know that in the space Aess(DA, RA, T) of forms that are p-essential for all p dividing M = M1M2 strong multiplicity one holds, i.e., each system of eigenvalues
of the t (p) for p t M occurs at most once, and the eigenfunctions are in one to one correspondence with the newforms in the space 52+2" (M) of elliptic cusp forms of weight 2 + 2v for the group ro(M) that are eigenfunctions of all Hecke operators (if -r is the trivial representation and R is a maximal order one has to restrict here to cuspidal forms on the quaternion side in order to obtain cusp forms on the modular forms side). This correspondence (Eichler's correspondence) preserves Hecke eigenvalues for p f M, and if cp corresponds to g E S2+2v (M) then the eigenvalue of g under the Atkin-Lehner involution w7, is equal to that of cp under wP if D splits at p and equal to minus that of cp under wP if Dp is a skew field. From (3.13) of [6] we know that if g having first Fourier coefficient 1 corresponds in this way to cp with (cp, cp) = 1 then (g, 601) (K)) = (g,g)(cp(yi) 0 (yj)) holds.
It is not difficult (see also [13]) to extend this correspondence to not necessarily new forms g E S2+2, (N) in the following way :
LEMMA 5.2. - Let N = M1M2 be a decomposition as above. Call g E
S2+2v (N) an M'-new form if it is orthogonal to all oldforms coming from g' E
S2+2v (M') for M' I N. Then Eichler's correspondence from above extends to a one to one correspondence between the set of all M1-new eigenforms of the Hecke operators for p {' N in S2+2v (N) that are eigenfunctions of all the wP for p I M2 with the set of all Hecke eigenfunctions in A(DA , RA x, T) that are eigenfunctions of all the involutions Vi,,. This correspondence is compatible with the Hecke action and the eigenvalues under the respective involutions
as above; it maps newforms of level M' I N (with M1 I M') to forms that
40
S. BOCHERER and R. SCHUl7_E-PILLOT
are p-essential precisely for the p I (M'/M1). The correspondence can be explicitly given (in a nonlinear way) by the first Yoshida lifting sending cp to the form
E
((Ay) 0 AyA 6(2v)(I=i))) ete
;
it then sends cp with (cp, cp) = 1 to g' having first Fourier coefficient one. Moreover, in this normalization it satisfies the scalar product relations from above, i. e. , if g corresponds to cp then (g, 0(2L,) (Ij)) (p, cp)2 _ (g" g`) (cp(yti) cp(yj)) holds.
Proof. Let Mbe a divisor of N and g E S2+2v (M') and let a be a function from the set S of prime divisors of N/M' (whose cardinality we denote by w(M/M')) to {f1}"(M/M'). Then the function gE :_ >s'CS 91 fpES e(p)WP) in S2+2v (N) is an eigenfunction of the Atkin-Lehner involutions w p for the p I (N/M') with eigenvalues e(p). Similarly, fix a maximal order R C D and an Eichler order R = R(Ml, M2) C_ R, let M2 I M2 and let R(M1i M2) be the Eichler order of level M1M2 in D containing R and contained in R. Let e be a function on the set of prime divisors of M2/M2 as above. Then to cp E Aess (DA x, (R(Mj, M2))x , T) we construct as above a unique W" having
the same Hecke eigenvalues for p { N and the same ti -eigenvalues for p I M1M2 as cp such that cp` is an eigenfunction of the t`, for p I (M2/M2) with eigenvalues e(p). Given an Mi-new Hecke eigenform g in S2+2v(N) that is an eigenfunc-
tion of all the wP for p I M2 with eigenvalues eP we then associate to it the newform g of some level M' = M1M2 I M1M2 that has the same Hecke eigenvalues for p { N so that g = gE with e(p) = eP and apply Eichler's correspondence to get a cp E Aess(D' , (R(Ml, M2))x , From [2] we know that this can be normalized such that g is obtained from cp by Yoshida's lifting. We.then pass to the eigenfunction V' of all ii1P with the same eigenvalues as g for p I M2. The scalar product relation (up to normalization) follows
then in the same way as in [6], using the uniqueness of the given set of ii -eigenvalues and the fact that application of wP for p I M2 transforms e(2i) (Its) to e(2") (II,j), where yi, represents the double coset of gtxp 1 (this is an easy generalization of Lemma 9.1 a) of [2], see also 131). The same argument shows that Yoshida's lifting realizes this correspondence (using the well known fact that it gives the right Hecke eigenvalues for the p { N), and using the expression of g' as a Yoshida lifting we find the correct normalization of the scalar product relation as in [6]. We will need a version of the scalar product relation in Lemma 5.2 also for the case of newforms of level strictly dividing the level of the theta series involved.
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
41
LEMMA 5.3. - Let M1, M2 with M = M1 M2 dividing N be as before and
let g be a normalized newform of level M and weight k = 2 + 2v as in the previous Lemma; let S be the set of prime divisors of N/M. Put M2 = N/M2
and let R' = R(M1, M2) and R = R(Mi, M2) C R' be Eichler orders of levels M, N respectively in D (Mi) and consider a double coset decomposition
DA = Uh ,D. y R and corresponding quadratic lattices (ideals in D) Iz, relative to R as before. Let cp E AeSS (DX , (R' ) x, be the essential form corresponding tog under Eichler's correspondence (with (p, cp) = 1). Then (5.6)
(g,e(2l)(It,))
( rl (J
_ (9, 9)
co)(y1) °
S'CS pES'
( rl pES'
Proof. Let e be a function from the set of prime divisors of N/M to {fl}"(N/M) and let gE cp` be as in the proof of Lemma 5.2. Then- gE corresponds to (co')/ under Yoshida's correspondence, so we get (5.7)
(ge, e(2v)(Itia)) =
(EE,9E)
W,(yi) 0
(yA
by Lemma 5.2. For any S' C_ S the Petersson product (g, 91M ES' e(p) (wp)))
is the same as the product of g with the image of gI(11pEs' e(p)(Wp)) under the trace operator from modular forms for ro(N) to modular forms for Fo(M), hence equal to fl7,ES' e(p)p-"a9(p)(g,g), since for p { M the map g --> 9111;p composed with the trace just gives the (renormalized) Hecke operator. The same argument applies to (gyp, (11 ES' e(p)(wp)cp)) and
gives the same factor of comparison with (cp, cp) since g and cp have the same (renormalized) Hecke eigenvalues. Thus we have (g', g') (V', 0E)-1 = (g, g) (V,,p)-1, and summing up the identities (5.7) for all the functions e gives the assertion. LEMMA 5.4. - Let, f, 0,111 as in Section 2 be newforms of squarefree levels
Nf, No, N,1, with N = lcm(Nf, No, N p) and with weights ki = 2 + 2µi. Then in 5.5 the summand for M1, M2, M3 is zero unless M3 = 1, N = M1 M2 and M1 I gcd(Nf, No, N,,) hold
Proof. Since by Lemma 5.2 we can express 4,'O as Yoshida-liftings an easy generalization of Lemma 9.1 b) of 121 shows that f is orthogonal to all Obi") (K) for K in gen(Ml, M2, M3) for which Nf does not divide M1 M2 and analogously for 0, Vi. This establishes the vanishing of all summands
for which N M1M2. If there is a p I M1 not dividing gcd(Nf, Nk, N,,) then say p { Nf. The Petersson product off with e(21`1) (K) for K in gen(Mi, M2, 1) is then the
42
S. BOCHERER and R. SCHULZE-PILLOT
same as that of f with the form obtained by applying the trace operator from modular forms on I'°(N) to modular forms on I'°(N/p) to the theta series. But it is easily checked that this trace operator annihilates the theta series of K E gen(Ml, M2, M3) if p I Ml, see [81; the same argument is applied to 0,,o which shows the last part of the assertion. LEMMA 5.5. - Let f , ¢,
be cusp forms of weights k1, k2, k3 for ]P° (N) with
square free N as in Section 2, assume them to be eigenforms of the Hecke operators for p { N and of all the Atkin-Lehner involutions w p with eigenvalues e f (p), e,6 (p), e (p) but not necessarily newforms. Let M1 M2 = N (with Ml as always having an odd number of prime factors) and let R = R(M1 , M2) bean Eichler order in D = D (M1). Let V f, co , cps, be the forms in A(D' , R' , -r,,)
corresponding to f, 0,,o under the correspondence of Lemma 5.2 (with ki = 2 + 21ii for i = 1, ... , 3). Then the summand for Ml, M2, M3 = 1 in 5.5 is
(To(i(v) ®Vm(Yi) ®Wo
)2.
The latter expression is zero unless for all p I N one has p I Ml if and only if ef(p)e0(p)e,1,(p)
= -1.
Proof. The first part of the assertion is an immediate consequence of Lemma 5.2 and the decomposition of T as T (O) ®T(°). For the second part we
Ei
To notice that the expression 1 I 1`"») does not change if an involution wr, is applied to all three functions cp f, cpo, WV, since this only permutes the order of summation. On the other hand each summand is
multiplied with the product of the eigenvalues of cp f, cp,6, cpV, under Op-, which
in view of the relation between the wI, eigenvalues and the wr,-eigenvalues of corresponding functions proves the assertion.
Although the scalar product relation in Lemma 5.2 is not true if one omits the condition that g is an eigenfunction of all the involutions wp, the next Lemma shows that by an amusing newforms argument an only slightly
changed version of Lemma 5.5 (which is based on this scalar product relation) remains true without this condition. LEMMA 5.6. - Let f, 4,, 4, be normalized newforms of levels Nf, NO, NO
o f weights ki = 2 + 2pi ( i = 1, ... , 3) as in Section 2 and let Al, A2, A3 be pairwise disjoint subsets of the set of primes divisors of Nf = N/N f, NO, N'V' respectively such that Al U A2 U A3 is the set of all primes dividing precisely
one of the integers Nf,No,NV,. For rc= 1,...,3let
1 ;: = H.
PEAK.
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
43
Let a decomposition N = M1 M2 as before be given, fix a maximal order R in D = D(M1) and an Eichler order R = R(M1, M2) C R of level M1M2 = N
in D and consider a double coset decomposition DA" = Uh 1DxyiRA and corresponding quadratic lattices (ideals in D) Ii., relative to R as before. For each M2' I M2 letR(M1M2) = R(M1,M2) be the unique Eichler order of level MI M2 contained in R and containing R. Let cP1 E .Aess (DA'w , (R(N f)A , TN,, ))
be the form corresponding to f under Eichler's correspondence and define cP2, c23 analogously with respect to 0, 0. Men T ((f, 6(IL1)) (0, O(µ2)) (v, 6(I13)))
=
2-w(Scd(NI,N0,Np)) (f, f)
(0,
0) (01 0)
(5.8)
X (To CL ei 1
0WA2
i=1
Proof. This is an immediate consequence of Lemmas 5.2 and 5.3: upon inserting the scalar product relations from these lemmata into the left hand side of (5.8) we obtain a sum of terms of the type h
(To((i)A;cP1(yi)
WA3W3(yi)))2
(9
i=1
with arbitrary subsets A' of the sets of primes dividing N/Nf, N/NO, N/N,,) respectively. Let p be a prime dividing two of the levels, say p I Nf, p I No. Then since applying @v,-, to all three of the cp,, only changes the order of summation, the involution w1, for p E A3 may be pulled over to cP2, W3 which are eigenfunctions of wµ. The terms with the set A3 and those with A3 \ {p} give therefore the same contribution. Let now p be a prime dividing only one of the levels, say p I Nf. If p ¢ A2 U A3 then the component of V2 0 c03 in the Tµ, -isotypic component of 0 is an oldform with respect to p, hence orthogonal to the p-essential form cPi. Since To (9'l 0 cp2 ®V3) is proportional
to the scalar product of coi with this component of cP1 0 co such a term gives no contribution; the same argument applies if p E A2 fl A3 holds. If p is in precisely one of A2, A3 then the same argument as in the first case shows that p may be shifted to either one of these sets without changing the contribution of the term. Taking together both cases we find that all terms appearing are of the shape 7'o
(\
uIn, cP1(yi) ®wn 2cP2(yi)
2
i=1
with each term appearing 2"(N/ gcd(NI,N,6,N0)) times, which in view of 5.1 implies the assertion. Collecting all the information obtained we arrive at the main theorem :
44
S. BOCHERER and R. SCHULZE-PILLOT
THEOREM 5.7. - The value of the triple product L function L(f, 0,V), s) at the central critical value s = k'+ 22+k3 - 1 is (5.9)
1 a'25+4a+3b-cn(gcd(Nf,N+,N,,))R.5+9a'+4b
(- )
X(f, f)*0)(4),
(a'+1)1'
214+61214 1(/'2+1)14 1('3+1)1° T1
(T0(Ei=1 '@A 01(yi)
where the notation is as in Lemma 5.6 and To is (as explained in the beginning of this section) the up to scalars unique rational invariant trilinearform on the representation space Uµ0 2 0Uµ0 2 0 Uµ02 (with pi = ki - 2) and takes values in the coefficient fields of f, 0, Eli respectively on the polynomials OA-, cp,,, (yi )
(fork=1,...,3). It should be noted that the rational quantity on the right hand side can be interpreted as the height pairing of a diagonal cycle with itself in the same way as in [ 111. One has just to replace (for rc = 1, ... , 3) the group Pic(X) of [ 11 ] with the group Pic(V,) from (14] obtained by attaching to each yi in the double coset decomposition of D, used above the space of Rt -invariant polynomials in U(°) . Our functions cp,, may then be interpreted as elements
of Pic(VK,). One may then form the tensor product of these three groups and obtain an analogue of the diagonal cycle 0 from [11] by using our Gegenbauer polynomials from above and proceed as in loc. cit. Manuscrit recu le 26 juin 1995
ON THE CENTRAL CRITICAL VALUE OF THE TRIPLE PRODUCT L-FUNCTION
45
References [11 S. BOCHERER. - Uber die Fourier-Jacobientwicklung der Siegelschen Eisensteinreihen II, Mathem. Z. 189 (1989),81-110. 121 S. BOCHERER, R. SCHULZE-PILLOT. - Siegel modular forms and theta
series attached to quaternion algebras, Nagoya Math. J. 121 (1991), 35-96. [3] S. BOCHERER, R. SCHULZE-PILLOT. - Siegel modular forms and theta
series attached to quaternion algebras II, Preprint 1995. [41 S. BOCHERER, R. SCHULZE-PILLOT. - Mellin transforms of vector val-
ued theta series attached to quaternion algebras, Math. Nachr. 169 (1994), 31-57. [51 S. BOCHERER, R. SCHULZE-PILLOT. - Vector valued theta series and
Waldspurger's theorem, Abh. Math. Sem. Hamburg 64 (1994), 211233. [61 S. BOCHERER, T. SATOH,T. YAMAZAKI. - On the pullback ofa differential
operator and its application to vector valued Eisenstein series, Comm. Math. Univ. S. Pauli 41 (1992), 1-22. [71 M. EICHLER. - The basis problem for modular forms and the traces of the Hecke operators, p. 76-151 in Modular functions of one variable I, Lecture Notes Math. 320, Berlin-Heidelberg-New York 1973. [81 J. FUNKE. - Spuroperator and Thetareihen quadratischer Formen, Diplomarbeit Kiiln 1994. [91 P. GARRET=. - Decomposition of Eisenstein series : Rankin triple products, Annals of Math. 125 (1987), 209-235. [101 P. GARRET=, M. HARRIS. - Special values of triple product L-Functions,
Am. J. of Math. 115 (1993), 159-238. 111] B. GROSS, S. KUALA. - Heights and the central critical values of triple product L -functions, Compositio Math. 81 (1992), 143-209.
112] M. Harris, S. Kudla. - The central critical value of a triple product L :function, Annals of Math. 133 (1991), 605-672 [131 K. HASHIMOTO. - On Brandt matrices of Eichler orders, Preprint 1994. 1 141 R. HATCHER. - Heights and L-series, Can. J. math. 62 (1990), 533560. [151 H. HIJIKATA, H. SAITO. - On the representability of modular forms by
theta series, p. 13-21 in Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Tokyo 1973. 1 161 T. IBUKIYAMA. - On differential operators on automorphic forms and invariant pluriharmonic polynomials, Preprint 1990. [17] H. JACQUET, R. LANGLANDS. - Automorphicforms onGL(2), Lect. Notes
46
S. BOCHERER and R SCHULZE-PILLOT
in Math. 114, Berlin-Heidelberg-New York 1970. [18) W. U. - Newforms and functional equations, Math. Ann. 212 (1975), 285-315. [19) P. LITTELMANN. - On spherical double cones, J. of Algebra 166 (1994),
142-157. [20) H. MAAt3. - Siegel's modular forms and Dirichlet series Lect. Notes Math. 216 Berlin, Heidelberg, New York : Springer 1971. [21) T. ORLOFF. - Special values and mixed weight triple products (with an appendix by Don Blasius), Invent. Math. 90 (1987), 169-180. [22) T. SATOH. - Some remarks on triple L -functions, Math. Ann. 276 (1987), 687-698. [23) H. SHIMIZU. - Theta series and automorphic forms on GL2, J. of the Math. Soc. of Japan 24 (1972), 638-683. [24) G. SHIMURA. - The special values of the seta functions associated with cusp forms, Comm.Pure Appl. Math. 29 (1976), 783-804. [25) G. SHIMURA. - The arithmetic of differential operators on symmetric domains, Duke Math. J. 48 (1981), 813-843.
S. Bicherer and R. Schulze-Pillot Fakultat fur Mathematik and Informatik Universitat Mannheim Seminargebaude AS D-68131 MANNHEIM
Number Theory Paris 1993-94
Abelian extensions of complete discrete valuation fields Ivan B. Fesenko
Introduction In 1968 Y. Ihara [Ih] proposed to study class field theory of the padically complete field Q(t)p that is the quotient field of IimZ[t]/p`Z[t]. This field in modern terminology is a two-dimensional local-global field. Ihara considered cyclic extensions of degree p of this field. He suggested that its
class field theory could "explain arithmetically" the map j - 1I3 which associates to each j E IFp`ep, j # 0; 1 (p # 2; 3) the subgroup II; C QP generated by a'/a where (1 - au)(1 - au) is the numerator of the zeta function of E with E running all elliptic curves defined over finite extensions of 1Fp(j) with modulus j.
Later that work of Ihara stimulated two completely different series of works on abelian extensions of complete discrete valuation fields with very general residue field produced by H. Miki ([M21, 1977, also [M1]) and K. Kato ([Kal-Ka7], 1977-1982).
The first direction describes some types of abelian extensions of a complete discrete valuation field with imperfect residue field via study of the group of principal units without using cohomological methods. In the second direction abelian extensions of an n-dimensional local fields are described in terms of the Milnor Kn-groups. The latter theory is based in particular on Galois cohomology groups calculations. Independently A. N. Parshin from different motivations suggested and then developed higher local theory in positive characteristic by using quotients of Milnor K-groups endowed with some proper topology and applied it to a description of abelian
coverings of two-dimensional arithmetic schemes ([P1-P5], 1975-78, 85, 90).
The aim of this work is to sketch the present-day scenery of local class field theories including [Kur], [Kol-Ko2], [Sp), [F1-F5]. I am grateful to Y. Ihara for sending me a copy of [Ih] together with its translation into English.
48
I.B. FESENKO
Parts of the first part of the work were prepared during a seminar on higher class field theory in Max-Planck-Institut fur Mathematik organized by W. Raskind and the author in spring 1994. I am thankful to W. Raskind, Y. Koya and M. Spie13. Special thanks to T. Fimmel for his stimulating questions and discussions. An essential part of this work was written while I was staying in the University of Sydney in August 1994. Its hospitality is gratefully acknowledged. I am grateful to K. F. Lai and other participants of my lectures there for stimulating atmosphere.
Higher local theories First we introduce main objects describing abelian extensions of multidimensional local fields : Milnor K-groups and topological K-groups (1° 6°). We follow [P1-P51, [Ka3-Ka41, [F1,F2,F4]. Then we consider higher local class field theories (7° - 100).
1. - Multidimensional fields Given a two-dimensional smooth projective scheme X over a finite field of characteristic p one can attach to a point x E X and a curve y C X passing smoothly through x the quotient field of the completion (OX,x)y of the localization at y of the completion OX,x of the localization
at x. This is a two-dimensional local field over a finite field (which is itself considered as a 0-dimensional local field). More generally, an ndimensional local field F is a complete discrete valuation field with residue
field being (n - 1)-dimensional. Due to classical structure theorems F is noncanonically isomorphic to k,n((t,,,+ 1)) ... ((tn)) where km is a coefficient
field corresponding to the (n - m)-th residue field of F and either m = 0 or char(k,,,,) = 0, char(k,,,,_1) = p. In the latter case the group of principal units of F with respect to the discrete valuation of rank n - m is divisible, and so is not of interest for class field theory. The field km for m # 0 is called a mixed characteristic field, it is a natural higher analog of a p-adic fields. Lifting prime elements from F and residue fields kn_1, ... , ki to the field F one obtains an ordered system of local parameters tn, ... , t1 (tn is a prime element of F).
If a field M has mixed characteristic and OM is its ring of integers with respect to the discrete structure of rank m and t,,, is a main local parameter, then the quotient field M{ {t}} ofl4imOM[[t]][t-1]/t, OM[[t]][t-1]
is an (m + 1)-dimensional mixed characteristic field. In general a mixed characteristic field is a. finite extension of a field like Q1,{{t1}} ... {{tm_1}} ([FV, sect. 5 Ch. II], for more details see [Zh]).
It occurs that the Milnor K-groups are not the most suitable objects to be related with abelian extensions of an n-dimensional local field F.
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATTON FIELDS
49
It is more convenient to work with quotients
of Milnor K-groups endowed with a special topology (A,,,, (F) is the intersection of all neighbourhoods of zero). Arithmetical homomorphisms from Milnor Kgroups (like a reciprocity map) usually factorize through such quotients.
2. - Topology on the multiplicative group It is natural to expect compatibility of theories of a field and its residue field (lifting of extensions of the residue field as unramified extensions of the field and the border homomorphism in K-theory). Thus, a genuine topology on a multidimensional field has to take into account topologies of its residue fields. Denote by 0 the field in F corresponding to the last finite residue field ko when char(F) = p and the image in F of the ring of Witt vectors of ko corresponding to the homomorphism ko -f k,,,._1 (which is uniquely determined, see [FV, sect. 5 Ch. II]) when char(F) = 0. The ring 0 contains the set of canonical liftings R from ko, so called multiplicative representatives. For a 2-dimensional local field F with local parameters t2, tl define a basis of open neighborhoods of 1 as 1 + t20F + ti0o[[t1]] (e.g. [Ka3]). Then every element a E F* can be expanded as a convergent with respect to the just defined topology product a = t22t1
JJ(1 + i9 ,it2ti)
with 0 54 0, Oi,4 E R, a1, a2 E Z.
In the multidimensional case one can define topology by induction on dimension. Let F be an n-dimensional local field with char(kn_1) = p. Fix a lifting (and thus a set of representatives S) of kn_1 in F compatible with the residue morphism : ko -* 0o, the residues ti E kn_1, 1 < i < n - 1, of local parameters go to t, in F. Given the topology on the additive group kn_1, introduce the following topology on the additive group F. First, an element
a E F is said to be a limit of a sequence of elements a,, E F, v -> +oo, if and only if given any writing a = E2 Ov,tit'y, a = >ti (1t ; with 9* E S. for every set {U2, -oo < i < +oo} of neighbourhoods of zero in kn_1 and every io for almost all v the residue of Bv,ti - Oi belongs to Ut for all i < io.
Second, a subset U in F is called open if and only if for every a E U and every sequence a E F having a as a limit almost all av belong to U. This determines a topology T on F. Then a is a limit of av if and only if the sequence a converges to a with respect to the topology T. By induction on dimension one verifies that limit is uniquely determined, each Cauchy sequence with respect to the topology T converges in F, the limit of the sum of two convergent sequences is the sum of their limits.
50
I.B. FESENKO
If a subgroup _A = {a = Nitn : a E F, Bi E Si C S} is open, then all sets of residues S. are open subgroups in kn_1. Note that the topology T on the additive group is different from that
introduced by Parshin in [P4] for n > 2
:
for example, the set W =
F\ {t2ti c+t2 ati : a, c > 1} in F = lFl,((t1))((t2)) is open in the just defined topology, i.e. for each convergent sequence x, - x E W almost all x, belong to W. If for some open subgroups Ui in the additive group of 1Fr((ti)) such that Ui = 1F1,((tl)) for i > a the group {x = > ait2 : x E F, ai E Ui} were contained in W, then for any positive c such that ti E U_a we would have t2 t1 ° + t-a ti E W, a contradiction. However, a sequence of elements in F coverges to an element of F with respect to T if and only if it converges with respect to the topology introduced by Parshin. The group F is not a topological group for n > 2 with respect to T. For example, if W'+ W' C W, then W' is not open with respect to T.
If char(kn_i) = p, then define the topology T on F* as the product of the induced from F topology on the group of principal units VF = 1 + (tn, ... , ti) OF, the discrete topologies on the cyclic groups generated by ti and the cyclic group of non-zero multiplicative representatives of ko
in F. A general principle on higher dimensions states that there are two essential breaks in objects and methods that are to be involved : from 1 to 2 and from 2 to >2. For a 2-dimensional local field its multiplicative group F* is a topological group and it has a countable basis of open subgroups. In the case of at least three dimensional field both assertions don't hold. For example, if W' . W' C 1 + Wt3 + OLt3 for L = F((t3)), then W' is not open in 1 + OLt3. If char(kn_i) = p, then every element a E F* can be expanded into a convergent product : a = tom,....
H fI(1 + 9i,,,...,iji ... ti' ).
Local parameters tn, ... , t1, multiplicative representatives 0 and principal units 1 + Btn- ... ti' are topological generators of F* : each element a E F* can be expanded into a convergent product of some of them. F* is a semi-topological group and sequential group (the multiplication is sequentially continuous). This topology on the multiplicative group is different from that introduced by Parshin in [P4] and then refered to in [Fl-F4]. It is easy to check that for n > 3 each open subgroup A in F* with respect to the topology introduced in [P4] possesses the property : 1 + t! OF C (1 + tnOF)A. However, the subgroup in 1 + tnOF topologically generated by 1 + Ht;" ... tl' with (in, ... , i1) # (2, 1, ... , 0), in > 1 (i.e. the sequential closure of the
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATION FIELDS
51
subgroup generated by these elements), is open in r and doesn't satisfy the above-mentioned property. If char(F) = char(k,..) = 0, char(k,ii_1) = p, then define the topology T on F* as the product of the trivial (anti-discrete) topology on the divisible part of F*, the discrete topology on the cyclic groups generated by tti with i > m and the just defined topology on k,;,,. For class field theory sequential continuity seems to be more important than continuity. This is a hidden phenomenon in dimension 1 and 2, where continuity is the same as sequential continuity. For other definitions and details see (Ka3), [F1,F2,F4), [MZh].
3. - Topology on K-groups Let A be the finest topology on K,M(F) for which the map (F*)'"' _KM (F) is sequentially continuous with respect to the product of the introduced above topology on F* and for which the subtraction in KM (F) is sequentially continuous. Put K,;EP(F) = KM(F)/Am.(F)
with the quotient topology where Am(F) is the intersection of all neigh-
borhoods of 0 with respect to A (hence is a subgroup). When m = 1, Ki°P(F) = F* algebraically and topologically. Note that for two principal units e, 77 E F* the following holds in KM (F) :
{E,7]}={1-E,1-(1-E-i)(1-7))}+{7),1-(1-E-')(1-77)} with the principal unit 1 - (1 - E-1)(1 - i) of higher order than that of e, 17. This allows one to continue this process in K2 P(F) and finally to write {E, i} as a sum of symbols in the form 1 pi, tti } with principal units pi and local parameters ti. Note also that {8, H'} = {B, c} = 0 for (q - 1)-th roots of unity 0, 0' and a principal unit E. Therefore, every element x of K,,,.(F) can be written as a sum of a fixed number of terms in the form {ai} some
local parameters} with a; E F* plus an element of A, (F). This definition of topological K-groups is somewhat implicit, but it is the most convenient for an initial study of them. Note that if there is a symbolic sequentially continuous homomorphism from the tensor product of m copies of F* to a group G on which the topology is defined by means of a set of subgroups, then it induces a continuous homomorphism from to G.
Later in 5° we will show that A,..(F) coincides with 11>1lKM(F) for m < n and is a divisible group. The structure of topological K-groups of multidimensional local fields is almost completely known (in contrast to the Milnor K-groups). The
52
I.B. FESENKO
simplest way to describe it is to introduce at first Artin-Schreier Witt Parshin, Vostokov and higher tame pairings, and then to apply explicit formulas defining them. The role of the Artin-Schreier-Witt pairings in the one-dimensional case is known from the theory of class formation in characteristic p [KS]. Recall that the Vostokov pairing [V2] has appeared as a multidimensional variant of his explicit formulas [V 1 ] for the Hilbert norm residue symbol in number local fields in case the residue field being of odd
characteristic. A general philosophy due to Shafarevich [Sh] as reflection of similarities between Riemann surfaces and algebraic number fields is to find an explicit formula for the pr-th Hilbert symbol, then forget about class field theory, and using the pairing correctly defined by the explicit formula develop independently and explicitly class field theory for Kummer extensions. For higher local fields both Artin-Schreier-Witt-Parshin and Vostokov pairings are first applied to determine structure of the quotient
group K.°P(F)/pr (in characteristic p for arbitrary r). Then they serve as implements to construct Artin-Schreier-Witt and Kummer extensions which correspond via these pairings to open subgroups of finite index in K OP(F). Coincidence of both pairings with the corresponding pairings induced from class field theory enables one to deduce existence theorem.
4. - Pairings of the topological K-groups Let F be an n-dimensional local field of characteristic p. For al, ... , a,t E F*, and a Witt vector (/30, ... , Qr) E Wr(F) put
(a1,...,a,,,, (/o...... r)]r =Trko/Fn(-Yo...... r) where ko is the last finite residue field of F and the i-th ghost component ^Y(i) of ('Yo, ... , yr) is defined as resko (3(t)ai 1da1 A ... A an 1dan,). This is a sequentially continuous symbolic in the first n coordinates map. Hence it defines the Artin-Schreier-Witt-Parshin pairing :
x Wr(F)/(Frob - 1)Wr(F) -> Wr(IFn) where Frob is the Frobenius map.
Let F be an n-dimensional mixed characteristic local field and let a primitive pr-th root of unity ( be contained in F. Let Xi i ... , XX, be independent indeterminates over the quotient field of Oo. For an element
a=tnn ...ti1)
of F`, with 9 # 0, 9i...... i1 E 1 put a(X) = XXI, ... X119 fJ(1 + O ,,...,i,Xn ... Xi').
53
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATION FIELDS
The formal power series a(X) E Oo((Xi)) ... ((X,,)) depends on the choice of local parameters and the choice of the decomposition of a. Denote z(X) = ((X), s(X) = z(X)P' - 1. Define the action of the operator A on B's and on Xi as raising to the p-th power. For a E F* put I(a) = p-' log a (X)P-'. Now for elements a,,. - -, an+l E F* define (aj,..., a,,+,) as n+1
E(-1)n+1-=1(at) dal n . I
i=1
al
.
.A
d ati-1
ai-1
n
p_l d
a° ao
1
A
.
.
.A
p-1 d a'+ 1 ao
i+1
I.
n+1 /
Let 1Pr denote the cyclic group generated by C. Define a map Vr: (F*)n+1 -4
IL r r
as
Vr(al,...,an+1) = (1,
-y =
Too/zPres4,(a1i...,an+l)/S(X)
This is a very deep result [V21 that Vr doesn't depend on the attaching formal power series to elements of F (for another proof involving syntomic cohomologies of Fontaine-Messing see [Ka8]). The map V, is sequentially continuous and symbolic. It defines the Vostokov pairing :
Kn'(F)lpr x
F*/pr
- µp-
-Let F be an n-dimensional local field and let the cardinality of the group it of all roots of unity of order prime to p be q- 1 (in other words, ko = 1F.). For
an element a E F* and its writing as above put v(j) (a) = aj for 1 < j < n. For elements a1,.. . , an+l of F* define c(al, a2, ... , an+l) as the (q - 1)-th root of unity whose residue is equal to the residue of ail ... ant (-1)b in the last residue field ko, where b = > v(-) (ati)v( ) (aj)biand bj is the determinant of the matrix obtained by omitting the j-th row with the sign (- 1)3-1 from the matrix A = (v(') (aj)), and is the determinant of the matrix obtained by omitting the i-th and j-th columns and s-th line from A. The map c is well defined and is sequentially continuous and symbolic. Therefore, it induces the tame symbol - the pairing
<
Ktor(F)/(q - 1) x F*/(q - 1) -> p. The tame symbol is the composition
K,n'+1(F) a' Kt°P(kn-1)
a' ...
K1(ko) _ IF*
54
I.B. FESENKO
where a are border homomorphism in K-theory, c.f. [FV, Ch. IX].
5. - Structure of Kt°P-groups Note that for two principal units E, q E VF the following holds in K2 (F) :
{E,77}={1-E,1-(1-E-1)(1-i)}+{g,1-(1-E-1)(1-y)} with the principal unit 1 - (1 - E-1)(1- y) of higher order than that of c, rl. This allows one to continue this process in KZ P (F) and finally to write {E, ?]}
as a sum of symbols in the form I pi, ti } with principal units pi and local parameters ti. Note also that {e, 8'} = {8, c} = 0 for (q-1)-th roots of unity 8, 0' and a principal unit E. Therefore, K,t°P(F) is topologically generated by the symbols
(1) {t...... ti b (2) {0, t,z, ... , ti+1, ti-,,. . tive 0, 1 i n,
.
,
tl} with a nonzero multiplicative representa-
(3) 11+0t'-...t' '
ti+1, ti-i, , tl } with a multiplicative representative 0, 1 < i n. Topological relations among these generators (modulo p'' for each r
in the case of char(F) = p, modulo pr in the case char(F) = 0 and a primitive pr-th root of unity belongs to F, modulo p if char(F) = 0 and a primitive p-th root of unity doesn't belong to F) are revealed using the Artin-
Schreier-Witt-Parshin, Vostokov and tame pairings, see [P1,P3,P4], [FlF2]. Simulateneously one verifies that all the pairings are nondegenerate. In particular, K°P (F) is isomorphic to the product of the cyclic group generated by It1, ... , tn}, n copies of the cyclic groups of order q - 1, and the subgroup VK,°P(F) generated by principal units VF. In the case of char(F) = p an extended theorem of Parshin [P4] claims
that there exists an isomorphism and homeomorphism 0 : fj £j -> VK,,°bP(F) with the sequential topology on rj Ej. Here J consists of ji,... , j,,,,_1 and runs all (m -1)-elements subsets of n}, m < n+ 1. Ej is the subgroup of VF generated by 1 + Stn' tl', 0 E 00, with restrictions that ri doesn't divide gcd(i1, ... , in) and the smallest index 1 for which it is prime top doesn't belong to J. This provides an explicit and satisfactory description of the topology on K,1,°P(F) in the positive characteristic case.
Note that in this theorem one should the topology on the multiplicative group defined in 2° instead of that defined in [P4] is used. Let r be the finest topology on the Milnor K, -group such that for every anti) in F*, 1 < j <, m, 1 < i < r, the sum r >, 1 and every \ \ {aitl1 , am }. Denote by T,..(F) Ea(i) , a 1,} converges to i{ l,v+ 1
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUA77ON MELDS
55
KM (F) can be written as a sum of a fixed number of element in the form {ati} some local parameters} with ati E F* plus an element of T,,,.(F). One has f1l,1lKM(F) C T,,,(F), because (1) ll,11KM(F) C VKM(F) where VK,` (F) is the group generated by principal units VF of F, (2) lx can be written as the sum of symbols {a`} {some local parameters} and
an element of T,,,,(F), (3) for any a E VF the sequence
v -> +00,
converges to 1. By induction on dimension one can prove that for m <, n the following holds. (1) Let r >, 1, m > 2, and let U be a neighbourhood of 1 in 1 + tnOF
with respect to the discrete valuation of rank 1. Then for al E VF, a2i ... , a,n E F* there exist elements /3j E VF which sequentially continuously depend on al,. .. , a,n such that {a1i... , an} can be written as E{Qj, tj1, ... , tj,,,_1 } mod prVK,M(F)+{U}KiM_1(F). Here {U}Kirz_1(F) is the subgroup of KM (F) generated by U. (2) The quotient topology on VKM(F)/(pTVKM(F) +\{U}KM_1(F)) of
the product of the topologies on VF, d = (mN_
1
I via the surjec-
tive homomorphism (0j) - E{Qj, t71f.. ., tt7m-1 t} mod ,TVKo(F) + {U}K,1`n'1_1(F) is equivalent to the quotient topology of T.
(3) In characteristic p the topology z is equivalent to A, T,n(F) = A,,, (F)
and the space VKM(F)/T,n(F) with the quotient topology of r is homeomorphic to fi Ej, (3j) -+ >{/j, t1,... , tjm_, } mod T .. (F). (4) T,,,,(F) = ni>_1lI,M(F) and is a divisible group. (5) The homomorphism (VF)d -.. VKX (F)/Tm(F), (/3j) --.. E{Qj, t1, ... , tj,,,-, } mod T,n(F) is surjective and the quotient topology of
the product of the topologies on (VF)' is equivalent to the quotient topology of 7- on KX(F) IT,. (F).
(6) T,n (F) coincides with A,n (F) and is the intersection of all open subgroups of finite index in I (F). The sequence 0 - A,n(F)
Km(F) -
0 splits.
The proof goes as follows. We can assume that char(kn-i) = p. Let a = 1 + Ot mod to 1 OF with i > 0 and let tn-1,...,ti1 n-1
for the residue of 0 and ti and 0 <_ i 1 , .
. .
, in-1 < p'. Then for /3 E VF
tntn-1 ..
t11 } +{1v sv} V
with 7v E 1 +t;i 1OF, fiv E OF sequentially continuously depending on a, /3. Furthermore, putting rl = Ninr tiltntinn-i . . . tll we get that -{1 1
56
I.B. FESENKO
can be written mod pr KM (F) as
{1 -0 - 1)/(1 - 77)" 3} + i{1 - rl(,Q - 1)/(1 - rl), tn} + ... + .. . +ii{1 - 77(,Q - 1)/(1 - 71), ti}.
If a,,3 E VF, ¢ 1 + tnOF, then one can apply the induction assumption for kn_1 and the sequentially continuous lifting from k,,_1 to F. This implies (1). Then (2) follows from definitions. To prove (3) it is sufficient to notice that there is a surjective map cp : flj E.j -> VKM(F)/Tm(F) analogous to Parshin's map ' . Since the composition of cp and the surjective map VK,M(F)/A,,,,(F) coincides with V), T,,,,(F) = A,.. (F). VKX (F)/Tm(F) As in (2) one deduces that cp is a homeomorphism. To deduce (4) one applies (2) and then in characteristic p proves that nupIc (F) + {U}Kin_1(F) = pK,M(F) using injectivity of the differential symbol
dF: I
(K) /p - Q',
dai al
dam am
which is a part of Bloch-Kato-Gabber's theorem [BK].
That implies T,n (F) = n,, i u<; !(F). From (3) it follows that K P (F) doesn't have a nontrivial p-torsion, hence T,,,.(F) is a p-divisible group. If char(F) = 0 and a primitive p-th root of unity belongs to F, then the p-torsion of K,I;?(F)/T,,,,(F) is generated by the p-torsion in F* [F1,F2] (the proof uses Kato's theorem which claims that the homomorphism from KM(F)/p to H"Z(F,Z/p) is an isomorphism). Then T,,,(F) is a divisible group.
For al E VF, a2i ... am E F* there exist elements Qj E VF which sequentially continuously depend on a1, ... , a,,, such that {al,.. . , a,,.} can be written as E10 j, t;, , ... , t;,,,_, } mod T,,,(F). Then, due to the definition of the topology of F*, the preimage in (VF)' of an open set in I< (F) /Tm, (F)
is open there. Since the intersection of all open subgroups in (VF)`' is trivial, (6) follows
from (5) and the observation that a subgroup in I
(F) is open in r if and
only if it is open in A.
Let p be the finest topology on I (F) for which the map from (F*)n to JC (F) is sequentially continuous and the intersection of all neighbourhoods of zero in K,M(F) contains n,>11KM(F) (this topology was used in [F1,F2]). The previous statements imply that p is finer than 'r and the intersection of all neighbourhoods of zero in K;n (F) with respect to p coincides with n,,iiiI'i (F). On the level of subgroups all three topologies A, p and r coincide : a subgroup in 1(!! (F) is open in A if and only if it is open in p if and only if it is open in r.
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATION FIELDS
57
For char(F) = 0 I. B. Zhukov found (applying higher class field theory) a complete algebraic description of 1V°I' (F) in several cases (private communication). In particular, if TPKt°P(F) is the topological closure of
the p-torsion in K,,t,P(F) and F has a local parameter t,, algebraic over Q P, then V I P (F) /TPKQ°P (F) possesses a topological basis of the form {e, tn_1, ... , tI } with a running free Zr-generators of the group of principal
units of the algebraic closure of Q. in F modulo its p-primary torsion. For another approach to topologies on K,M see [Ka6].
6. - Norm It follows easily from 5° that for a cyclic extension L/F of a prime degree K,',"P (L) is generated by L* over the image of Iin°Pl(F). In characteristic p there is a very simple way to define the norm mapping on topological K-groups :
(1) for a cyclic extension L/F of a prime degree introduce NL/F K °I' (L) -, K z°I' (F) as induced by the norm on KI ; (2) for an arbitrary abelian extension L/F define the norm decomposing L/F in cyclic extension of prime degree. Correctness of this definition follows from an application of the ArtinSchreier-Witt-Parshin and tame pairings. The norm on K,, P(L) is dual to the map induced by the fields embedding F -+ L, for details (one should replace K2 by K1 there) see [P4].
For an arbitrary multidimensional local field define the norm on Kt°P(F) as induced from the norm on Milnor K-groups. Compatibility of the just defined norm with induced from the Milnor K-groups follows then from 5°.
Hilbert Satz 90 plays a very significant role in K-theory. For a general type of fields and arbitrary cyclic extension it is still only known for K2. If F is a higher local field, then Hilbert Satz 90 holds for K;01'. The proof uses the description of the torsion in K;,°'' in 5° and small Hilbert Satz 90 if L/F is of a prime degree 1 with a generator or, then the sequence iF/L®(1-0r)
J ttOP(F)/l ® 1Ct0f'(L)/1
NL/F
) K °P(L)/l -
+ K,ti°P(F)/1
is exact, where iFIL is induced by the fields embedding. The latter theorem is verified by explicit calculations in 1f,,,, P/1-groups whose structure is completely known due to the tame and Vostokov pairings (adjoin if necessary a primitive 1.-th root of unity ( and then return without problems as IF(() : Fl is prime to 1,). Similar calculations show that the index of the norm group NL1F1C°P(L) in K."1'(F) is IL : F1 when IL: F1 is prime [F1, F2]. Now we review 4 approaches to higher class field theory: of K. Kato [Ka 1Ka'7], Y. Koya [Kol-Ko2], A. N. Parshin [PI-P51 and the author [F1,F2,F4].
58
I.B. FESENKO
7. - Kato's approach For a field F, K. Kato introduced remarkable groups Hm(F) as follows. (1) Hm(F) = lin H' (F, po('n-1)) for a field of characteristic 0, where µj
is the group of all 1-th roots of unity in FSe ), µis the (m - 1)-th tensor power, 1 > 1 and the homomorphisms of the inductive system are induced by the canonical injections µ®(m-1) divides P;
/twhen l
(2) Hm(F) = limH'n(F,µ®("'-1)) ®UrHP (F) for char(F) = p > 0, where 1I runs all positive integers prime to p, r runs all positive integers.
Here H,';r(F) = Wr(F) 0 (F* 0 .
0 F*) /J, where J is the subgroup
,n-1 times
generated by the following three types of elements a) (Frob(y) - y) 0 NI ®... 0 with y E Wr(F), Ni E F*; b) (0'...,0,)31,0,...,0)(01) 0 /3i 0 ... 00,n-1 with 0 <, i < r ; i times
Nqi
c) y ®01 0 ... 0Q,,,-1 with
= 0, for some i
j.
Equivalently one can put Ha;r (F) to be H'(F, Wr1l' dog) where WrS2F i g is the logarithmic part of the De Rham-Witt complex.
For any field F the group H1(F) is isomorphic to the group of all continuous homomorphisms Gal(F'h/F) -+ Q/Z and H2(F) is isomorphic to Br(F). For an n-dimensional local field F a celebrated theorem of Kato claims that there exists a canonical homomorphism H'+' (F) ^- Q/Z. This is an analog of the classical theorem describing the Brauer group of a local field with finite residue field. The proof of Kato's theorem is easy for the prime to
p part where it follows from typical arguments involving the HochshildSerre spectral sequence. The proof is more difficult for the p part and relies in particular on relations among quotients of Milnor K-groups, Galois cohomology groups and subquotient modules in the module of differentials of fields of positive characteristic. Some ingredients are the theorem of Kato
on the residue symbol KM(F) /p --i Hn(F,Z/p) mentioned in 5° and the study of the cohomological residue Hn+1(F,µ®1'0) --> see (Ka3-Ka4], [R]. In fact, many results established by Kato in [Ka3-Ka4] hold for arbitrary complete discrete valuation fields. Using the canonical pairing Hn(kn_1,µ®(n-1))
H1(F) x KM (F) -> H1(F) x Hn(F) , Hn+1(F) one obtains the higher local reciprocity map WM: KM(F) -+ Gal(Fah/F).
0/71,
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATION FIELDS
59
It describes finite abelian extensions L/F in a sense that %F induces an isomorphism of KM(F)/NL/FKM(L) onto Gal(L/F). Kato has also proved existence theorem which describes norm subgroups [Ka6I. His approach is different from that of 10°.
8. - Koya's approach The previous theory can be treated as a generalization of Tate's approach in classical class field theory. For one-dimensional fields the notion of formation of classes seems to be the most standard way to define the reciprocity map. There is an important obstruction for its generalization to higher-dimensional fields. Already for (> 1)-dimensional fields the Galois descent for K ,',°P fails if L/F is a finite Galois extension, then iF/L: Kt,°1'(F) -> Kn11'(L) induced by F -+ L isn't in general injective, and iF/LK;°P(F) doesn't in general coincide with the Gal(L/F)-invariant elements of Kn'"(L) (the same is true for Milnor K-groups). :
For instance, consider K = Q1,(() with a primitive Irth root of unity
. Let w be a p-primary element in K, i.e. a principal unit such that K1 = K(w) is the unramified extension of degree p over K. Let t be a transcendental element over K and let F = K{{t}}. Consider the totally ramified (with respect to the 2-dimensional structure) extension L = F() of degree p. Put F1 = FK1, L1 = LF1. Then, according to properties of K2 of a local field (see, e.g., [FV, Ch. IX)) for a prime element 7r = 1 - ( of K the symbol iK/K, {w, ir} is a divisible element in K' (KI ), since iK/K, {w, 7r}
belongs to pKM(Ki). Hence iF/F,(iK/F{w,lr}) = 0 in K'2' (FI), but from explicit calculations it follows that iK/F{w,7r} V NL/FK2 P(L). Since
a{, 7r} - {, 7r} is a multiple of {(, 1 - } = 0 for a generator a of L/F, the symbol {, 7r} is a-invariant but it doesn't belong to iF/LK2 P(F). Y. Koya found a class formation approach to higher class field theory using bounded complexes of Galois modules and their modified hypercohomology groups IHI instead of respectively Galois modules and their modified (Tate) cohomology groups [Ko 1-Ko2). His generalized Tate-Nakayama
theorem claims that if for a finite group G and a bounded complex A of G-modules there is an element a E IHI 2 (G, A) such that for every prime 1 and Sylow 1-group G, of G the group 19 (G1, A) is trivial and the group IQ 2 (G1, A) is generated by the symbol rest/G, (a) of order IGl1, then for ev-
ery subgroup H of G and i E Z the cup-product with rest/H(a) induces an isomorphism of the Tate cohomology group Hi-2 (H, Z) onto 19 t (H, A). Koya's generalized axioms of class formation for a profinite group G and a bounded complex A of G-modules are the following : (1) III t(U, A) = 0 for every open subgroup U in G and i = 1;
60
I.B. FESENKO
(2) for every open subgroup U in G there is a canonical isomorphism invU: III 2 (U, A) --> Q/Z
such that invv oresu/v = I U : V linvu for every pair of open subgroups V C U in G. For a 2-dimensional local field F it follows from results of S. Saito [S] and Koya [Ko 1 ] that the shifted Lichtenbaum complex 7G(2)F [2] satisfies the
generalized axioms of class formation. Then for every open normal subgroup U = Gal(FseP/L) of GF = Gal(FSeP/F) the finite group Gal(L/F) and the complex A = T_ 2 has not yet been constructed, one can't extend this approach for higher dimensions directly. Recently M. Spiels ISpI proved that for an n-dimensional local field F the shifted complex Z'(n)F[n], where Z'(n) either is the decomposable part of motivic cohomology G7 ® [-n] studied by B. Kahn in IKnI or is T>oZn(FseP, 2n - .) with Z (FSeP, ) being the Bloch complex [B], satisfies Koya's class formation axioms. Thus there is an isomorphism
KM(F)/NL/FKM(L) -
Gal(L/F)ab
9. - Parshin's approach A. N. Parshin was the first who suggested a K-theoretic approach to higher class field theory. His celebrated theory for multidimensional local fields of characteristic p [P l-P5] is more accessible than that of Kato. In fact,
we need to say only few things in addition to those in 2°-4°, 6° in order to describe the reciprocity map in characteristic p. Recall that for an n-dimensional local field F of characteristic p the torsion subgroup TK1°P(F) in K,°P(F) is isomorphic to (ko)n, ko = IFq (see 5°). Put K(F) = K' P(F)/TIf..t°P(F). Let L/F be a cyclic extension of degree pm with a generator Q. Then the sequence ZL/F _
1-0, -
NL/F
0 , k(F) -> K(L) -> K(L) -f K(F) is exact. The proof is a la [KS] : denote the limit
lim Wr(F)/(Frob - 1)Wr(F)
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATION FIELDS
61
with respect to usual maps (ao,... , a,-1) -> (0, ao, ... , a,._1) by W(F). Then the Artin-Schreier-Witt-Parshin pairing via extended Pontryagin duality [P5, §1,§21 provides a duality between k(F) and W(F). Therefore, the exactness of the above sequence can be easily deduced from from the exactness of the sequence
W(F) -i W(L) -+ W(L)
W(F) -> 0.
Note that the above sequence for k-groups is exact even for Milnor Kgroups of an arbitrary field F of characteristic p due to a deep theorem of 0. T. lzhboldin [Izh].
Now it is straightforward to show that k is class formation in the category of p-extensions of F and in this way the p part of class field theory follows. Nondegeneracy of the tame pairing of 4° and the Kummer
theory provide the prime to p part map KKOP(F) -> Gal(F( °-/)/F), that of the Artin-Schreier-Witt-Parshin pairing (the p part map Kn°P (F) -+ Gal(FabP/F) where Fabp is the maximal abelian p-extension of F). There is the third map which transforms the symbol {t1,... , tom.} for a system of local parameters ta, . . . , t 1 in F to the lifting of the Frobenius automorphism
on F ®rq 1F"P. All three maps are compatible, and their stitching is the reciprocity map
Kn°P(F) - Gal(Fab/F). Thus, the whole construction of the reciprocity map in the Parshin theory is cohomology free. In contrast to the Milnor K-groups used in the previous theories, the group KK°'(F) describing abelian extensions is completely known, see 5°- 6°.
10. - Explicit approach Finally we describe main ideas of the approach to local class field theories in IF1,F2,F4]. This approach is a generalization of two explicit constuctions of the reciprocity map and its inverse one for classical local fields due to M. Hazewinkel [H1-H3] and J. Neukirch [N1-N21, essential ingredients are the tame, Artin-Schreier-Witt-Parshin, Vostokov pairings and topological K-groups. For an n-dimensional local field F denote by vF: Kt°P(F) --> Z the composition
K
Ko(ko)=Z
where a are the border homomorphism in K-theory. An element IIF of K OP(F) which is mapped to 1 is called prime. Its role in higher class field theory is in many respects similar to the role of a prime element of a classical
62
I.B. FESENKO
local field. Given a system of local parameters tl a prime element can be written as {t1, ... , t,,.} + e with e E ker VF. Let F be the maximal purely
unramified extension of F i.e. the unramified extension with respect to n structure corresponding to ko P / k. The Galois group of F/F has a canonical generator - the lifting of the Frobenius automorphism from Gk, which is called by the same name. The inverse map to the reciprocity map can be explicitly described as follows : let L/F be a finite Galois extension, attach to an automorphism a the element NE/FIIE mod NL/FKI P(L) where 1IE is any prime element of Kt,P(E) and E is the fixed field of a lifting of the a on Gal(L/F) such that its restriction on Gal(F/F) is a positive integer power of the Frobenius automorphism. This is a direct generalization of the Neukirch definition in the classical case. The main result is that the map just defined doesn't depend on the choice of lifting of or and the choice of a prime element, and induces an isomorphism of groups YL/F:
Gal(L/F)b
KttOP(F)/NL/FKtn"P(L)
[F1,F2,F4[.
The proof is essentially based on Hilbert Satz 90 and the norm index calculation for extensions of a prime degree in 6°. It is convenient to introduce the second map acting in inverse direction from topological Kgroups to the Galois group as a generalization of the Hazewinkel description of the reciprocity map for classical local fields. However, in complete extent this can be done only in characteristic p. Put K' (F) = lim Kt, P (F') with F' running finite subextensions of F'
ink For the fields of positive characteristic the Galois descent for the topological Ku-groups holds (see, for example, [F21). Given a finite Galois extension
L/F linearly disjoint with F/F denote by V(LF) the subgroup in K, P(L) generated by elements oa - a with or E Gal(L/F), a E VKn,°P(L). Then the sequence
1 -f Gal(L/F) -°-> Kt°P(L)/V(LF)
N-iF)-
I(.` (P) -.0
Is exact where c(U) = OIIL - HL modulo V (L F) doesn't depend on the choice of IIL [F41.
This allows one to define for a finite Galois extension L/F linearly disjoint with F/F a generalization of the Hazewinkel homomorphism 'P L/F : K'"P(F)/NL/FKn°P(L) --> Gal(L/F)ab
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATION FIELDS
63
as follows. Given an element e E ker VF write it as NL/F,y with r) E Kn°P(L).
Then for a lifting cp E Gal(L/F) of the Frobenius automorphism the element cpy - y belongs to the kernel of NL/F and according to the description of this kernel can be written as o11L -1IL modulo V (LI F) with Q E Gal(L/F). The generalized Hazewinkel map attaches the automorphism Q-l ILnF°b to e mod NL/FK;,°P(L). It is a well defined homomorphism. The generalized
Neukirch and Hazewinkel maps are inverse to each other and thus are isomorphisms. The whole theory here is cohomology free similar to the Parshin theory. In the case of characteristic zero all essential problems are concentrated
in p-extensions. There is a class of p-extensions which are very close to extensions in positive characteristic (so-called p-extensions (or ArtinSchreier towers) which are towers of subsequent cyclic extensions of degree p generated at each step by roots of an Artin-Schreier polynomial p(X) - a = XI' - X - a. One can prove that for a cyclic p-extension L/F linearly disjoint with F/F a weak Galois descent holds: the homomorphism VF maps the Gal(L/F)-invariant elements of K,',°P(L) onto IL : FEZ [F4, sect. 31. A generalized Hazewinkel map %PL/F for an arbitrary extension in the case of characteristic zero doesn't exist, see 8°. However, it can be defined by the same rule as in the positive characteristic case above for a finite Galois p-extension L/F linearly disjoint with F/F which is an ArtinSchreier tree (AST) that means that every cyclic intermediate subextension in L/F is a p-extension. Then for an AST-extension L/F the composition WL/F ° YLnFab/F is identity. AST-extensions are "dense" in the class of all p-extensions : for a finite Galois p-extension linearly disjoint with F/F there exists a p-extension Q/F linearly disjoint with F/F such that QnL = F and any intermediate cyclic extension in LQ/Q is an AST-extension. This allows one to prove that YL/F is an isomorphism [F4). For another proof when three assertions : Hilbert Satz 90, IIK' P (F) : NL/FK °P(L) I = IL : F1, YL/F is an isomorphism are verified for a cyclic extension L/F by simultaneous induction on degree, see (F1]. Now for an n-dimensional local field F passing to the projective limit for W L/F when L/F runs all abelian subextensions in Fab/F we obtain the reciprocity map
Kt°P(F) -> Ga1(Fb/F). It is compatible with the reciprocity maps defined in 7°-10°.
The reciprocity map TFP is injective and its image is dense in Gal(Fab/F). The maximal divisible subgroup A1,(F) of KM(F) coincides with the intersection of all open subgroups of finite index in KM (F) by 5°; the latter is the kernel of TM due to existence theorem : the lattice of open
64
I.B. FESENKO
subgroups of finite index in K,,P (F) is in an order reversing bijection with the lattice of the finite abelian extensions L/F, L --> NL,FK,0P(L) IF1,F21. Using the description of the topology on the Milnor K-groups one can
verify that for a finite Galois extension M/F the preimage of an open subgroup of finite index in If 'P(F) is an open subgroup of finite index in K' OP (M) and NMIFK.°P (M) is an open subgroup of finite index in K,t,°P(F). Then it is sufficient to prove existence theorem for a prime index.
The abelian extension attached to an open subgroup is constructed then as corresponding to the annihilator of the open subgroup via the ArtinSchreier-Witt-Parshin, Vostokov and tame pairings (again, if necessarily, adjoining a root of unity and then descending). For another approach to existence theorem see [Ka6]. There are several works on class field theory of a local field with a global residue field in terms of K2-idele groups : [Ka5], [Ko31, [Kuc1. In the next part we describe totally ramified abelian p-extensions of a complete discrete valuation field with arbitrary residue field of characteristic p in terms of the group of principal units.
Totally ramified abelian p-extensions and the group of principal units Let F be a complete discrete valuation field with residue field F of characteristic p. In this part we deal with reciprocity maps describing abelian totally ramified p-extensions of F in terms of subquotients of the group of principal units of F [F3,F51. We indicate then their relations with Miki's [M2] and Kurihara's results [Kur2].
11. - Perfect residue fields
_
Let F be a local field with a perfect residue field F of characteristic p > 0.
Let p(X) denote as above the polynomial XI' - X. Put s = dimFp F/V(F). We will assume that ,c 0, the case is = 0 when the field F is algebraically p-closed may be treated as a limit of class field theories of local fields with nonalgebraically p-closed residue field tending to F.
To describe the maximal abelian extension Fah/F one must study abelian prime to p-extensions and abelian p-extensions. Totally tamely ramified abelian extensions over F are easily described by the Kummer theory, since any such extension L/F is generated by adjoining a root for a suitable prime element it in F and in this case a primitive 1-th root of unity belongs to F. `
The description of the maximal unramified abelian p-extension follows from the Witt theory. Thus, the nontrivial part is study of abelian totally ramified p-extensions of F. A variant of their description in terms of con-
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATION FIELDS
65
stant pro-quasi-algebraic groups as a generalization of geometric Serre's class field theory was furnished by M. Hazewinkel ([H 1-H21). We describe
another approach to abelian totally ramified p-extensions which is cohomology free and of explicit nature [F3[. Let F denote the maximal abelian unramified p-extension of F. The Witt
theory shows that Gal(F/F) ^ Z'. Let L/F be a Galois totally ramified pextension, then Gal(L/F) can be identified with Gal(L/F). Let
Gal(L/F)` = Homzy (Gal(F/F), Gal(L/F)) denote the group of continuous homomorphisms from the profinite group Gal(F/F) which is a 7LI,-module (a a = .a, a E Z7,) to the discrete
Zr-module Gal(L/F). This group is isomorphic (non-canonically) with Gal(L/F)®. Now let L/F be of finite degree. Let X E Gal(L/F)` and E. be the fixed field of all T, E Gal(L/F), where T. I F = cp, T,Q I L = x(ep) and cp runs a topological 7L1,-basis of Gal(F/F). Then EX n F = F, i.e., EX/F is a totally ramified p-extension. Let UF and U1,F be the groups of units and the group of principal units of F respectively. Let 1rX be a prime element of
E. Put TL/F(X) = NE,1FirXNL/F1rL1 modNL/FUL,
where irL is a prime element in L. The group UFINL/FUL is mapped isomorphically onto the group U1,F/NL/FU1,L (multiplicative representatives are mapped to 1). So we obtain the map T L/F : Gal(L n Fab/F)- -> U1,F/NL/FU1,L
which is well defined and is another generalization of the Neukirch map of [N 1-N2[.
Note that there is an analog of the exact sequence in 10° N- -v
1 --> Gal(L/F)ab -> U1 L/V(LjF)
-LIF
+ U1 F, -a 1,
where V(LIF) is generated by ea-1 with e E U1 L. Now define the generalized
Hazewinkel map `WL/F as follows. Let e E U1,F and 0 E Gal(F/F). Write
e = NL/Fii with ri E U1 L. Let cp E Gal(L/F) be a lifting of 0. Then 77-1cp(q) = irLa(irL1) mod V(LIF) for a suitable a E Gal(L/F)ab where rrL is a prime element in L. Set x(q5) = 61 LnFab Then X E (Gal(L n Fab/F))-.
Put TL/F(e) = X. The main result is that TL/F and TL/F are inverse isomorphisms with natural functorial properties [F31. Thus, the quotients
66
I.B. FESENKO
U1,F/NL/FU1,L still as in classical theories describe abelian extensions. However they are roughly rc times larger than the Galois group of L/F. Passing to the projective limit one obtains the reciprocity map
TF : U1,F -f Homzp (Gal(F/F), Gal(Fbp/F)), where U1,F is the group of principal units, Fabp is the maximal abelian p-extension of F. By using extended theory of additive polynomials one can describe for a fixed prime element -7r of F those open subgroups of finite index in U1,F (normic subgroups) which are norm groups NL/FU1,L for finite abelian totally ramified p-extensions L/F such that 7r belongs to NL/FL*. Existence theorem in the perfect residue field case claims that for a fixed prime 7r in F the lattice of abelian extensions L/F such that x E NL/FL* is in order reversing bijection with the lattice of normic subgroups in U1,F [F3].
We note that there is a synthesis of the theories of 10° and 12° a description of totally ramified with respect to n-dimensional structure abelian p-extensions of an n-dimensional local field with last residue field being perfect of characteristic p [F4].
_
12. - General residue field case
Let F be a complete discrete valuation field with arbitrary residue field F which isn't separably p-closed. Denote again by F the maximal unramified abelian p-extension of F, i.e. the unramified extension corresponding to the maximal abelian p-extension Fabp of the residue field F.
Let L/F be a totally ramified finite Galois p-extension. Note that if e = NL/F/0 with 3 E UL, then one can write Q = Bri with 0 E UL, y E U1 L and then e' = NL/Fy E U1,F n N-E1FU1,i is uniquely defined mod NL/FU1,L Thus, the quotient group (UF n NZ/-UL)/NL/FUL is mapped isomorphically onto (U1,F n NL1FU1,L)/NL/FU1,L bye - e.
In the same way as in the perfect residue field case introduce the generalized Neukirch map
TL/F: Gal(L n F'b/F)- -* (U1,F n Ni/FU1 L)/NL/FU1,L. Assume that the residue field of F is not perfect. Denote by .7' = P(F) a complete discrete valuation field which is an extension of F such that Fperf = U;/1FP e(FIF) = 1 and the residue field of T is the perfection of the residue field of F (F isn't uniquely defined). In the same way define
a=P(F).
67
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATION FIELDS
For a E Gal(L/F) put c(a) = irL-1a7rLmodV(LJF), where irL is a prime element in L, and V(LIF) is the subgroup of U1 L generated by the elements e-1a(e) with e E U1,£, or E Gal(L/F), Z = La. Then the sequence
1 - Gal(L/F)ab - U1,i/V(LF)
NL/F NN/FU1,i
analogous to 10° and 11° is exact. Now we introduce a reciprocity map acting in inverse direction with
respect to TL/F. Let e E U1,F n Ni/FUi,i and ¢ E Gal(F/F). Let ri E U1 L be such that NL/Fri = e. Then for an extension p E Gal(L/F) of 0 one can write i)-1tp(ii) c((7-1) for a suitable or E Gal(L/F)ab where lrL is a prime element in L. Set x(c) = or. Then X E (Gal(L n
Fab/F))-. Put 'L/F(e) = X. The generalized Hazewinkel map `AIL/F (U1,F n Ni/FU1,L)/NL/FU1,L -f (Gal(L n Fab/F))- is well defined and a :
homomorphism [F51. The composition `1'L/F o TLnFRb/F is identity.
Put G = L.F. The maps Tn/.F and Tc/.F defined in 11° are compatible with their descendants for L/F : the diagram
Gal(LnFab/F)I
TL/F
(U1,FnNi/FU1 L)/NL/FU1,L TG/T
Gal(Ln.Fab/.F)--------
L/F
t AL/F
--p(Gal(LnFab/F))-
t I
%PL/.F
U1,.F/NL/FU11L
-4(Gal(Gn.Fab/F)))
(AL/F is induced by the inclusion) is commutative. Since XFL/.F is injective, we deduce that AL/F is surjective and leer W L/F = leer AL/F = (U1,F n NL1F,U1 1 n NL/FU1,G)/NL/FU1,L
In other words, TL//F induces the isomorphism (U1,F n NL/FU1 L)/(U1,F n NL/FU1 L n NN/FU1,G) -* (Gal(L n
In contrast to all previous class field theories a new problem comes on to the stage. The objects which describe abelian extensions in this case are not very simple especially because of the term Ni/FU1,i. And for a finite abelian totally ramified p-extension L/F there is no a priori as in other class field theories induction on degree (*)
NM/FU1,M n NE/F,Ul E = NM/F(U1,M n NE/MU1 E)
68
I.B. FESENKO
for every subextension M/F C E/F in L/F . One can show that the property (*) holds if and only if U1,F n Ni/FU1,L n NL/FUI,c = NL/FU1,L and if and only if TL/F and TL/F are isomorphisms [F51. If L/F is a cyclic extension, then the property (*) holds, thus W L/F is an isomorphism [F51.
If the residue field of F is imperfect, one can show that T L/F is an isomorphism in the following cases : (1) L is the compositum of cyclic extensions MM over F, 1 < i < m, such that all the breaks of Gal(MM/F) with respect to the upper numbering are not greater than every break of Gal(MM+1 /F) for all 1 < i < m - 1; (2) Gal(L/F) is the product of cyclic groups of order p and a cyclic group. Merits of the theory just exposed with respect to higher local class field theories are more simple structure of the objects in comparison to K-groups and more independence of a concrete type of the residue field. The main
demerit is that only totally ramified abelian extensions are covered, and not abelian extensions with inseparable residue field extension. Miki in [M2] has shown without explicit introduction of reciprocity maps that for a totally ramified cyclic extension L/F of degree m and for a finite abelian unramified extension E/F of exponent m the group (FnNEL/FUEL)/NL/FUL is canonically isomorphic to the character group of Gal(E/F). In the description of the Galois group of a totally ramified p-extension L/F one can take instead the maximal unramified abelian p-extension F/F any its subextension F/F whose Galois group is a free abelain profinite p-group. Then the group HomzP (Gal(F/F), Gal(L/F)) becomes smaller (what's nice), but NL/FUi,L n U1,F isn't the possible largest subgroup and isn't too far from NL/FU1,L (which is bad). One can consruct in the same way as above the homomorphism
TL/F: Holnz1 (Gal(F/F), Gal(L/F)) -4 (U1,F n NL/FU1,L)/NL/FU1,L
for an abelian totally ramified p-extension L/F. Assume now that the residue field F is a formal power series field of n - 1 indeterminates over a finite field ko. Denote a lifting in F of a system of local parameters of F by ta_1 i ... , t1. Then 7rF, ta_1, ... , t1 form a system of local parameters of F as of an n-dimensional local field over k. Denote by F the maximal abelian unramified p-extension of F corresponding to the 7LP-extension of ko. Let L/F be a finite Galois totally ramified p-extension with respect to the discrete valuation of rank 1. Then the following diagram
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VAWA77ON FIELDS
69
is commutative rL/F
Hom;n (Gal(F/F), Gal(L/F)) -+ (U1 F n Ni/FUi7i)/NL/FU1,L TL/F
Gal(L/F)
(U1,F n NL/FU1,L)/NL/FU1,L
t
YL/ F
Gal(L/F)
K' P(F)/NL1FKn°P(L)
where the homomorphism YL/F is the generalized Neukirch map in 100; the right one is induced by e --> It,, ..., t,,-l, a}.
13. - Some applications The description of the kernel of T L/F for cyclic extensions has numerous applications. First, one can show that for an abelian totally ramified p-extension E/F the norm groups NL/FU1,L are in bijection with subextensions L/F of the extension E/F. A deeper result is that for a complete discrete valuation field F with non-separable-p-closed residue field the norm group NL/FL* is uniquely determined by an abelian totally ramified p-extension L/F [F51: for abelian totally ramified p-extensions L1, L2 over F the equality of their norm groups NL1/FL*
= NL2/FL2 holds if and only if L1 = L2. This generalizes the classical assertion to the most possible extent. In the case of imperfect residue field, one needs additional information in comparison with the perfect residue field case about the structure of
norm subgroups. Existence theorem seems to be very difficult even to formulate. This is natural in view of the description of the norm groups in multidimensional class field theory where one uses all power of topological K-groups. However, for cyclic extensions of the fields with the absolute ramification index 1 there is a satisfactory description of norm groups. For a complete discrete valuation field F of characteristic 0 with residue field F of characteristic p > 2, absolute ramification index 1 and a fixed prime element it introduce the function &,7,: WW(F) --> Ui,F/UI,F
by the formula
((ao, ... , a.-1)) _
fl E(i(' o_
1-i
n
ir)1'= mod Ul F.
70
I.B. FESENKO
Here E(X) = exp(X + Xy/p + XP'lp2 +...) is the Artin-Hasse function, and ati is a lifting of ati E F in the ring of integers of F. Then cyclic totally ramified extensions L/F of degree pn, such that a fixed prime element x of F belongs to NLIFL*, are in one-to-one correspondence with subgroups
£n,,,(VWn(F)Frob(ao,...,an-i))UiF in U1,F, where (ao,... , an-1) is invertible in Wn(F), P = Frob- 1, and Frob is the Frobenius map [F5]. This was first discovered by Kurihara [Kur2] for x = p. He proved that there is an exact sequence
1 -> Hl (F, Z/pn)nr --> H1(F, Z/pn) -> Wn(F) - 1 with nice functorial properties, i.e. there is a canonical connection between
Witt vectors and cyclic p-extensions (in the case of e < p - 1 any cyclic p-extension has separable residue field extension, see (M1]). The approach of Kurihara is based on the study of the sheaf of the etale vanishing cycles on the special fiber of a smooth scheme over the ring of integers of F and of filtrations on Milnor's K-groups of local rings. The class field theories described above seem still to demonstrate "a vivid and lively picture of the great and beautiful edifice of class field theories", at least in the local case.
Manuscrit recu le 7 aout 1995
ABELIAN EXTENSIONS OF COMPLETE DISCRETE VALUATION FIELDS
71
References [B] S. BLOCH. - Algebraic cycles and higher K-theory, Advances in Math.
61, (1986), 267-304. [BK] S. BLOCH, K. KATO. - p-adic etale cohomology, Inst. Hautes Etudes Sci. Publ. Math. 63, (1986), 107-152.
[F1] I.B. FESENKO. - Class field theory of multidimensional local fields of characteristic 0, with the residue field of positive characteristic,
Algebra i Analiz 3 (1991), no 3, 165-196; English transl. in St. Petersburg Math. J. 3 (1992), no 3, 649-678. [F2] I.B. FESENKO. - Multidimensional local class field theory, II Algebra
i Analiz 3 (1991), no 5, 168-190; English transl. in St. Petersburg Math. J. 3 (1992), 1103-1126. [F3) I.B. FESENKO. - Local class field theory : perfect residue field case, Izvestija Russ. Acad. Nauk. Ser. Mat. 57 (1993), no 4, 72-91; English transl. in Russ. Acad. Scienc. Izvest. Math. 43 (1994), 65-81.
[F4) I.B. FESENKO. - Abelian local p-class field theory, Math. Ann. 301 (1995), 561-586. [F51 I.B. FESENKO. - On general local reciprocity maps, to appear. [FV] I.B. FESENKO, S.K. VosroKov. - Local Fields and Their Extensions : A Constructive Approach, AMS, Providence, R.I., 1993. [HI ] M. HAZEwINKEL. - Abelian extensions of local fields, Thesis, Amster-
dam Univ., 1969. [H21 M. HAZEWINKEL. - Corps de classes local, H. Demazure, P. Gabriel, Groupes Algebriques, T. 1, North Holland, Amsterdam, 1970. [H31 M. HAZEWINKEL. - Local class field theory is easy
,
Adv. Math. 18
(1975), 148-181.
[Ih] Y. IHARA. - Problems on some complete p-adic function fields (in Japanese), Kokyuroku of the RIMS Kyoto Univ. 41, (1968), 7-17. [Izh] O.T. IZHBOLDIN. - On the torsion subgroup of Milnor K-groups, Dokl.
Akad. Nauk SSSR 294, (1987), no 1, 30-33; English transl. in Soviet Math. Dokl. 37, (1987).
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K. KATO. - A generalization of local class field theory by using Kgroups, I, Proc. Japan Acad. 53, (1977), 140-143.
[Ka2] K. KATO. - A generalization of local class field theory by using Kgroups, II, Proc. Japan Acad. 54, (1978), 250-255. [Ka3] K. KATO. - A generalization of local class field theory by using Kgroups, I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26, (1979), 303-376. [Ka4] K. KATO. - A generalization of local class field theory by using Kgroups, II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, (1980), 603683. (Ka51 K. KATO. - A generalization of local class field theory by using Kgroups, III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29, (1982), 31-43. [Ka61 K. KATO. - The existence theorem for higher local class field theory, Preprint IHES, 1980. [Ka7] K. KATO. - Galois cohomology of complete discrete valuation fields,
Algebraic K-theory, Part II (Oberwolfach, 1980), Lecture Notes in Math.vol. 967, Springer, Berlin and New York, 1982, 215-238. [Ka8] K. KATo. - The explicit reciprocity law and the cohomology ofFontaine-
Messing, Bull. Soc. Math. France 119, (1991), 397-441. [Kn] B. KAI-IN. - The decomposable part of motivic cohomology and bijectiv-
ity of the norm residue homomorphism, Contemp. Math. 126, (1992), 79-87. [Ko 1 ] Y. KOYA. - A generalization of class formation by using hypercohomol-
ogy, Invent. Math. 101, (1990), 705-715.
[Ko21 Y. KoYA. - A generalization of Tate-Nakayama theorem by using hypercohomology, Proc. Japan Acad., Ser. A 69 (1993), no 3, 53-57. [Ko3] Y. KOYA. - Class field theory of higher semi-global fields, Preprint (1995).
[Kuc1 J. KucERA. - Uber die Brauergruppe von Laurentreihen- and rationalen Funktionenkorpern and deren Dualitat mit K-Gruppen, Dissertation Univ. of Heidelberg (1994). [Kurl] M. KuruI-IARA. - On two types of complete discrete valuation fields, Comp. Math. 63, (1987), 237-257. [Kur2] M. KURIHARA. - Abelian extensions of an absolutely unram fled local field with general residue field, Invent. Math. 93, (1988), 451-480.
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[KS] Y. KAWADA, I. SATAKE. - Class formations, II J. Fac. Sci. Univ. Tokyo
Sect. IA Math.7 (1956), 353-389.
[M1] H. MIKI. - On Zp-extensions of complete p-adic power series fields and function fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21, (1974), 377-393. [M2] H. MIKI. - On unramified abelian extensions of a completefield under a discrete valuation with arbitrary residue field of characteristic p , 0 and its application to wildly ramified 7Gr,-extensions, J. Math. Soc. 29 (1977), no 2, 363-371. [MZh] A.I. MADUNTS, I.B. ZHUKOV. - Multidimensional completefields : topol-
ogy and other basic constructions, Trudy St. Petersburg Mat. Obshchestva 3, (1994) ; English transl. in Proceed. St. Petersburg Math. Society, AMS Translation Series, 2 166, (1995), 1-34. [N1] J. NEUKIRCH. - Neubegrdndung der Klassenkorpertheorie, Math. Z. 186, (1984), 557-574. [N2] J. NEUKIRCH. - Class Field Theory, Springer, Berlin etc., 1986.
IP11 A.N. PARSHIN. - Class fields and algebraic K-theory,Uspekhi Mat.
Nauk 30, (1975), n 1, 253-254; English transl. in Russian Math. Surveys.
[P2] A.N. PARSHIN. - On the arithmetic of two dimensional schemes, I,
Distributions and residues), Izv. Akad. Nauk SSSR Ser. Mat. 40, (1976), no 4, 736-773; English trans]. in Math. USSR-Izv. 10, (1976). [P3] A.N. PARSHIN. - Abelian coverings of arithmetic schemes, Dokl. Akad.
Nauk SSSR 243 (1978), no 4, 855-858; English trans]. in Soviet Math. Dokl. 19, (1978). [P4] A.N. PARSHIN. - Local class field theory, Trudy Mat. Inst. Steklov. 165
(1985), 143-170; English trans]. in Proc. Steklov Inst. Math. 1985, no 3, 157-185. [P5] A.N. PARSHIN. - Galois cohomology and Brauer group of local fields,
Trudy Mat. Inst. Steklov. 183 (1990), 159-169; English transl. in Proc. Steklov Inst. Math. 1991, no 4, 191-201. [R] W. RASKIND. - Abelian class field theory of arithmetic schemes, Proc.
Symp. Pure Math. Amer. Math. Soc., Providence, R.I. 58, I, (1995), 85-187.
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[S] Sh. SAITO. - Arithmetic duality on two-dimensional henselian rings,
Arithmetic duality on two-dimensional henselian rings, Preprint Univ. Tokyo, (1988).
IShI I.R. SHAFAREVICH. - A general reciprocity laws, Mat. Sb. 26 (68) (1950), 113-146; English transl. in Amer. Math. Soc. Transl. (2) 4, (1956), 73-106. [Sp] M. SPIE13. - Class formations and higher dimensional local class field theory, preprint (1995). [V 1 ] S.V. VosTOtcov. - Explicit form of the law of reciprocity, Izv. AN SSSR.
Ser. Mat. 42 (1978), 1288-1231; English transl. in Math. USSR Izv. 13, (1979). [V2] S.V. VosTOxov. - Explicit construction in class field theory of a multidimensional local field, Izv. AN SSSR. Ser. Mat. 49 (1985), no 2, 238-308; English transl. in Math. USSR Izv. 26, (1986).
[Zh] 1. B. ZHUKOV. - Structure theorems for complete fields, Trudy St. Petersburg Mat. Obshchestva 3 (1994) ; English trans]. in Proceed. St. Petersburg Math. Society, AMS Translation Series, 2 66 (1995), 175-192. Ivan B. FESENKO
Department of Mathematics University of Nottingham University Park NG7 2RD Nottingham UK
Number Theoiy Paris 1993-94
Obstructions de Manin transcendantes David Harari
Manin a introduit en 1970 (1111) une obstruction au principe de Hasse
qui a permis d'expliquer tous les contre-exemples explicites connus a cc jour a cc principe. L'idee de cette obstruction (dont on rappellera la definition en 1.3) consiste, quand on s'interesse au principe de Hasse pour une variete X definie sur un corps de nombres k, a faire intervenir le groupe de Brauer Br X de X. Supposons X propre, lisse, et geometriquement integre sur k. Notant
X = X x k k (oil k est une cloture algebri_que fixee de k), on dispose du sous-groupe Br 1 X = ker (Br X --+ Br X) de Br X . Rappelons que Br Y est toujours une extension d'un groupe fini par un groupe divisible (17], corollaire 3.4). Le groupe Br 1X/Br k est isomorphe au groupe de cohomologie galoisienne H' (k, Pic X) cc qui permet souvent en pratique de le calculer et d'obtenir au moyen de ses elements des contre-exemples au principe de Hasse. On trouve en particulier de tels contre-exemples parmi les surfaces fibrees en coniques comme les surfaces de Chatelet (voir par exemple [21 et [61). Ces dernieres sont definies par des equations affines du type y2 - az2 = f (x), ou f est un polynome de degre 4. Les elements transcendants de Br X (c'est-a-dire qui ne sont pas dans Br 1X) sont a priori plus difficiles a representer simplement; les exemples d'obstruction de Manin obtenus jusqu'a present ne les faisaient pas intervenir cc qui aurait pu faire penser que c'est seulement le groupe Br ,X qui est important pour 1'etude de cette obstruction. Le but de cet article est de montrer que cc n'est pas le cas.
Nous donnons en effet (theoreme 1 et proposition 2) un exemple d'obstruction de Manin pour une k-variete X dont le groupe de Brauer 1. algebrique" Br 1X est trivial (c'est-a-dire reduit a Br k). En particulier, l'obstruction est definie a 1'aide d'un element transcendant de Br X. Vers 1976, Colliot-Thelene et Sansuc (dans [41) ont formule une obstruc-
76
D. HARARI
Lion de Manin pour l'approximation faible (propriete arithmetique intimement liee au principe de Hasse). Nous donnons egalement (theoreme 2) un exemple d'une telle obstruction definie au moyen d'un element transcendant du groupe de Brauer. Notons que nos contre exemples s'inspirent des contre-exemples con-
nus sur les surfaces de Chatelet; ils sont obtenus a partir de fibres en coniques au-dessus du plan. On pourra trouver des resultats generaux sur l'obstruction de Manin au principe de Hasse pour les families de varietes
_
dans [9].
D'autre part, le groupe de Brauer geometrique Br T des varietes considerees dans cet article est fini. La question de trouver un contre-exemple au principe de Hasse ou a l'approximation faible s'appuyant sur la partie divisible de Br X reste donc ouverte.
1. - Rappels et notations 1.1. - Symboles de Hilbert Soit F un corps de caracteristique differente de 2 et Br F son groupe de Brauer. Pour tout couple (f, g) d'elements de F*, on note (f, g) le symbole de Hilbert de f et g : c'est un element du sous-groupe 2Br F = H2(F, Z/2) constitue des elements de Br F tues par 2. Il s'obtient en faisant le cup-
produit des classes de f et g dans F*/F*2 = H'(F,Z/2). Le symbole de Hilbert (f, g) est bilineaire en ,f et g ; it est trivial si et seulement si 1'equation x2 fy2 - gz2 = 0 a une solution non triviale (les inconnues x, y, z etant a
-
valeurs dans F). C'est le cas en particulier si f ou g est un carre dans F*, ou encore si g = - f . Quand f n'est pas un carre dans F*, le symbole (f, g) est trivial si et seulement si g est une norme de 1'extension F(V/J)/F.
En particulier, quand F est un corps complet pour une valuation discrete a corps residuel fini ou quand F = R, le groupe 2BrF est isomorphe a Z/2 (1121, 10.7 et 13.3). Lorsque F = Qr, avec p premier ou F = R, on identifiera ce groupe a {1, -1}. On rappelle les regles de calcul suivantes ([ 13], theoreme 1 page 39) :
PROPOSITION 1. - Soit f = pau et g = rev (od sont dans Z et u, v inversibles dans ZI,) deux elements de Q p. Pour tout element inversible x de
Qp, posons [x]r, = 1 si x est un carre modulo p et [x]P = -1 sinon. Pour tout element inversible y de Q2, notons e(y) et w(y) les classes respectives
modulo2de(y-1)/2et(y2-1)/8.Alors: Sip 0 2, on a : (1,9) _ (-1)aaE(P)[uR]P[va]P
OBSTRUCTIONS DE MANIN TRANSCENDANTES
77
Sip= 2, on a : (f,g) = (-1)E"A> Si a et b sont deux elements de R*, on a (a, b) = 1 sauf si a et b sont tour les deux strictement negatifs.
Notons en particulier que si h est un element de Q de valuation strictement plus grande que celle de g, on a (f, g) = (f, g + h) si p # 2 et le meme resultat vaut pour p = 2 a condition de supposer que h/g est de valuation 2-adique au moins 3.
1.2. - Groupe de Brauer non ramifd Soit maintenant k un corps de caracteristique zero et V une k-variete
geometriquement integre de corps des fonctions F = k(V). On note Brnr (F/k) le groupe de Brauer non ramifie de V, c'est-a-dire le sous-groupe de Br (k(V)) constitue des elements dont les residus en tous les anneaux de valuation discrete de corps des fractions k(V) et contenant k sont triviaux. D'apres le theoreme de purete de Grothendieck (18], corollaire 6.2), ce n'est autre que le groupe de Brauer Br W = He (W, G1 ) d'un modele projectif lisse W de V ; c'est un invariant k-birationnel (181, corollaire 7.3).
(Rappelons que quand R est un anneau de valuation discrete dont le corps residuel is est parfait, on dispose du residu Br K --> H1 (n, Q/Z), ou K est le corps des fractions de R. Pour une definition de cette fleche, on pourra se reporter au paragraphe 2 de [81).
Fixons une cloture algebrique k de k et notons k(V) le corps des fonctions de la k-variete V = V xk k. Un element de Br (k(V)) sera dit transcendant s'il n'est pas tue par la fleche Br (k(V)) -* Br (k(V)). Dans le cas contraire, it sera dit algebrique. On emploiera la meme terminologie Br (k(V)) (qui est aussi pour les elements du sous-groupe le sous-groupe Br V quand V est propre et lisse sur k).
1.3. - Obstruction de Mania On fixe desormais un corps de nombres k dont on note Ilk 1'ensemble des places. Pour toute place v de k, on note jv : Br kv - Q/Z l'invariant de la theorie du corps de classes local. Soit W une k-variete geometriquement integre, projective et lisse. On dit que W contredit le principe de Hasse si W a des points dans tous les completes de k mais n'a pas de point rationnel. Quand W (k) 54 0, on dit que W verifie l'approximationfaible si W(k) est dense dans [J W(kv) (muni de vES
la topologie produit des topologies v-adiques) pour tout ensemble fini S de places de k. On dit qu'il y a obstruction de Manin (ou de Brauer-Manin) au
78
D. HARARI
principe de Hasse pour W si pour tout point (Pv) de II W(kv), it existe vEtik
un element A de Br W tel que :
E jv(A(Pv)) # 0 dans Q/Z. vEf k
Il s'agit bien d'une obstruction au principe de Hasse d'apres la loi de reciprocite du corps de classes global. Notons aussi que quand v est reelle ou non archimedienne, l'invariant jv induit l'isomorphisme de Or kv sur Z/2 (lequel s'injecte dans Q/Z). La condition que W est propre (jointe a A E Br W) assure qu'en dehors d'un nombre fins de places v (les places reelles et les places de mauvaise reduction de W ou de A), on a A(PP) = 0 pour tout point P. de W (kv) (151, 3).
On a de meme (quand W (k) # 0) une obstruction de Manin a l'approximation faible definie par la condition qu'il existe un point (Pv) de 11 W (kv) et un element A de Br W tels que E j, (A(Pv)) # 0 dans Q/Z. vEnk
vEOk
Plus precisement, on dit dans ce cas qu'il y a obstruction de Manin a l'approximation faible associee a A. De meme, quand un element A de Br W verifie E jv(A(Pv)) # 0 pour vES2k
tout point (P1),, de fJ W (kv), on dira qu'il y a obstruction de Manin au vEnk
principe de Hasse associee a A pour W. Bien entendu, un element constant (c'est-a-dire provenant de Br k) de Br W ne peut fournir d'obstruction de Martin.
Si A est en outre un element transcendant de Br W, nous parlerons d'obstructions de Manin "transcendantes".
Rappelons que si X est une k-variete lisse et U un ouvert de Zariski non vide de X, alors U(kv) est dense dans X(kv) pour la topologie vadique ([2], lemme 3.1.2). On en deduit que si U est un ouvert lisse d'une
k-variete V et A E Br,,r (k(V)/k) c Br U est tel que E jv(A(PP)) # 0 vEI k
pour tout (P1)v de ff U(kv), alors tout modele projectif lisse de V est un vEQk
contre-exemple au principe de Hasse (cette propriete est independante du modele choisi d'apres le lemme 3.1.1 de [2]). De meme, si pour un certain
element (Pr)v de fi U(kv), on a E jv(A(PP)) # 0, alors tout modele vES2k
vEnk
projectif lisse de V est un contre-exemple a 1'approximation faible. Pour cette derniere propriete, it faut bien noter que si on a seulement A E Br U
OBSTRUCTIONS DE MANIN 7RANSCENDANTES
79
(ce qui n'empeche pas la somme E j,(A(P,)) d'etre finie pour certains vEnk
(P)), on ne peut rien conclure : it est essentiel que A appartienne au groupe de Brauer d'un modele projectif lisse de V.
Par abus, nous dirons donc qu'une k-variete V verifie le principe de Hasse si c'est le cas pour un modele projectif lisse W de V. On parlera de meme d'approximation faible, ou d'obstruction de Manin pour V.
2. - Un exemple d'obstruction de Manin transcendante au principe de Hasse Notation : Soit R = P/Q (avec P et Q polyniimes non nuls premiers entre eux, en n variables a coefficients dans k) une fraction rationnelle. Par abus de langage, nous parlerons de la k-variete algebrique definie dans Ak par 1'equation R = 0 pour designer l'ouvert Q # 0 de l'hypersurface de Ak d'equation P = 0. TiiEOUEME 1. - Soit V la Q-hypersurface de . g2
definie par l'equation :
- g(t)z2 = [.f (x)2 + X7 j [l + g(t)2 - g(t) (f (x)2 + 2 + 2)],
avec : p nombre premier congru a -1 modulo 4; s
P x) = (xs Alors :
1)
etg(t) _ - (t22 +11) - 1.
+
1. L'element A = (g(t), (f (x)2 + 27/p)) de Br Q(V) est un element transcendant de Br,,,. (Q(V)/Q). 2. It y a obstruction de Manin au principe de Hasse associee a A pour V.
Preuve : pour simplifier un peu les notations, posons F(x) = f (x)2 + 27/p et G(x, t) = 1 + g(t)2 - g(t)(f(x)2 + 27/p + 2). Soit U l'ouvert de V defini par g(t)(y2 - g(t)z2)(t2 + 1)(x2 + 1) # 0; on a A E Br U (car g(t) et F(x) ne s'annulent pas sur U) et U est lisse. Remarquons que 1'element B = (g(t), G(x, t)) de Br U est egal a A car le symbole de Hilbert (g(t), y2 - g(t)z2) est trivial. Nous allons prouver les deux lemmes suivants : LEMME 1. - Soit I un nombre premier impair distinct de p, alors U(QI) et pour tout point I-adique Mi de U(Ql), on a A(MI) trivial.
0
80
D. HARARI
LEMME2.-Ona: 1. L'ensemble U(Qp) est non vide et pour tout pointp-adique Mp de U(Q ), on a A(Mp) non trivial. 2. L'ensemble U(Q2) est non vide et pour tout point 2-adique M2 de U(Q2), on a A(M2) triviaL
Preuve du lemme 1 : soft MI = (X, Y, Z, T) dans U(Q1), notons v la valuation l-adique sur Qj. On a A(M1) = (g(T), F(X)) et v(1/p) = 0. Si v(f(X)) < 0, on a (d'apres la proposition 1) A(M1) = (g(T), f(X)2) donc A(M1) est trivial. Supposons donc v(f (X)) > 0. Alors, si v(g(T)) > 0, on a B(M1) = (g(T), 1) et si v(g(T)) < 0, on a B(M1) = (g(T), g(T)2) donc dans ces deux cas A(M1) (qui est gal a B(M1)) est trivial.
Si enfin v(f (X)) > 0 et v(g(T)) = 0, alors la seule possibilite pour que A(M1) = B(M1) soit non trivial est qu'on ait v(F(X)) et v(G(X, T)) impaires et que g(T) ne soit pas un carre modulo 1; mais ceci implique que
F(X) et G(X, T) sont nuls modulo l (puisque F(X) et G(X, T) sont dans Z1) donc en particulier (g(T) - 1)2 nul modulo 1, donc g(T) = 1 modulo l ce qui contredit le fait que g(T) ne soit pas un carre modulo 1. Ainsi, dans tous les cas on a bien A(M) = 0. Pour montrer que U(M1) est non vide, it suffit de trouver T et X dans
Q tels que (g(T), H(X,T)) soit trivial, ou H(X,T) = F(X)G(X,T) car on aura bien alors Y et Z dans Qi tels que y2 - g(T) Z2 = H(X,T). Choisissons deja T de valuation < 0, on obtient g(T) congru a -1 modulo 1. Si 1 ne divise pas 27 + 4p, on choisit X de valuation < 0 ce qui donne v(f (X)) > 0 et done v(F(X)) = 0, puis G(X, T) congru a 4 + 27/p modulo l donc v(G(X, T)) = 0 et finalement v(H(X, T)) = 0 donc (g(T), H(X, T)) trivial. Supposons donc que 1 divise 27 + 4p, alors si l 54 3, on a 1 + 27/p
inversible dans Q, donc en prenant X = 0 (c'est-a-dire f (X) = 1), on obtient v(F(X)) = 0 et G(X, T) congru a 5 + 27/p (donc a 1) modulo 1 et on a encore v(H(X,T)) = 0 puis (g (T), H(X, T)) trivial. Enfin, si 1 = 3 avec 27 + 4p divisible par 3 (snit p congru a I modulo 3), on prend T = 0 ce qui donne g(T) congru a 1 modulo 3 et ainsi g(T) est un carre modulo 3 et on aura automatiquement (g(T), H(X, T)) trivial. Ceci acheve la preuve du lemme 1. Preuve du lemme 2 : soient X et T dans Q p, alors comme -1 West pas un cane modulo 1), on a (en notant v la valuation p-adique) v(f (X)) > 0 et
donc v(F(X)) = -1, tandis que v(g(T)) = 0 et g(T) est egal a -1 modulo p ce qui implique (avec la proposition 1) que (g(T), F(X)) = [-1]I, est non trivial donc pour tout point MI, de U(QI,) on a A(MI,) non trivial.
D'autre part on a bien U(QI,) non vide car si l'on prend T et X quelconques tels que ,f (X) et g(T) soient non nuls, on aura v(f (X)) > 0
OBS7RUC77ONS DE MANIN TRANSCENDANTES
81
et v(g(T)) = 0; ainsi v(F(X)) = -1 et v(G(X,T)) = v(27/p) = -1 donc v(H(X, T)) sera paire et (g(T), H(X, T)) trivial et on pourra bien trouver Y et Z verifiant y2 - g(T)Z2 = H(X,T). D'ofi le premier point. Soit maintenant T dans Q2, on a (en notant maintenant v la valuation 2-adique) v(T2 + 1) < 1 (car -1 n'est pas un carre modulo 4) donc g(T) =
-1 + 25 a, avec a dans Z2. Si v(f (X)) < 3, on a v(2'/p) > v(f (X)2) +3 donc d'apres la proposition 1 on a (g(T), F(X)) = (g(T), f(X)2 ) qui est trivial. Si v(f(X)) > 3, on a G(X,T) congru a 22 modulo 25 dons (g(T), G(X, T)) = (g(T),22) est encore trivial.
Enfin U(Q2) est non vide car en prenant X = T = 0, on trouve (en appliquant la proposition 1) (g(T), H(X, T)) = (-1, 5) qui est bien trivial. Preuve du theoreme 1 : d'apres les lemmes 1 et 2, l'ouvert lisse U a des points Bans tous les completes de Q (l'existence de points reels est evidente). D'autre part, on a A(M1) trivial pour tout nombre premier 1 distinct de p et tout QI-point MI de U donc d'apres le theoreme 2.1.11 de [91, 1'element A de Br U est en fait dans (Q(V)/Q). Or A(MA) est trivial pour tout
point reel M,,. de U (parce que pour tout X de ][8 on a F(X) > 0) et le lemme 2 dit que A(MI,) est non trivial pour tout Q -point Mr de U. Ainsi, pour tout point (Mi,) de fl U(Q.,,), (on note S21'ensemble des places de Q), vES2
j (A(M.,,))
on a
0; de ce fait (d'apres les remarques a la fin de 1.3), it y
1,En
a obstruction de Manin associee a A au principe de Hasse pour tout modele projectif lisse W de V (on a bien A E Br.W puisque A E (Q(V)/Q)). Il
reste enfin a prouver que A est un element transcendant de
(Q(V)/Q). Notons K = Q(x,t) le corps des fonctions rationnelles en deux variables sur Q. La variete V est fibree en coniques (via (x, t)) audessus de AZ-; la conique generique C a pour corps des fonctions K(C) _ Q(V). Comme C est une conique, le noyau de la fleche Br K - Br (K(C)) est engendre par la classe de l'algebre de quaternions associee a C (131, proposition 1.5), c'est-a-dire par 1'element (g(t), F(x)G(x, t)) de Br K. Ainsi, (Q(V)/Q) (c'est-a-dire si si A n'etait pas un element transcendant de (_Q(V))) A s'annulait dans Br (Q(V)))), 1'element (g(t), F(x)) de Br K (dont l'image serait nul ou dans Br (K(C)) est precisement l'image de A dans Br egal a (g(t), F(x)G(x, t)). Ainsi, it nous suffit de prouver que les elements (g(t), F(x)) et (g(t), G(x, t)) de Br K sont non triviaux.
1 Le recours a ce resultat n'est en fait pas indispensable : on peut montrer
que A est non ramifie par des calculs algebriques de residus tout a fait similaires aux calculs arithmetiques du lemme 1.
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D. HARARI
La fonction g(t) West pas un carre dans Q(t), notons L le corps K( g(t)). 11 est clair que F(x) West pas une norme de 1'extension L/K (vu que F(x) ne depend que de x et nest pas un carre dans K) donc on a deja le resultat pour (g(t), F(x)). Posons -y = 27/p+ 2 et u(t) = g(t)/(1 +g(t)2 - -yg(t)); notons egalement P(x, t) = (x2 + 1)2 - u(t). On a G(x, t)/P(x, t) = (1 + g(t)2 - yg(t))/(x2 + 1)2 donc, comme (g(t),1 + g(t)2 - yg(t)) est trivial dans Br K (vu que la fonction 1 + g(t)2 - -yg(t) = (1 + g(t))2 - g(t)(y + 2) est une norme de 1'extension L/K), on a (g (t), G(x, t)) = (g(t), P(x, t)) dans Br K. Notons Z la courbe d'equation P(x, t) = 0, c'est-a-dire (moyennant l'abus de notation du debut du paragraphe 2) la Q-variete define daps le plan affine par 1'equation : (x2 + 1)2 = a(t).
(1)
On peut voir cetteequation comme une equation du quatrieme degre en x a coefficients dans Q(t), ce qui definit un plongement de Q(t) dans le corps
des fonctions M de Z, qui est le corps de rupture de cette equation (il est donc de degre 4 sur Q(t)). Ainsi M est engendre sur 0(t) par -1 - u(t) donc la seule extension quadratique de U(t) que contient M est obtenue en adjoignant une racine carree de u(t) a Q(t) (sinon M serait biquadratique sur Q(t)). Pour prouver que P(x, t) (qui nest pas un carre dans K) West pas une norme de 1'extension L/K, it suffit de prouver que 1'element g(t) de Q(t) West pas un carre dans M car M est le corps residuel du plan affine au point generique de la courbe Z d'equation P(x, t) = 0 (1'idee sous-jacente a tout ce calcul est de montrer que 1'element (g(t), P(x, t)) de Br K a un residu non nul au point generique de la courbe Z). Il s'agit donc de voir que
g(t)/u(t) = 1 + g2(t) - yg(t) nest pas un carre dans Q(t), ce qui resulte aisement de ce que -y nest pas egal a 2 ou -2. Ceci acheve la preuve du theoreme 1.
3. - Un exemple d'obstruction de Manin transcendante a l'approximation faible 11 est plus simple d'obtenir des obstructions a 1'approximation faible (voir la fin du paragraphe 6 de [9] pour une remarque generale a ce sujet);
1'exemple ci-dessous a servi de point de depart pour la construction de 1'exemple du theoreme 1
:
TIIEOREME 2. - Soit V la Q-variete de A4 define par 1'equation :
y2 -tz2 = (x2+ 1)(1+t2 -t(x2+ 1 +2)), P
p
OBSTRUCTIONS DE MANIN TRANSCENDANTES
83
ou. p est un nombre premier impair. Alors :
1. L'element A = (t, x2 + 1/p) de Br (Q(V)) est un element transcendant de Br., (Q(V)/Q). 2. La variete V possede un k-point lisse mais it y a obstruction de Martin d t'approximation faible associee a A pour V.
Preuve : notons encore U l'ouvert t(y2 - tz2)
0 de V. Un calcul
similaire a celui du lemme 1 montre que pour tout nombre premier impair l distinct de p et tout QI-point MI de U, on a A(MI) trivial donc on a deja A E Brnr. (Q(V)/Q) (d'apres le theoreme 2.1.1 de [91; on pourrait aussi avoir recours a un calcul algebrique de residus). La transcendance de A se prouve par le meme argument que dans le theoreme 1, en utilisant le fait que x2 + 1/p et 1 +t2 - (1/p+ 2)t ne sont pas des carres dans Q(x, t), ce qui permet de voir que les elements (t, x2 + 1/p) et (t, 1 + t2 - t(x2 + 1/p + 2)) de Br (Q(x, t)) ne sont pas triviaux (pour le deuxieme, on voit en effet que t n'est pas un carre dans le corps des fonctions de la courbe 1 + t2 - t(x2 +
1/p+2)) =0). Choisissons t dans Z non divisible par p et qui nest pas un cane modulo p, puis x dans Z, alors les valuations p-adiques de x2 + 1/p et 1 + t2 - t(x2 + 1/p + 2) sont toutes deux -1 et on a donc (t, x2 + 11p) non trivial mais par contre le symbole (t, (x2 + 1/p) (1 + t2 - t(x2 + 1/p+ 2)) est trivial ce qui fournit un Q1,-point Mr de U tel que A(MP) soit non trivial.
Enfin, en prenant t = 1, on trouve un point rationnel M de U (c'est evident en faisant le changement de variables y' = y + z et z' = y - z, la fibre en t = 1 etant meme Q-rationnelle) et on a A(M) trivial. On obtient un point de fi U(Q,) en prenant pour M, le point M aux places v vEcI
autres que p et le point MP a la place p et on a bien E jv(A(Mv)) 34 0 d'ou 'En le theoreme 2.
Remarque : on peut en outre voir que dans cet exemple, l'obstruction de Manin a ('approximation faible est la seule pour un modele projectif
lisse X de V (cela signifie que tout point (Pv) de fi
E
verifiant
vEO
0 pour tout a de Br X est dans l'adherence de X (Q) pour la
vED
topologie produit des topologies v-adiques). On obtient ce resultat en fibrant
V (via t) au-dessus de la droite affine et en appliquant le theoreme 4.2.1
de (91, vu que toutes les fibres sont geometriquement integres et que l'obstruction de Manin au principe de Hasse et a l'approximation faible est la seule pour ces fibres (161, theoreme 8.11). Nous verrons du reste au paragraphe suivant que Br X est engendre par A modulo les constantes,
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D. HARARI
ce qui fait qu'un point (Pv) de 11 X(Q) vEcl
jv(A(P,,)) = 0.
et seulement s'il verifie vEil
4. - Complements 4.1. - Calcul du groupe de Brauer Nous allons montrer que dans les deux exemples que nous avons consideres, le groupe de Brauer non ramifie de V est en fait engendre par A (modulo les constantes) ce qui fait qu'aucun element algebrique du groupe de Brauer ne peut induire d'obstruction de Manin. PROPOSITION 2. - Soit V la Q-variete du theoreme 1 (resp. du theoreme 2). (Q(V)/Q) est engendre par A modulo les constantes. Alors, le groupe
Preuve : nous faisons la preuve pour la variete du theoreme 1 (1'autre preuve est analogue) ; gardons les notations de la preuve de ce theoreme et considerons V comme fibree (par t) au-dessus de la droite affine. D'apres la proposition 5.1 de [2] (voir aussi [ 101, proposition 2.1.1), le groupe de Brauer non ramifie de la fibre generique est engendre (modulo les elements
qui viennent de Br (Q(t))) par la classe de A car les polyniimes F(x) et G(x, t) sont irreductibles sur Q(t), ainsi, un element A' de Brn, (Q(V)/Q) (qui doit a fortiori titre non ramifie sur la fibre generique) est egal a A modulo un element Ao de Brnr (Q(V)/Q) qui provient d'un element A de Br (Q(t)). Mais pour tout point m de Al , 1'element A0 doit titre non ramifie au point generique de la fibre en m (notee V,,,,), laquelle est geometriquement integre; de ce fait 1'element A de Br (Q(t)) est non ramifie en m car si K,n,
est le corps des fonctions de V . et k,,, le corps residuel de m E A , la fleche Hl (k,n, Q/Z) -> Hl (K,n, Q/Z) (induite par l'inclusion k,n, C K,n) est injective puisque kn est algebriquement clos dans K,n ; or le residu de Ao au point generique de Vn n'est autre que l'image dans Hl (K,n, Q/Z) du residu de A en m (par fonctorialite du residu, cf [3], corollaire a la proposition 1.1).
On en conclut que A (qui nest ramifie en aucun point de la droite affine) est dans Br Q d'apres la suite exacte de Faddeev (cf [9], preuve du lemme 4.1.1) et A0 est constant. Ainsi Brn,. (Q(V)/Q) est reduit a A modulo les constantes. Ceci acheve la preuve de la proposition 2.
4.2. - Remarques geometriques Pour conclure, disons quelques mots sur la geometrie de la variete du theoreme 2; notons deja que cette variete est un revetement double de qui est ramifiee le long d'une surface quartique.
85
OBSTRUCTIONS DE MANIN TRANSCENDANTES
PROPOSITION 3. - Soit V la Q-hypersurface de A d'equation : y2
- tz2 = (x2 + 1)(1 + t2 - t(x2 + 1 + 2)) p p
(oft p est un nombre premier impair).
_
Alors la uariete V est Q-unirationnelle mais V = V XQ Q nest pas rationnelle. Le groupe de Brauer non ram fie de la Q-uariete V est engendre par ('element A = (t, x2 + 1 /p).
Preuve : posons t = u2, nous obtenons une nouvelle hypersurface V' de A, qui domine V, et qui est rationnelle sur Q : en effet en faisant
les changements de variables y' = y - uz et z' = y + 'z on voit que V' est Q-birationnelle au produit d'une droite et d'une hypersurface de 3 d'equation y'z' = H(x, t) ou H est un polynome non nul. D'autre part, on a vu (theoreme 2) que le groupe de Brauer non ramifie de V etait non nul (il contient 1'element non nul q) ce qui implique la non rationalite de V (a cause de la nullite du groupe de Brauer non ramife d'une extension transcendante pure de 0). Pour montrer que Brnr (Q(V)/Q) est engendre par A, on considere encore V comme fibree par t au-dessus de la droite affine. La fibre generique est une surface deChatelet sur le corps Q(t) (qui est C1 d'apres le theoreme de Tsen donc Br (Q(t)) = 0); comme le polynome
G(x, t) = 1 + t2 - t(x2 + 1/p + 2) est irreductible sur Q(t), le groupe de Brauer non ramifie de cette fibre generique est engendre par (t,x2 + 1/p) (ceci resulte encore de la proposition 5.1 de 121). Comme Brnr (Q(V)/9) en
est un sous-groupe et qu'il contient bien A, le groupe Brnr (Q(V)/Q) est engendre par q (ainsi Brnr (Q(V)/Q) est isomorphe a Z/2). Remarque : la variete V est fibree en coniques au-dessus du plan affine ; ceci est a rapprocher du celebre contre-exemple d'Artin et Mumford (Iii). Le
groupe de Brauer y etait utilise (via la torsion du groupe H3(X, Z)) pour donner un exemple de C-variete X projective et lisse, unirationnelle mais non rationnelle. Darts [3], on trouvera des variations sur cet exemple, avec le point de vue du groupe de Brauer non ramifie (qui apparait dans les travaux de D. Saltman), ainsi que des exemples utilisant des invariants cohomologiques de degre superieur a 2. Manuscrit recu le 22 mars 1994
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BIBLIOGRAPHIE
[1] M. ARTIN, D. MuMFORD. - Some elementary examples of unirational
varieties which are not rational, Proc. Lond. Math. Soc. 25, 75-95 (1972). [2] J.-L. COLLIOT-THELENE, D. CORAY, J.-J. SANsuc. - Descente etprincipe
de Hasse pour certaines varietes rationnelles, J. reine angew. Math. 320, 150-191 (1980). [3] J.-L. COLLIOT-THELENE, M. OJANGUREN. - Varietes unirationnelles non
rationnelles : au-dell de l'exemple d'Artin et Mumford, Invent. Math. 97, 141-158 (1989). [4) J.-L. COLLIOT-THELENE, J.-J. SANsuc. - Trois notes, C.R. Acad. Sci.
Paris 282, 1113-1116 (1976) ; 284, 967-970 (1977) ; 284, 1215-1218 (1977). [5]
J.-L. COLLIOT-THELENE, J.-J. SANsuc. - La descente sur les varietes rationnelles II, Duke Math. J. 54, 375-492 (1987).
[6) J.-L. COLLIOT-THELENE, J.-J. SANsuc, Sir Peter SWINNERTON DYER. -
Intersection of two quadrics and Chatelet surfaces, J. reine angew. Math. 373 (1987) ; 374 (1987). [7] A. GROTHENDIECK. - Le groupe de Brauer, II, dans Dix exposes sur la cohomologie des schemas, Masson-North-Holland, Amsterdam 1968. [8] A. GROTHENDIECK. - Le groupe de Brauer, III : exemples et comple-
ments, dans Dix exposes sur la cohomologie des schemas, MassonNorth-Holland, Amsterdam 1968. [9] D. HARARI. - Methode des fibrations et obstruction de Manin, Duke Math. J. 75, 221-260 (1994). [ 101 D. HARARI. - Principe de Hasse et approximation faible sur certaines
hypersurfaces, a paraitre aux Annales de la Faculte des Sciences de Toulouse. [I I ] Yu. I. MANIN. - Le groupe de Brauer-Grothendieck en geometrie diophantienne, dans Actes du Congres Intern. Math. (Nice 1970), Tome 1, 401-411, Gauthiers-Villars, Paris 1971.
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[ 12] J.-P. SERRE. - Corps locaux, Hermann, Paris 1968. [ 131 J.-P. SERRE. - Cours d'arithmetique, PUF, Paris 1970. David HARARI
(e-mail : [email protected]) E. N. S.
Departement de mathematiques et informatique 45 rue d'Ulm 75005 Paris FRANCE
Number Theory Paris 1993-94
On Selmer Groups of Adjoint Modular Galois Representations Haruzo Hida*
0. - Introduction Let p be an odd prime. Starting from a modular Galois representation cp into GL2(II) for an irreducible component Spec(]) of the spectrum of the universal ordinary Hecke algebra of prime-to-p level N, we study the Selmer group Sel(Ad(cp) 0 v-1)IQ of Greenberg [G] for the adjoint representation of Ad(V) on the trace zero subspace V(Ad(cp)) of M2(l) and the universal
character v unramified outside p deforming the trivial character of cQ = Gal(Q/Q). The Pontryagin dual of Sel(Ad(cp))/Q is basically known to be a torsion II-module of finite type by a result of Flach [F] and Wiles [W] under a suitable assumption on V. The key point of the proof is to show for an
arithmetic height 1 prime P, the subgroup Sel(Ad(cp)) [P] killed by P is finite. Our Selmer group Sel(Ad(cp) ® v-1) is naturally a module over II[[I']]
for r = Im(v) (= Zr). However, it is well known that for the augmentation ideal P of II[[r]], Sel(Ad(cp) ® v-1)[P] has non-trivial II-divisible subgroup, and hence the co-torsionness of Sel(Ad(cp) ® v-1) over II[[F]] does not follow
from the co-torsionness of Sel(Ad(cp))/Q over II. In this paper, under a suitable assumption, we prove a control theorem giving the following exact
sequence: 0 --> Sel(Ad(cp))/Q -> Sel(Ad(W) ®
v-1)r
--> 11
--> 0,
where II* is the Pontryagin dual module of II on which r acts trivially. Actually this assertion is valid for more general 2-dimensional representations cp not necessarily modular (theorems 2.2 and 3.2) and also for Sel(Ad(cp))/F for a general number field F. Although the above exact sequence does not directly yield the co-torsionness of Sel(Ad(cp) 0 v-1), when cp is modular,
we can deduce it from the co-torsionness of Sel(Ad(c ))/q using the fact * The author is partially supported by a grant from NSF
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H. HIDA
that the p-th Hecke operator T(p) is transcendental over 7GI, in the universal ordinary Hecke algebra (see Theorem 3.3). For these, we consider the universal ordinary deformation ring RF of cp restricted to Gal(Q/F) as in [W]. Then we have natural projection irF : RF -4 II, and we can identify Sel(Ad(cp))/F with the module (called a Mazur module) of 1-differentials of RF as in [MT] which gives a tool of proving the above control theorem. To help the reader to understand the formal but subtle argument dealing with various deformation rings, we added to the main text a lengthy Appendix which describes a general theory of controlling deformation rings.
1. - Control of differential modules In this section, we describe how a group action on a ring induces a group action on its differential modules.
1.1. Functoriality of differential modules. We start with a noetherian
integral domain A with quotient field K. Let H be an A-algebra, and A : H -f B be an A-algebra homomorphism. The differential module is then defined by C1 (A; B) = TorH (Im(A), B) =' Ker(A) ®H,x B (Ker(A)/ Ker(A)2) ®H,,\ B.
See [H2] Section 6 and [H3] Section 1 for a general theory of these modules
including above isomorphisms. Suppose that we have two surjective A0 , T --L+ B with,\= p o 0. Anyway, these algebra homomorphisms : H modules are torsion modules over A if B is of finite type as an A-module. Then we recall Theorem 6.6 in [H2] : PROPOSITION 1.1. - Suppose the surjectivity of O and p. Then we have the
following canonical exact sequences of H-modules : Tori (B, Ker(EI,)) -> Cl (0; T) ®T B --> Cl (A; B) --> Ci (µ; B) -' 0 ;
Proof : we have an exact sequence of H-modules :
0 -> Ker(0) --> Ker(A) --L Ker(p) - 0. Tensoring B over H with the above sequence, we obtain the desired result.
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTA77ONS
91
We now suppose that a finite group G acts on H through A-algebra automorphisms. Thus the finite group G acts on Spec(H). We consider the following condition :
Spec(T) is the fixed point subscheme of G in Spec(H) .
(Nt)
Let a be the augmentation ideal of Z[G]. Then the condition (Nt) is equivalent to
(Nt') Ker(O) is generated over H by g(x) - x for x E H and g E G, that is, Ker(O) = HaH.
Let or E G. Then, under (Nt), the action of a - 1 induces an A-linear map : Ker(A) -+ Ker(H). If x, y E Ker(A), then a(xy) - xy = (a(x) - x)(a(y) - y) + x((Y(y) - y) + y(a(x) - x)
(a(x) - x)(a(y) - y) - O mod Ker(A) Ker(b) . Note that Ci(0;T) ®T,f,, B = (Ker(B)/ Ker(A) Ker(B)) ®H,, B. Thus the Alinear map induces a B-linear map [a - 1] : C1 (A; B) - Cl (0; T) OT,,,, B. Under (Nt), a(x) - x for x E H and a E G generates Ker(O) over H. Now
assume that (Sec)
A has a section t : B -> H of A[G] - modules.
Then for each y E H, we can write y = x ® tA(y) for x = y - tA(y) E Ker(A), and hence a(y) - y = a(x) - x E (a - 1) Ker(A). Thus [a -1](x) for or E G and x E Ker(A) generates Cl (B; T) ®T,j, B over B, and ®oEG [a - 1] : ®vEGC1(A; B) -* C1(N; T) ®T,1,. B is surjective.
This shows that the image of Cl (0; T) ®T,µ B in C1 (A; B) is equal to aCi(A; A), and we have COROLLARY 1.1. - Suppose (Nt) and (Sec). Then we have
Ci(Ei;B) =C,(A;B)laCi(A;B) =Ho(G,Ci(A;B)) where a is the augmentation ideal of Z [G] .
We now put ourselves in a bit more general setting where it is not necessarily surjective. We write Bo for Im(lt) and consider the following three algebra homomorphisms :
BO®AB -'B
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where m(a®b) = cab. Since Ker(µ®id)' is a surjective image of Ker(µ)3 ®AB, the natural map : C1 (A; B) ®A B -> C1 (A 0 id; B ®A B) is surjective. When B is flat over A, the map is an isomorphism of B ®A B-modules. Similarly,
the natural maps Cl(A;Bo)®AB-->CI(A®id;Bo ®A B), (Exti)
C1(µ; Bo) ®A B -* C1(µ (D id; Bo ®A B) and
Cl(0;T)®AB->C1(0;T ®A B) are all surjective and are isomorphisms if B is flat over A. By Proposition 1.1, we get an exact sequence, writing B' for Bo ®A B, (Ext2)
TorB' (Ker(m), B) --> C1(µ ®id; B') ®B' B -4 Ci (m
id); B) -+ Ci (m; B) --+ 0 .
We get from the short exact sequence : 0 -> Ker(p) -> T --+ Bo -+ 0, an exact sequence : TorA (Bo, Bo) -> Ker(u) ®A Bo - T OA Bo -+ Bo OA Bo -+ 0, and as a part of it, we know the exactness of the following sequence :
Tori (Bo, Bo) , Ker(µ) OA Bo -> Ker(µ ® id) -+ 0 . Applying ®T' B to the last sequence, writing T' for T ®A B, we have another exact sequence : (Ext2')
TorA (Bo, Bo) ®B' B - (Ker(µ) OA Bo) Or B C1(µ®id;B')®B'B->0
and
(Ker(µ) ®A B0) ®T' B = (Ker(µ)/ Ker(IL)2) OA Bo ®B' B
= ((Ker(µ)/ Ker(µ)2) ®B,, Bo) OA B ®B' B _ (Ker(/i)/ Ker(la)2) ®Ba B = C1(µ; Bo) ®BO B
This combined with (Ext2) shows the exactness of (Ext3)
C1(lz; Bo) ®B,, B -+ Ci (m o (p ®id); B) -+ Ci (m; B) -+ 0
.
If Bo is A-flat, then TorA(BO, Bo) = 0. When B = Bo, the above sequence is nothing but the well known exact sequence for the closed immersion µ
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of X = Spec(Bo) into Y = Spec(T) over S = Spec(A) : Ker(µ)/ Ker(µ)2 -µ`SZY/S -> OX/S --, 0.
2. - Control Theorems of universal ordinary deformation rings We fix a prime p > 3. For a number field X in Q, we write QX = Gal(Q/X) for the absolute Galois group over X. Let ID be a valuation ring finite flat over Z P with residue field F. We consider a p-ordinary deformation problem V = DX defined on the category CNL j) of complete noetherian local
ID-algebras with residue field F. Morphisms of CNLL are assumed to be local ID-algebra homomorphisms. See [T] and Appendix for a general theory of such deformation problems. Let (Rx, px) be the universal couple of the deformation problem DX of representations of 9X. We study how the Galois action controls R.X.
2.1. Deformation problems. Let p be a continuous representation of cE into GL2(F) for a number field E. We consider the following condition for an algebraic extension F/E : (AIF)
p restricted to 9F is absolutely irreducible.
We assume (AIF). For each prime ideal C, we write FF for the C-adic completion of F and GF1 for the absolute Galois group over Ft. Let C be an integral
ideal of E prime to p and write CIE(Cp) for the strict ray class group of E modulo Cp. We also pick a character x : CIE(Cp) --* IJ" such that the order of x is prime to p. We write C(x) for the conductor of x and assume that CI C(x). By class field theory, we may regard x as a character of QE. We write xq for the restriction of x to the decomposition subgroup GEq at each prime q. Let M be a finite set of primes outside p. We assume that M contains all prime factors of C(x) outside p. We write M (X) for the set of primes in M dividing C(x) and put M' = M - M(x). We write E for the union of M and the set of all prime factors of p. Then for q E E, write I. (resp. Nq) for the inertia subgroup of cEq (resp. the p-adic cyclotomic character H restricted to cEq ). Here we normalize cyclotomic characters so that they take the geometric Frobenius at each unramified prime ideal ( to the norm of the ideal 1. Under this convention, we consider a deformation problem of p on CNLL. A deformation p : CJE -+ GL2(A) of p is called of type D = DE if p satisfies the following five conditions (UNR), (x,), (Reg,,) for each prime pip, (xq) for each prime q E A4 (X) and (.Nq) for each prime
qEM': (UNR)
7r is unramified outside E ;
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We have an exact sequence Of 9Ep -modules: 0 --> V(pl;p) - V(p) --> V(P2,p) --a 0 with V(p2,p) A-free of rank 1, p1,p unramified and p2,p mod m = xp mod m on the inertia subgroup Ip. (xp)
Writing pti for pti mod m, we assume (Regp)
Pi,p # P2,p
We assume the following conditions for q E M : (Xq) As Iq - modules, V(p) = V(id) ® V(Xq) with V(Xq) A-free of rank 1 for gIC(x) , V\(q) For q E ,/Vl, we have an exact sequence, of cEq-modules, non-split over Iq . 0 -> V(Pi,q) -* V(P) -> V(P2,q) - 0
where V(p2,q) is A-free of rank 1, pti,q (i = 1, 2) is unramified and P1,gP2,q = Nq.
Since the order of x is prime to p, Xq for q E M(x) is non-trivial. To make our deformation problem DE non-empty, we assume that p satisfies the above five conditions. The contragredient of this deformation problem
is studied in [W] and is denoted by D = (Ord, E, iJ, M) (for E = Q) there. As shown in [W], the problem D is representable, and hence DE is also representable. See [T) and Appendix for the proof in more general case. To apply the argument in [T] and Appendix to our situation here, we note the following facts : for the maximal extension FE of F unramified outside E. the Galois group G = Gal(FE/E) satisfies the condition (pF) in Appendix; any deformation of type DE factors through G; the group D E S
(resp. its subgroup I) is given by a choice of decomposition subgroups (resp. its inertia subgroup) at each p', and the condition (Regp) is the same as (RGD) in Section A.2.2 for the decomposition subgroup D of p in G. We write (RE, PE) for the universal couple for DE. Thus for each deformation p !;E -> GL2 (A) of type DE, there exists a unique local 0-algebra homomorphism cp : RE -> A such that p is strictly equivalent to cppE. Here we say p is strictly equivalent to p' if p(r) = xp'(r)x-1 for x E GL2 (A) = 1 + mAM2 (A) independent of T. We write p -- p' if p is :
strictly equivalent to p'. Let F be a finite extension of E. Write PF (resp. xF) for the restriction of p (resp. X) to cF. We consider the deformation problem DF of pF on CNL,
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given as follows. Let MF(X) (resp. MF) be the set of prime ideals dividing C(XF) and prime to p (resp. the set of primes dividing primes in M'). We write EF for the union of MF(X), M'F and the set of all prime factors ofp in F. A deformation p of pF is a continuous representation p : CcF -+ GL2 (A) with p mod MA = pF for an object A in CNLO. A deformation p of pF is of type VF if p satisfies the following five conditions (UNRF), (xP,F), (Regp,F)
for each prime P of F dividing p, (xQ,F) for Q E MF(X) and (NQ) for
QE.MF (UNRF) p is unramified outside EF ; (xp,I We have an exact sequence of GFP modules for each prime ideal Pp: 0 -* V(pl,p) - V(p) ---> V(p2,P) ---a 0 with V(p2,P) A-free of rank 1, pi,p unramified and p2,P mod m = xp mod m on Ip ; (RegP,F) Pi,p # 72,P for each prime ideal PIp.
where pt,P = pi,,P mod m. Writing XQ for the restriction of x to cFQ, we assume : (xQ,F) As IQ-modules, V(p) = V(id) ® V(XQ) with A-free of rank 1 for Q E MF, (NQ) For Q E MF we have an exact sequence, of 9F. -modules, non-split over IQ,
0-'V(P1,Q)-'V(P)-'V(P2,Q)->0 where V(p2,Q) is A-free of rank 1, pi,Q (i = 1,2) is unramified and P1,QP2,Q = A1Q.
Then DF is representable under (AIF). We write H = Gal(FE/F). Then H is a normal subgroup of G with G/H = A = Gal(F/E). We take SG = {D}p1v and SH = {Dp}pl1, for the decomposition subgroups D? for each prime "?". Then to the quadruple (G, H, Sc, SH) the theory described in Appendix (Sections A.2.1-3) applies, and it is easy to deduce the
representability from the argument in Section A.2.3. Hereafter, assuming (AIF), we write (RF, pF) for the universal couple representing the problem DF.
2.2. Controlling universal deformation rings and Mazur modules. We now suppose that F/E is cyclic of degree d. Since PE restricted to H = Gal(FE/F) is a solution of the deformation problem DF, we have a non-trivial algebra homomorphism a : RF -> RE such that apF PE-
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H. HIDA
If p is a deformation of type DE, then we have a unique cp : RE -> A cpapF. Let us write fF (resp. such that PPE -_ p. Then pI H ~ (PPE I H 21 for the deformation functor of the problem DF (resp. DE). Then the Galois group A = Gal(F/E) naturally acts on .FF as follows : taking o E G and define p°(g) = p(ugor-i) for p E YF(A) and g E H. We take c(a) E GL2(0) such that c(o) = p(c) mod m,, and define pt°1 = c(a)_'p°c(o) E .FF(A). The strict equivalence class of p[°i is well defined depending only on the class of or in A, and in this way A acts on FF and RF through 0-algebra automorphisms. We define a new functor .FFr by .Fr(A) = p}. Since cpa is the unique homomorphism bringing l p E .FF (A) pt°] PF down to PIH, the deformation subfunctor : FE,F(A) = {PI H E .FF(A) I p E.F(B) for a flat A-algebra B in CNLD}
is representable by (Im(a), aPF) under (AIF), as long as apF can be extended to an element of.F(B) for an algebra B flat over Im(a) in CNLD. The argument proving this is the same as the proof of Theorem A.2.3. To check the extendibility of apF, we introduce the following assumptions. Let FO be the maximal subfield of F such that d' = [Fo : E] is prime to p. Let S be the set of primes of E ramifying in F. For each prime q of E, we write Iq (resp. I(q)) for the inertia group of q in G (resp. A). We also write D(q) for the decomposition subgroup of A at q. We assume for q outside p (TRq)
II(q) is prime to p
.
For pip, we assume either (TR.p) or (Exp)
Every character of I(p) n Gal(F/Fo) with values in A" can be extended to a character of A having values in B" for a flat extension B of A such that it is unramified outside p .
These conditions correspond the conditions (TRD) and (ExD) in Section A.2.3. If I(p) n I(q) n Gal(F/Fo) = {1} for any two primes p dividing p and an arbitrary q, then (Exp) is satisfied. In particular, if F is a subfield of Qom, (Exp) is satisfied, where Q is the unique Zp-extension of Q. Take 7r E .FF°. (A). By Corollary A. 1.2 combined with the argument in A.2. 1, there exists a faithfully flat A-algebra B in CNLD such that 7r extends to a representation irE G -> GL2(B) with 7rE - p mod mo. By (TRq),
the unramifiedness at q ] p in S - M of p implies the unramifiedness of irE, where S is the set of primes ramifying in F/E. We now look at the restriction of IrE to the inertia group I = Iq for primes q in M or dividing
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p. By (TRq) and the fact that the decomposition group at q acts through conjugation on the maximal tame quotient of the inertia subgroup by the cyclotomic character N, the conditions (XQ) (resp. (NQ)) for ;5 implies (xq) (resp. (.N)) for irE at q = Q fl E. We look at the restriction of 7rE to Ip. Since the characteristic polynomial of p(a) for an element a of Ip has two distinct
roots in IF by (Regp) and (xp), that of irE(a) again has two distinct roots a and b in B by Hensel's lemma. Then writing V for Ker(TrE(a) - a id), V (7rE) /V is B-free of rank 1, and on V, Dp acts by a character 1]p. Replacing
a by b if necessary, we may assume that 1Jp is trivial on I(p). If p satisfies (TRp), the argument is the same as q above. Now suppose (Exp). Since rIp is trivial on I(p), 77p factors through the p-primary quotient of I(p). Thus by (Exp), we can lift rip to a p-power order character 1; of A unramified outside p. Replacing 7rE by FE 01 -1, we may assume that 7rE satisfies (xp) because Z; - 1 mod m. Thus rrE is a deformation of type VE, and we get PROPOSITION 2.1. -Assume (AIF), (TRq) for q outside p and one of (Exp)
or (TRp) for pip. Then it E FF (A) can be extended to an element rrE of .F(B) for a faithfully flat A-algebra B in CNL0. Moreover the functor FE,F is represented by (Irn(a), apF).
For each integral ideal C of a number field X, we write ClX (Cpe) for the strict ray class group modulo Cpe. We allow e = oc, and ClX(Cp°°) = 1l im C1X (Cpe). Then by class field theory, there exists an abelian extension e
X,,,/X unramified outside Cp such that Gal(X,,./X) = Clx(Cp°°). We consider the character det(pF) : GF --> R. By (xQ), the restriction of this character to the inertia subgroup IQ factors through a finite quotient. Thus there exists an integral ideal C prime to p of F such that det(pF) factors through ZF = C1F(Cp°°), and hence there exists an algebra homomorphism : i7[[ZF]] -> RF taking z E ZF to det(pE(z)). We take C maximal among the ideals satisfying the above condition. We write AF for the image of 0[[ZF]] in RF which is an object in CNLO.
We now modify the deformation problem DE on CNLD and create a new one VA defined over the category CNLA of complete noetherian local AE-algebras with residue field F by adding the following condition to the conditions of VE : (det)
det(p) for each deformation p : CJE -+ GL2(A) of type DE
coincides with det(pE) composed with the inclusion i : AE -p A.
For any deformation p : QE -+ GL2(A) of type DA, it is automatically a deformation of type BE. Thus we have a unique 0-algebra homomorphism
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: RE - A such that OpE p and q(det(pE)) = det(p), which implies that is actually a AE-algebra homomorphism. Therefore (RE, PE) represents the new problem VA. We consider another deformation functor FA,E,F defined on CNLA by FA,E,F(A) = {PIH E .FA,F(A)Ip E 1'A(B) for a flat A-algebra B in CNLA} .
By Lemma A.2.1 in Appendix, actually .EA,E,F(A) = {pI H E .FA,F(A)IP E .FA(A)}. By the argument proving Theorem A.2. 1, we can conclude that this functor is represented by (Im(a)AE, PEI H), where we write Im(a)AE
for the image of Im(a)®oA in RE. Here is the argument : let p and p' be two deformations of P over E of type DA with values in GL2 (A). Suppose that p p' on H. Under (AIF), p = p'® i;' for some character of Gal(F/E) (see Corollary A.2. 1). Since in our deformation problem, the determinant is fixed, we have S2 = 1. Since p = p ®Z; and is quadratic, if # 1, Z; mod m is non-trivial, because p is odd. By [DHI] Proposition 4.1, p is an induced representation of a character of Ker(b), which violates (AIF). Thus is trivial. The algebra Im(a) may not be a A-algebra for A = AE. We thus find
HomA_aig(Ir(a)AE, A) = {iIH I ir E F(A), det(ir) = det(pE)}/ .F'A(A) = HomA_alg(RE, A) for A-algebras A.
Thus under (AIF), we have Iin(a)AE = RE. THEOREM 2.1. - Assume (AIF) and one of (Exp) or (TRp) for pI p, (TRq)
for q outside p. Then we have Im(a) AE = RE. If a prime factor p of p rams totally in the maximal p-extension of E in F, then Im(a) = RE.
Proof : we only need to prove the last assertion. Here we give a short argument restricting ourselves to our special case. See Appendix for the treatment in more general cases. We argue similarly as above by replacing the deformation problem DA by DE. Thus we pick two deformations p and p' of p over E of type DE with values in GL2(A). Suppose that p p' on H. Under (AIF), p = p' ® for some character of Gal(F/E). If . # 1, by our assumptions and (xp), pl,p = p'2,p and P2,p = pi,p. Since t; mod m is trivial (because 1; is of p-power order), this contradicts to (Regp). Thus 1; = 1 by the total ramification of p, and we find
A) = .TE,F(A) D Homo_alg(RE, A) for ID-algebras A,
which shows the result, because Im(a) C RE.
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Let a be the ideal of RF generated by [Q] (.x) - x for all x E RF and a generator a E A. Then it = PF mod a : H --> GL2 (A) for A = RF/a satisfies 7rEa] ti it and hence, by proposition 2.1, 7r extends to a representation irE in F(B) for a faithfully flat A-algebra B in CNLID. On the other hand, we
have the Galois representation PE attached to RE with a o pF .:; pE on H. By the universality of (RE, PE), we have an ID-algebra homomorphism 0 : RE - R such that 0 o PE zz iE We conclude from OapF .^s OPE ^ 7rE that Ba coincides with the inclusion map of A into B. Thus a is injective on A = R/a. This shows (Nt) in 1. 1 for (RF, Im(a)) in place of (H, T) there. See Theorem A.2.3 for a more general result of this type.
By definition, the ideal C defining AF is invariant under A, because p is A-invariant. Thus A naturally acts on AF. Identifying ZF = Gal(F... /F), we have the restriction map res : ZF -> ZE (= Gal(E,,,,/E)). This induces a A-equivariant algebra homomorphism res : AF --f AE. We now take a closed ID-subalgebra AF in RF which is stable under the action of A. We can take AF to be AF, but some other choice is also possible. The map a : AF , a(AF) coincides with res on the image of AF and is equivariant under the Galois action. Let II be a normal integral AF-algebra which is a member of CNLD. Let it : RE -+ II be a AF-algebra homomorphism. We put AF = m(7ra (DA' id) : RF®A' II - II and ILF = Tn(7r OA' id) : RE®A' !
II
for the multiplication in : II®AF,1 -4 II, where 1o = Im(7r). Here "®" indicates
the completion of the algebraic tensor product under the adic topology of the maximal ideal of the algebraic tensor product. Since the condition (Nt) is insensitive to tensor product, Ira 0 id : RF®A' II -> RE®aFII again satisfies (Nt) if a is surjective. Note that RF®AFII is an II- algebra and hence AF has a trivial section of II-modules. Thus we get from Corollary 1.1. THEOREM 2.2. - Assume (AIF) and one of (Ex);, or (TRp) for pip, (TRq) for q outside p. Let II be a normal integral AF-algebra in CNL,D and 7r : RE --> II be a AF-algebra homomorphism which is a morphism in CNL. Then we have,
for A = Gal(F/E), (i) Spec(Im(a)) is the maximal subscheme of Spec(RF) fixed under the action of A ; (ii) If a is suijective, then Cl (j4,F,; II) = C1(AF,; II)/(a - 1)C1(AF; II),
where a is a generator of A. Suppose that a is surjective and II is a AF.-module of finite type. We now take a subalgebra AE of RE containing a(AF). Note here that C1(it ; II) can
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,
be different from C1(/>,E; II) for FiE = m(T ®AE id) : RE ®A I --> II. We now
compute the difference using the following diagram : 0
A®®a(A') II
AE ®A,II --III
id ®A
4
RE ®a(AF) II
RE ®AE 1-41.
We have an exact sequence, writing T = RE ®AE I, Tor(II, Ker(!4E)) -' CI (id ®0; T) ®T II ---> C1(A'; II) -- C1(04 ; II) -4 0I
By the diagram, since the multiplication map has a section of AE-modules, we have C1 (id ®0; T) = C1 (0; AE ®A, II) ®AE RE and Cl (0; AE ®AE II) = Cl (id ®0; T) ®T II
.
Since 0 is a scalar extension to I of the multiplication map : AE ®a(A) AE -' OA' 1, and we have an AE. Thus we see that C1(0; AE ®AE II) QA' exact sequence : (Ext4) Tort (II, Ker(ii )) -' SAE/a(AF) ®AE II Cl (A' ; II) /(a - 1)C1(AF) -II) - C1(IL' II) -> 0.
Then as seen in [H3) Lemma 1.11, if AF = i7[[WF]] and AE = t7[[WF]] for a p-profinite subgroup WX of ZX, we have SZAE/AF = AE ®Z WE/ res(WF), and hence Ci (id ®0; T) ®T I = I ®Z WE/ res(WF). We write AX and AX for
A' and µ'' when AX = Ax. Assume that AF contains AF. By a similar argument using the following diagram with exact rows : 0
nF®AFII
AF®AFII -*II
00 RF®AFIIid_--' RF®AFII-*II,
we get the following exact sequence : (Ext'4)
S1AF/AF ®AF II - Cl (AF; II) -* Ci (AF; II) --4 0 .
Here II may not be a AF-module of finite type, but we assume it is AFmodule of finite type. The exactness of the above sequence follows from the compactness of these modules and [EGA] IV, 0.20.7.18.
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ON SELMER GROUPS OF AAIOINT MODULAR GALOIS REPRESENTATIONS
3. - Selmer groups Keeping notation introduced in the previous section, under a suitable assumption, we deduce the control theorem for the Selmer group Sel(Ad(cp) ® v-')IQ from the control theorem of the deformation rings (The-
orem 2.2). Then assuming the transcendence of coi,p(gr) over Zp for the geometric Frobenius 4n at p and the co-torsionness of Sel(Ad(cp))/Q over 1, we prove the co-torsionness of Sel(Ad(cp) 0 v-')IQ over II[[r]].
3.1. Selmer groups. Here we suppose that E = Q; so, instead of writing primes in E as q, we use the Roman character q for that. Let II be an integral normal local domain complete under mp-adic topology for the maximal ideal mo. We assume that II is an algebra over 0 and II/mg = F. Let cp : 9Q --f GL,, (1) be a continuous Galois representation. Thus E = Q in the notation of the previous section. Let V (cp) = IIn be the representation space of cp and W(W) be a subspace of V (W) stable under 9(QP . We define two Galois modules V(cp)* = V ®p 1* and W(W)* = W(cp) 011*, where II* is the Pontryagin dual module of 1. Let be the unique Zr-extension. We identify Gal(Q /Q) with r = 1 + pZ . by the cyclotomic character N. Here N satisfies N(Oq) = q for the geometric Frobenius Oq at q. We write v = v,,, for the inclusion of Gal(Q /Q) into ID[[]P]]. Let F,,,, be the subgroup of r of index p and write Q,n for the fixed field of rm. We write the projection 7rm : ID [[r]] - ID [[r/r,,L]] and put v,n = 7r,n o v. We put for m = 1, 2, ... , oo , V (7o 0 v,1L1)
((p)®,L) ("m' ), W (W ® v1/L')
W(W)
V (vml)
,
where V (v,-,,') = 0[17/17,n] for finite m.
LEMMA 3.1. - We have the following isomorphisms of II[[r]]-modules for
m = 1,2,...,(X,
H1V(p)*) -
H1(9Q,V(7' ® v1IL1)*) ,
H'(II, n cQ,,,, V ((p)*/W (cc)*) - H1(I,, V (cp 0 vml)*/W (cp 0 vml)*) ,
II7Er/r,H1('YIq'Y-1,V(cc,)*/W(cP)*) - H1(II,V(co ® vLl)*/W(cP ® v;')*) forq p, where Iq is the inertia subgroup at q of cQ, II[[r]] acts through coeffictents on the right-hand side, and on the left-hand side I acts through coefficients but r acts through the group cQ, by conjugation.
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Proof : note that V(cp 0 v,,,,l)* is the induced representation of (into GQ, that is, V(cp)&DV(v;1) is isomorphic to the space of continuous functions Cont(r/rm, V(cp)*) on r with values in V(co)*, on which GG acts by go(y) = cp(g)0(g-ly) for ¢ E Cont(r/r.,,,,,V(cp)*). Thus by Shapiro's lemma, we have the first isomorphism in the lemma. The second isomorphism can be proven in the same manner because Ir/Ir fl GQ,,, = r/r,,,.. Since Iq C GQ_, the third isomorphism is obvious and induced by the first.
jective type) of V(cp)* restricted to
We now fix a representation p satisfying the conditions defining V. Let (RQ, pQ) be the universal couple representing the problem V for p. We assume that Spec(II) gives a closed subscheme of the normalization of the reduced part of Spec(RQ). Let 7r : RQ -4 I be the projection and tp = 7rpQ be the representation of GQ into GL2 (1). Since p is odd, we can decompose V (V OR cpV) = Endij (V (cp)) into the sum of trace zero space s 12 (V (cp)) and
the center Z(cp). We write the representation on z12(V(cp)) as Ad(cp), that is, V(Ad(cp)) = s[2(V(cp)). We have a filtration (Xr) and (N9) of the Galois representation of GQ,, for q E M' U {p} : V(pQ,1,q) C V(pt). This induces the filtration on 5(2(V(c,)) 0 C y+(Ad(V)) C V9 (Ad(cp)) C V(Ad(cp)) :
given as follows (see [1-161):
V; (Ad(s))
E
0} and
Vq (Ad(cP)) = {0 E s[2(V(cv))J0(V(PQ,1,q)) C V(PQ,1,9)}
Let F be an algebraic extension of Q. Suppose (AIF) and that p satisfies the condition defining DF. Then the above filtration stable under GFQ induces a filtration for each prime Q in MF U {PJp} : 0 C VQ (Ad(cp)) C VQ (Ad(cp)) C V(Ad(cp)). In other words, if GFQ, then VQ (Ad(cp)) = Q yt(Ad(cp)). For each II-direct summand W of vgQgo-1
V (Ad(cp)) or V(Ad(cp) ® v,-,,') for 0 < m < oo, we define W* = W OR P for the Pontryagin dual II* of II. When m = oo, for each II[[r]]-submodule W of V(Ad(cp) ® v-1) = V(Ad(cp))®k7V(v-1), we define W* = W ®II[[rll II[[r]]* for the Pontryagin dual II[[F]]* of II[[r]]. We also put VQ (Ad(cp)) ®v z') = VV (A([(cp))®0V(v;,,,
for m =0,1'...100.
We let the Galois group act on W* through W and write 4P for one
of Ad(W) and Ad(V) ®for m = 0,1, ... , oo. For each prime ideal Q of F and a subset L of N[F, we consider the following subgroup LQ of HI(GFQ,V(I')*) :
LQ = ker(H1(GFQ,V(4)*) -> H1(IQ,V(f)*/VQ (t)*)) for Q E LU {PIp} , LQ = ker(H1(9FQ,V(4))*) -> Hl(IQ,V(4))*)) for Q outside L U {PIp} .
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTA77ONS
103
Then associated Selmer group of 41, over F is defined by (Sel)
Selc(4')/F = flQker(H1(QF, V(()*) . H1(cFQ,V(f)*)/LQ))
The Selmer group defined in [G] Section 4 is equal to SelO(Ad(W))/F, which we write simply as Sel(Ad(cp))/F. By a general theory due to Greenberg ]G] p. 217, the Pontryagin dual Sel*(Ad(cp))/F of Sel(Ad(cp))/F is of finite type over II. In this case, starting from a modular Galois representation p, it is basically known by Flach [F] and Wiles [W) that Sel*(Ad(cp))/Q is an II-torsion module of finite type if Spec(l) gives an irreducible component of RQ. It has been conjectured by Greenberg that Sel* (Ad(cp))/F (resp. Sel*(Ad(cp)) 0 v-1)/F) is an II-torsion (resp. p[r]]-torsion) module of finite type if II is sufficiently large. If F/Q is a Galois extension, by definition, Gal(F/Q) naturally acts on Sel*(Ad(cp))/F, and the restriction map of cohomology takes Sel(Ad(cp))/E into Sel(Ad(cp))/F. By Lemma 3.1, for the subfield Q,, of Qom, we have the following commutative diagram for n > m : Selc(Ad(ca))/Q,n
Selr-(Ad(cp) 0 V,,,1)/Q I i,,,,,,
1 res
Se1c(Ad(cp))/Q,
where 2,n,,i,
- Sele(Ad(cp) ®Vn 1)/Q
is induced by the natural inclusion V(Ad(cp) 0
C
V(Ad(cp) ®v7L 1)* induced by the dual map of the projection iD[r/rn] -+ Since the formation of Galois cohomology commutes with injective limit of coefficients, we get the following version of a result of Greenberg [G] Proposition 3.2 : (Sell)
lir Selc(Ad(cp))/Q,,, =tin Selc(Ad(cp) ®v;,,,l)/Q zn
rn
= SeIL(Ad(cp) 0 v-1)/Q = Sel,c(Ad((p))/Q_
3.2. Mazur modules and Selmer groups. We return to the situation in 2.1. We assume (AIF) and the conditions defining DF for Ti. Take a Aalgebra homomorphism r : RF -> II and suppose that II is a AF-module of finite type. Let cp = 7rpF. We consider the scalar extension R = RF OAF II, which is naturally an II-algebra. Then we consider the module SZR/1 of mRadically continuous 1-differentials over II. Then HomR(IIR/0, M) = HomRF (ORF/AF, M) = DerAF (RF, M)
H. HIDA
104
for each topological II-module M of finite type or an injective limit of such modules. Here every homomorphism and derivation as above is supposed to be continuous under the ml-adic topology. We consider the ring RF[M] = RF G M with M2 = 0. Then for S E DerAF, (RF, M), we have an AF-algebra homomorphism t(5) : RF -+ RF[M] given by r i-4 r ® 6(r). Any AF-
algebra homomorphism, inducing the identity modulo M, is of the form t(S) : RF --> RF[M] for a derivation S.
We consider the deformation p 9F --+ GL2(RF[M]) of PF. Then we can write down p = pF ® u' for u' : ccF -> End1(M ® M). Define u(a) = u'(a)PF(a)-1. Then :
u(aT) = u
(aT)PF(aT)-1
= (PF(a)u (T) + u (a)PF(T))PF(aT)-1
= Ad(PF) (a)uu(T) + u(a) .
Note that det(l(Bu) = Tl^(u) fore E End1(M(DM). Thus by (det) in 2.2, it is a 1-cocycle, under the adjoint action Ad(pF) on Ad(M) = V (Ad(cp))OIIM in End1(M (D M), having values in Ad (M). One can check that the map : p '-* the cohomology class of u from the set of deformations of PF of type
DF is an injection. For primes Q in MF(x), by (xQ), the order of p(IQ) is prime to p, and thus (a) u(IQ) = 0. For Q E M' or Qlp, we have (b) u(QFQ) c Vj(Ad(cp)) OR M and (c) u(IQ) c VQ (Ad(cp)) OR M. Since cFQ normalizes IQ, the non-splitting of the exact sequence in (NQ) or (Regp) shows that (c) implies (b). It is obvious that if we are given a 1-cocycle it satisfying (a) and (c), p = pF 0UPF is a deformation of type DA. Thus we get a version of the results in [MT] Proposition 25 and [HT] Proposition 2.3.10 : THEOREM 3.1. - Suppose that II is a AF-module of finite type. We have HomR(SlR/II,1*) = C, (A; II)* = SelM,(A(I(W))/F
,
where A is given by in o (ir 0 id) for the multiplication m : 1[ OAF II -> II with lo = Im(7r) and "*" indicates the Pontryagin dual module.
3.3. Control theorem of Selmer groups of Ad(cp). We now assume that E = Q and F = Q3 C Q. Then it is easy to check that if p satisfies the condition of DE, then p restricted to CJF satisfies the conditions of DF.
We assume (AIQ,). Then we write ay (resp. Aj, m3) for the morphism a : Rj = RF -> Rj (resp. fl [[I'j]] in AF, the multiplication m : IIo OA; II --+I)We write pj for the universal representation realized on R;. Now let Spec(II) be an irreducible component of the normalization of Spec(RQ) and write
ON SEIMER GROUPS OF AAIOINT MODULAR GALOIS REPRESENTATIONS
105
: RQ --> II for the projection map. We assume that II is a torsion-free Aomodule of finite type. We apply the above theorem to 7rj = 7raj R,, , II. We write A. = mj o (7rj 0 id). Under (AIQ_), by Theorem 2.1, Im(7rj) = 7r
:
Im(7r) = lo independent of j. Then similarly to a,, we can construct an pj. Then aj,k algebra homomorphism aj,k : Rk -+ R; so that a3,k o pk induces a projection map C1(Irk;IIo)
Cl(lra;IIo),Cl(mk;II) -+Ci(m,;II) andCl(%k;II)
Ci(A,;II)
.
Since RF is topologically of finite type over Aj, all these modules are made of compact modules. Note that projective limit is an exact functor on the category of compact modules. Then we take the projective limit of these modules, and write them as C1(A ; II), Cl (ire; II) and Ci (m,,; II). Then we have an exact sequence from (Ext4) in 2.2, for the generator 'y, of I'j = Gal(Q /Qj), k > j and Tj = Rj ®A; II (Ext5) Tori' (II, Ker(pQ, )) -' II ®7 ri /rk --+ Cl (Ak; II)/('yi - 1)C1(.\k; II) -+ C1(A,;1) -+ 0 .
Since these are compact modules, after taking projective limit with respect to k, we still have the following exact sequence :
(Ext6)Tori'(II,Ker(µQ,)) --+II®zprf(-II) -+Cl(A; l[)/('y, -1)C1(A ;1) -+ C1 (a;; II) -+ 0 .
Suppose RQ is reduced. Then if RQ is a AQ-module of finite type, Cl(Ao; II)
is a torsion II-module of finite type. We now show that
Cl(,\.; II)/(-y - 1)Cl(,\.; II) contains actually a copy of II. For that, we look into the following exact sequence obtained from (Ext3) in 1.1 :
Ci (irj; IIo) off. II ' * CI (Ar; II) -* Ci (ni,,; II) -> 0 for j = 0,1,... , oo .
We study Ker(tj). By definition, A,, is isomorphic to D[[r,]]. We write
A for lD[[r;]]. By (Ext2-3) in 1.1, this module is a surjective image of Tori (Ker(rnj), II) if IIo is flat over A, where we write II° for II®®Aj II. Suppose for the moment that IIo is A-flat. We write A° for A ®Aj A. We first compute
ToriA' (Ker(rnj), A) when II = A. Identifying A with lJ[[T]], we see easily that A° - iJ[[T]][S]/((1 + S)`' - (1 +T)`") for q = rp'. Then Ker(mj) is a principal ideal generated by S - T, and decomposing (1 + S)" - (1 + T)9 = (S-T) f (S, T), we have an isomorphism Ker(mj) = A°/(f (S, T)). Thus we have an exact sequence : 0 -+ (S - T)A°
A° f
(S ,T)
A° -+ Ker(rnj) -+ 0
.
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H. HIDA
Tensoring A, we get Tory (Ker(rrtj), A) = 0 and llim Tori ° (Ker(mj), A)
= 0. Since II is a A-module of finite type, Ker(tj) is a torsion II-module in general. In particular, if C1(A0, II) is of II-torsion, C1(fro, bo) ®ll II is of II-torsion.
Writing I1 for IIo®A; II, we have projective systems of surjective homo-
morphisms IIj -> IIj_1 and Ker(mj) , Ker(mj_1). Writing II,,. for Jim II,, 7
we know that Ker(m..) = llim Ker(mj) and that Torl°° (Ker(m,,'), II) _ lim Tor (Ker(rna,), II). Note that 100 = IIo®oII. It is obvious that Ker(moo) is the ideal of the diagonal A in Spec(IIo) x Spec(1D) Spec(II). Obviously A is irre-
ducible and spanned by S - T. Thus Ker(ni,,) is II.-free. This implies that
Tore" (KerII)
= 0. Thus we have an exact sequence of 11[[r]]-modules :
0 -' C1(iroo; 1O)
II -' C1(Aoo; II) -+ Cl (nt,,;1) - 0
.
When 1o is not flat over A but A-torsion free, then 1o can be embedded into a A-free subalgebra V of II such that II'/1o is pseudo-null. Thus we can repeat the above argument in the category of II-modules with pseudo-morphisms. We then get the above exact sequence with pseudo-null Ker(t,,,,).
We now study C, (m,,;1o). Then, by definition, we have Cj(mj; IIo) _ QIlo/A,, where Honnp0(QH0,/Aj,M) is naturally isomorphic to the module DerA, (h o, M) of continuous derivations over A; for all compact IIo-modules
M. As we have seen (see also (H31 Lemma 1.11), we have Ker(moo) = (S - T)IIo®DIIo. This shows that C1(nt,,; IIo) = II0 and that Cl (mom; IIo) is a torsion 1[[r]]-module of finite type.
Suppose either that RQ is reduced and is a A-module of finite type or RQ is reduced and Spec(IIo) is an irreducible component of Spec(RQ). Then C, (A0; II) is of torsion and cannot contain a submodule isomorphic to II. Thus the inclusion of II into C1 (A,,.; II) composed with the projection from Ci(.j; II) onto II is a non-zero II-linear map, and it is injective. Writing M for we have a commutative diagram with exact rows : Sel; ,
M/(71' - 1)M
C1(Aj; II)
n
,
1
II ®zy r
M/('Y - 1)M
C1(.\o; II)
II ®z,, I'.i
0
-a
E°-f
0
0
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTA77ONS
107
where the last two vertical arrows are surjective and the first vertical map is injective induced by the inclusion : I'j C F. Thus ej is injective. As is remarked in Section A.1 (see (Al) after Corollary A. 1.3), the condition (AIQ_) is equivalent to (AIQ). We record what we have proven.
THEOREM 3.2. - Suppose (AIQ), the conditions of D for p and that I is a torsion free Ao -module of finite type giving the normalization of an irreducible
component of Spec(RQ). Let Se1*(Ad(cp) 0 v-1),Q be the Pontryagin dual module of the Selmer group Se1G(Ad(cp) ®v-1)/Q. We have the following two exact sequences of 1-modules : I®z,,I'j 4 Set .AA ,
(Ad(W)0v,-1)1Q1('Y''j 1)
Set M , (Ad(W)®v-1)/Q-'Cl
Ci(7r,,, Io) ®uo II -+
0 v-1),,
(Aj;1)-'0
-> 11 -4 0
with pseudo-null kernel Ker(t ), which vanishes when 11o is flat over A. Moreover suppose that RQ is reduced and either that RQ is a A-module of finite type or that Spec (10) is an irreducible component of Spec(RQ). Then c, is injective.
3.4. Cotorsionness of the Selmer group over ]1[[F]]. We write A'. for the subalgebra of Ro topologically generated over 0 by a(pF,1,p(0p)) for F = Q3, where qP is the geometric Frobenius element in Dp/Ip. Taking a unit u in 0 such that pQ,1,l,((pl,) = 711110 , we assume that, with the notation of (XP,F). (Ind) the subalgebra of No topologically generated over ID by cp1,p(ov) is isomor-
phic to the one variable power series ring ID[[X]] via cpi,p(O.) - u H X. Since it takes pQ,l,l,(l51,) to tpl,l,((pl,), pQ,1,1,(01,) - u is analytically in-
dependent over ID. Since p totally ramified in Q, a takes pF,1,p(gp) to pQ,1,P(op) Thus (Ind) implies
(Ind) the subalgebra of R; topologically generated over 0 by pF,1,P(op) is isomorphic to i7[[X]] via pF,1,P((bp) Ga1(Qj /Q).
-u-
X and is stable under
Thus A' = A is independent of j and R; is naturally an algebra over A'. We also suppose that 10 is a Ao-torsion free module of finite type. Since Io is an integral domain, Io is a A'-torsion free module. We write A' for the composite of Ira ®id : R;®A'I -4 Io®A,I and the multiplication : Io®A,I ` II, where "®" indicates the completion under the adic topology of the maximal ideal of the algebraic tensor product. We prove the following theorem in this section :
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H. HIDA
THEOREM 3.3. - Suppose (AIQ), (Ind), that 11 is a torsion free A-module of finite type giving the normalization of an irreducible component of Spec(RQ) (Ad(cp))/Q is a torsion 1-module. Then we have and that (i) 0 v-1)/Q is a torsion II[[r]]-module offinite type;
0 v-1)IQ into (ii) There is a pseudo-isomorphism of M x II for a torsion 1[[I']]-module M = Ci (A' ;1) such that M/(y - 1)M is a 00 torsion II-module and M is pseudo-isomorphic to C1(7r,,;10) ®IIo II;
(iii) If (Ad(cp))/Q is a pseudo-null 1-module and A' = II, then Sel*(Ad(p)) 0 v-')IQ is pseudo isomorphic to II, on which r acts trivially; (iv) If lo is formally smooth over iD, then we have the following exact sequence of 1[[[I']]-modules :
0 - Ci (7r,,; II) -' Ci (A'00; II) ` SIR/A' - 0 ,
where SIR/A, is the module of continuous 1 -derentials or equivalently is the mu-adic completion of QH/A' (which is a torsion II-module of finite type by (Ind)).
By the theorem,
0 v-1)IQ is a torsion 1[[F]]-module of finite
type for any subset L of M'. The theorem combined with Theorem 3.2 reduces the study of Sel*(Ad(cp)) 0 v-1),Q to the study of Ci(7r.; II) if IIo is formally smooth over iD. Here is a concrete case where the theorem applies. For a positive integer
N prime to p, let h°`d(N; ID) be the universal ordinary Hecke algebra for GL(2)/Q. Then h°rd(N; iD) is an algebra over iD[[I' x (Z/NpZ)"]]. The algebra structure is given so that h°''d(N; ID)/P"h°`d(N; ID) is isomorphic
to the ordinary Hecke algebra of weight ic + 1 of level Np, where P" is the prime ideal of A generated by 'y - N(y)r. Take a primitive character
z/' modulo Np and suppose that 0 has order prime to p. We take the algebra direct summand h(,0) of h"' (N; ID) on which (Z/NpZ) x acts by 0. Take a maximal ideal in of h(V)) with residue field F and write H for the m-adic completion of the Hecke algebra h(&). Then we have a unique isomorphism class of Galois representations p : G -i GL2 (H) as in [H 1 J (see also [DHI] Section 1) if p = p mod m is absolutely irreducible. In this case, p and p satisfy the requirement of the deformation problem DQ. Since H is reduced and A-free of finite rank (see [H 1 J and [H] Chapter 7), the reducedness and the finiteness of RQ over AQ follows from Wiles' result [W] Theorem 3.3 asserting that (RQ, pQ) = (H, p) under the assumption
(AIF) for F = Q(/(-1)(v')/2p). Thus under this condition, for each irreducible component Spec(1) of the normalization of Spec(H) and the
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTATIONS
109
for cp = irp is a torsion 11-module projection it : H -+ 11, of finite type. This also follows from a result of Flach [F] in some special cases. The condition (Ind) can be verified in this case as follows. Note that pQ,1,p(op) = T(p) in H. Thus specializes to an algebraic integer a,, in U modulo P,,, with lakl = p"/2 for any archimedean absolute value on Q(a,.) for infinitely many it. Thus cpl,p(q p) is transcendental over 0 and hence (Ind) is satisfied. Thus in this case Sel,(Ad(cp) 0 v-1) is a torsion 11[[r]]-module of finite type.
We give two proofs of the theorem. The first one is just a repetition of the
argument in the previous section replacing Aj by A', which is easy but we need to assume an additional assumption that 10 is a A'-module of finite
type. The other one works in general, but we need to use the theory of imperfection modules in [EGA] IV.0.20.6. Anyway we look into the following exact sequence obtained from (Ext3) in 1.1 for j = 0, 1, . . . , oo : CI(1rj;11o)0l011
where m' : 110®A'11 -* 11 is the multiplication map and A, : Rj®A'11 --+ II is the
composition m' o (iraj 0 id). Note here that the first term C1(irj;11o) ®1° II is
the same as the case studied in 3.3. We study Ker(Lj). As we remarked, here we assume that 110 is a A'module of finite type and will deal with the general case later. Then Ker(ij) is a surjective image of Toryy (Ker(m'), II) if 110 is flat over A and is a surjective
image up to pseudo-null error in general (see (Ext2) and (Ext2')), where we write II' for 11®®A' II. Note that this fact holds independently of j. We have the long exact sequence for M' = Tori° (Ker(m'%, IIo) and 1l = 110 ®A' 110
0-+M'-->I®1'I-4I-*f11°/A,-4 0 obtained out of the following short exact sequence :
0-*I-+II0-+10-+0. Thus M' is an 110-module of finite type. Since 111°/A' is a torsion h omodule, localizing at a prime P outside Supp(111o/A'), we have Ip/Ip =
0, and by Nakayama's lemma, Ip = 0, which implies Mp = 0. Thus Supp(M') C Supp(1 0/A,), which shows that M' is a torsion 110-module of finite type. Thus Cl (x,,.; h o) ®1° 11 is pseudo-isomorphic to Ci (A'00 ; II) as 11[[r]]-modules. By Theorem 2.2, we have C100;11)/(7 - 1)C1 (A,00;11) = C1 (,\O'; II)
110
H. HIDA
Thus if Cl (Ao;11) (Ad(cp)) /Q is of II-torsion, Cl(irojo) Ono land hence C1(As; II) are II-torsion modules of finite type. The Krull dimension of CI (A'; II) over II satisfies dimii(C1(A'; 1)) < dim(1[). By (EGA] IV.0.16.2.3.1, we have
dim(II[[r]]) = dim(II) + 1 > dimg1[r]](Cl(A/.; II)/('r -1)Ci
,0; II)) + 1
> dimg[[r]] (C1(A0; II)).
Thus C1 (A 00 ;1[) is a torsion II[[r]]-module, and hence Cl (ir,,.;1o) ®IIo II
is a torsion II[[I']]-module of finite type. Then Theorem 3.2 tells us that Sel*, (Ad(cp) 0 v-1)/Q is a torsion ]1[[r]]-module of finite type.
A principal ingredient of the second proof is the theory of imperfection modules in [EGA] IV § 0.20.6; so, we recall the theory here and generalize it in the case of compact adic rings. Here i7 is a valuation ring finite and flat over Z P with residue field F. Let A be a base ring which is an object of CNL = CNLD. If X -+ Y is a morphism in CNL, we write Sty/X for the mX-adic completion of the module of one differentials of Y over X. Then Sty/X is a Y-module of finite type and hence is compact. We consider local C. Then by [EGA] algebra homomorphisms in CNL : A -+ A - B 0.20.7.18 we have an exact sequence S2B/A®BC v*. AC/A u*) SZC/B --..0 ,
where "®" indicates the mc-adic completion. We define the imperfection module TC/B/A following [EGA] 0.20.6.1.1 by TC/B/A = Ker(S2B/A®BC -v*+ S C/A)
Then we have the following commutative diagram with exact rows (see [EGA] 0.20.6.16) :
0
SZA/A®AC - (S2A/A®AC) ® (SlB/A®BC)
SiB/A®BC -> 0
t (vu).®id
1 u.Oid 0
n
-> S1B/A®BC -+
Po
DC/A
(flC/A ®
-0,
where, writing f = zu* ®id and g = v., jl (x) = x®f (x), p1(y(Dz) = z- f (y), jo(x) = g(x) ® x and po(y ® z) = g(z) - y. We put B/A/A
= Ker(1lA/A6AC -4 SlB/A®BC)
.
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTATIONS
111
Then by the snake lemma, we have an exact sequence (cf. [EGA] 0.20.6.17) : (El) TB/A/A -> TC/A/A --> TC/B/A -> StBIA®BC -> 1C/A --> SZCIB -> 0 .
0
Now suppose that v is surjective, and hence SZCIB = 0. By [EGA] 0.20.7.20, we have another exact sequence : (El')
Ker(v) / Ker(v)2 -> IB/A®BC ` fCIA ` 0
.
If B/ Ker(v)2 -+ C has a section of A-algebras (for example, if C is formally smooth over A), then 0 -> Ker(v)/ Ker(v)2 -> QB/A ®B C -> DC/A -` 0 is exact (cf. [EGA] 0.20.6.10). Since taking MB-adic completion is a left exact
functor, we have the exactness of 0 -> Ker(v)/ Ker(v)2 -+ SZB/A®BC ` iC/A -4 0 .
This shows that if B/ Ker(v)2 -+ C has a section of A-algebras, then TCIBIA - Ker(v)/ Ker(v)2 and the following sequence is exact :
(E2) 0->TB/A/A-+TCIAIA -4 Ker(v)/Ker(v)2-->SZB/A®BC11C/A 40 .
Let K be a finite extension of the quotient field L of A = i7[[t]] for a variable t. Let II be an A-subalgebra of K integral over A with quotient field K. Thus dim(II) = 2. Since A is a Japanese ring, II is an A-module of finite type. Thus we have a surjection v : A[[X1, ... , X,.]] -> I. This yields an exact sequence Ker(v)/ Ker(v)2 -> ®iIIdXi - QR/A --+ 0 . Since SZIIK/L = 0, nII/A is a torsion II-module of finite type. We also have the following exact sequence :
0 --f Tp/A/D - 1A/O ®A II `
o f/A --+ 0
.
Since any continuous derivation of A can be extended to K, the image of S1A/fl ®A II = IIdT = II in SZII/O is II-free of rank 1. Since QII/A is a torsion II-module, S1AID ®A I has to inject into fZ111 . Thus T11AIO = 0. Let S E MI,
112
H. HIDA
and suppose s is analytically independent over U. Let A = i7[[s]] C II. We consider 1111/A. Then we have, taking a surjective algebra homomorphism
v': A[[X,,...,Xr]]-->1, Ker(v')/ Ker(v')2 -* ®1IIdX; ` QII/A ` 0
.
If t and s are analytically independent over iJ, then II becomes integral over the power series ring ID[[T, S]] = 0 [[t, s]], which is impossible because
dim(II) = 2. Thus t and s are analytically dependent, and the evaluation map v" : i7[[T, S]] --+ II at (t, s) has non-trivial kernel P. The prime P cannot have height 2 or more because dim(Im(v")) = 2. Thus P is of height 1, and it is therefore generated by a single element f (T, S) because U[[T, S]] is a unique factorization domain. Then we have r3f
57, (t, s) dt +
OfaS
(t, s)ds = 0 in
and
D[[t,31]/ - (U[[t, s]l dt ® D [[t, sl ]ds)lOT (-(t, s)dt + Lf as (t, s)ds)
Suppose as (t, s) = 0. Then as (T, S) is divisible by f (T, S), that is, = fg for g E U[[T, S]] and hence (t, s) = (g + f )(t, s) = 0.
i
Repeating this argument, we find as (t, s) = 0 for all n, and hence f (T, S) E U[[T]] because U is of characteristic 0. This is in contradiction 0. Similarly we know to the analytic independence of t. Thus as (t, s)
that 2L (t, s) # 0. Since ds and dt has a linear relation, ds is II-linearly independent. We thus conclude for an analytically independent s, 01/01[.11 is a torsion II-module. We now look at the following exact sequence : 0 - TII/0113]]/0 -' D[[sll/D®D[[311II -' 2II/D -' fiII/DQsll -' 0 .
I via ds H 1.
Since s is analytically independent,
Since S2II/0[[Y11 is a torsion II-module and Q11D ®II 1K is of dimension 1, has to inject into This shows that TII/D[[sll/ = 0. We now consider the situation where we have a surjective ID [[s]] -algebra homomorphism 7r : R -> II for an object R of CNLD. By (E1), we have the following exact sequence : 0
TR/D[[s]]/D
TII/R/D
-' S2R/D[[sll®RII -'
0
and TII/R/D - Ker(7r)/ Ker(ir)2 = CI (7r; II) if II is formally smooth over 0. By the above result, this yields a short exact sequence : 0-4 Ti/R/D -> SR/D[[sll®RII -> S21/0[[4]1 - 0
.
ON SELMER GROUPS OFADJOINTMODULAR GALOIS REPRESENTA770NS
113
Now we study how large the difference of TIl/RID and Cl (ir, I) = Ker(rr)/ Ker(ir)2, when II is not formally smooth over i7. The key point here
is that Tp/R/D is independent of the choice of s. We pick t' E R so that 7r(t') = t as above. We regard R as an 0[[t]]-algebra through the algebra homomorphism of U[[t]] into R taking t to t'. Then we have again an exact sequence : 0 - TI/R/il -" SLR/DIltll®RII --> SLl/b[[t]l '-: 0 .
We have a surjective II-linear map r : Ci(Or,11) -+ Tu/R/U from (E1) and (E1'). Let II' = II OD[[t]] II and m : II' -+ II be the multiplication. In this case,
if II is flat over U[[t]], by (Ext2 and Ext2'), Ker(r) is a surjective image of Tory (Ker(m), II), which is a torsion II-module of finite type, because I is an 0[[t]]-module of finite type. Even if I is not flat over ID[[t]], one can embed II into an i7[[t]]-flat module with pseudo-null cokernel. Thus Ker(r) is a surjective image of Tory (Ker(r), II) up to pseudo-null error. The error is annihilated by a power mII for a positive M independently of R and the choice of t' with -7r(t) = t (but depending on I and t). Thus without any assumption, Ker(r) is a torsion II-module of finite type killed by a non-trivial ideal a of II independent of R.
We now give the second proof. Here we do not assume the integrality of II over A'. Since the result over II is just a scalar extension of the result over IIo, we only need to prove the assertions (i)-(iv) replacing II by IIo. Thus
hereafter, we assume that Im(7ro) = II and discard the assumption that II is integrally closed. Thus hereafter, we write II instead of IIo for Im(iro).
We pick t E II so that II is integral over 0[[t]]. We take tj E Rj so that aj,k(tk) = tj and 7ro(to) = t. Then we apply the above theory to R = R;,
s = pF,l,p(i') E Rj for F = Q3 and t' = tj. Then we have an exact sequence of compact modules : C1(7rj , II)
TII/Ri/f7 -+ 0
,
where Ker(r3) is the image of a torsion II-module X = Tory (Ker(m), II) of finite type (independent of j) up to a bounded II-pseudo-null error. Taking
the projective limit with respect to j, we find that r,,
:
Cl (ire; II) -+
Tl/R_/D =1im Tp/Ri/D is surjective because of the compactness of these r
modules and that is an II-torsion module of finite type. Thus ro, is an II[[[']]-pseudo-isomorphism. By our assumption : II = Im(7ro), we have SZR,/AZRII = Ci(a';II). By taking the projective limit of the exact sequences : 0 -> Tp/Rj/i.) -" SLR3/A ®RII (- Ci(a'.;II)) -- SLR/A' --+ 0 for
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A' = D [[s]], we get another exact sequence : 0 -' ,Y II/R-/D -' Ci (A' ;1) " SZfl/A, -+ 0 for R,, = 1jim R; .
Since QH/A, is a torsion I-module of finite type, TH/R,o/D is pseudoisomorphic to Ci(ir.;11) as 1[[r]] -modules. Thus Ci(7r,,.;11) is pseudoisomorphic to C, (A'00 ; II) as 1[[[I']]-modules. By Theorem 2.2, we have Ci (A'00 ; II)/(-r - 1)Cl (A'00;1) = C1 (A0; II)
As in the first proof, the assumption that Sel*, (Ad(cp))/Q is a torsion 11-module tells us that Ci (7ro;11) is of 1-torsion. Again by the exact sequence :
Ker(ro) ' Ci(7ro; II) -' Ci(,\o; II) -' S21/A, -> 0 ,
the II-torsionness of Ker(ro), Ci(7ro; II) and S21/A, tells us the same for Ci (.0;11). Then we conclude the assertions (i)-(iii) as in the first proof. If II is formally smooth over i7, from this immediately.
0 for all j. The assertion (iv) follows
Appendix : control of universal deformation rings of representations In this appendix, we give a general theory of controlling the deformation
rings of representations of a normal subgroup under the action of the quotient finite group.
A. 1. - Extending representations Let G be a profinite group with a normal closed subgroup H of finite index. We put A = G/H. In this section, we describe when we can extend a representation 7r of a profinite group H to G (keeping the dimension of ir). The theory is a version of Schur's theory of projective representations ]CR] Section 11 E.
A.1.1. - Representations with invariant trace Let 0 be a complete noetherian local ring over Z with residue field F. We consider the category CNL = CNL0 of complete noetherian local
0-algebras with residue field F. Any algebra A in this section will be assumed to be an object of CNL. For each continuous representation p : H -> GL,,(A) and cr E G, we define p°(g) = p(ogt-1). We take a representation 7r : H - GL,, (A) for an artinian local 0algebra A with residue field F. We assume one of the following conditions :
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ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTATIONS
(AIH) p = 7r mod mA is absolutely irreducible for the maximal ideal mA of A;
(ZH) The centralizer of p(H) as an algebraic subgroup of GL(n)IF is the center of GL(n).
Of course the first condition implies the second. There are some other cases where the last condition is satisfied; for example, (ZH) holds if the following condition is satisfied :
(RedH) p is upper triangular with distinct n characters pi at diagonal entries, and its image contains a unipotent subgroup U' such that U'/(U', U') = U/(U, U) for the unipotent radical U. LEMMA A.1.1. - Suppose (ZIq ). Then the centralizer of7r in GL,,, (A) is A'.
We assume the following condition : (C)
7r = c(a)-fir c(a) with some c(a) E GL,,,(A) for each a E G.
If we find another c'((r) E GL,,, (A) satisfying 7r = c'(a)-17rac'(a), we have 7r =
c'((7)-lc((r)7rc(a)-lc'(a)
,
and hence by Lemma 1.1, c((T)-1c'(a) is a scalar. In particular, for or, r E G,
c(aT)-fir' c(aT) = 7r =
c(T)-17frc(T) = c(r) 'c(a)-17ro' c(a)c(T)
,
and hence, b(a, T) = c(a)c(T)c(aT)-1 E A'. Thus c(a)c(r) = b(a, T)c(aT). This shows by the associativity of the matrix multiplication that (c(a)c(T))c(p) = b((Y,T)c((YT)c(p) = b(a,T)b(aT, p)c(oTp) and
c(a)(('(T)c(p)) = c(a)b(T, p)c(Tp) = b(-r, p)b(a,Tp)c(arp) ,
and hence b(a, -r) is a 2-cocycle of G. If h E H, then 7r(g) =
c(hT)-1ir(hTgr-1h-1)c(hr)
=
c(hr) -17r(h)c(T)7r(g)c(T)-17r(h)-1c(hr)
.
Thus c(hr)-17r(h)c(r) E A'. Thus if we let h E G act on the space C(G; M,,(A)) of continuous functions f : G - ,,,(A) by f I h(g) _ 7r(h)-1 f (hg), then c is an eigenfunction belonging to a character : H -->
A". Now we take g : G -> A" such that r7(hr) = l;-1(h)r7(T) for all
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116
h E H. For example, writing G = UTER HT (disjoint), we may define r7(hr) =1;-1(h). We replace c by gc. Then c satisfies that
c(hr) = ir(h)c(T) for all h E H. Since c(1) commutes with Im(7r), c(1) is scalar. Thus we may also assume (Id) c(1) = 1. Note that for h, h' E H, b(ha, h'-r) = c(ha)c(h'T)c(hah'7-)-1 = (ir)
7r(h)c(a)ir(h')c(T)c(ar)-17r(hoh'u-1)-1
_
-7r(h)7r°(h')b(a,T)7r(hah'a-1)-1
= b(a,T) .
Thus b is a 2-cocycle factoring through A.
is further a coboundary of (: A -+ A", If b(o,T) = we modify c by (-1c. Since ( factors through A, this modification does not destroy (ir). Then c(ar) = c(a)c(T) and c(hr)c(T) for h E H. Thus c extends E A" is a 7r to G. Let d be another extension of 7r. Then x(a) = character of G. Thus c = d ® x. c(o)d(a)-1
We consider another condition (Inv) Tr(7r) = Tr(7r°) for all a E G.
Under (AIF), it has been proven by Carayol and Serre [C[ that (Inv) is actually equivalent to (C). Thus we have THEOREM A.1.1. - Let 7r : H -p GL,, (A) be a continuous representation for a p-adic artinian local ring A. Suppose either (AIH) and (Inv) or (ZH) and (C). Then we can choose c satisfying (7r). Then b(a, T) = c(a)c(T)c(ar)-1 is a 2-cocycle of A with values in A', and if its cohomology class in H2 (A, A') vanishes, then there exists a continuous representation IrE of G into GL, (A) extending ir. Moreover all other extensions of 7r are of the form 7rB 0 x for a character X of A with values in A'. In particular if H2 (is, A") = 0, then any representation 7r satisfying either (AIH) and (Inv) or (ZH) and (C) can be extended to G.
COROLLARY A. 1.1. - If A is a p-group, then any representation 7r with values in GL, (IF) for a finite field of characteristic p satisfying either (AIH) and (Inv) or (ZH) and (C) can be extended to G.
This follows from the fact that IF' I is prime to p, and hence H2 (0, F') = 0. When A is cyclic, then H2(A,A") = A"/(Ax)d for d = CAI. If for a generator a of G, i; = E (Ax )d, then b is a coboundary of ((as) = i;2/d. By extending scalar to B = A[XJ/(Xd in H2 (G, B"), the c(ad)lr(ad)-1
class of b vanishes. Thus we have
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117
COROLLARY A. 1.2. - Suppose either (AIF) and (Inv) or (ZF) and (C). If 0 is a cyclic group of order d, then 7r can be extended to a representation of G into GL,, (B) for a local A-algebra B which is A -free of rank at most d.
Let p = Tr mod mA. We suppose that p can be extended to G. Then we may assume that the cohomology class of b(a, rr) mod mA vanishes in H2 (G, F'). Thus we can find ( : G -> A" such that a(a,T)
=b(a,T)C(a)S(T)S(vT)-I
mod MA - 1.
Then a has values in d,,,,(A) = 1 + mA. In particular, if theGm(A)/G_m(A)ISI. Sylow psubgroup S of G is cyclic, we have H2(S, G,,,,(A)) = Write for the element in G,,,,(A) corresponding to a. Then for B = the cohomology class of a vanishes in H2(S, (5,, (B)). This implies that in H2(S, B"), the cohomology class of b vanishes. A[X]/(X I SI
COROLLARY A.1.3. - Suppose either (AIH) and (Inv) or (ZH) and (C). Suppose A has a cyclic Sylow p-subgroup of order g. If p can be extended to G, then it can be extended to a representation of G into GL,, (B) for a local A-algebra B which is A -free of rank at most g. We now prove the following fact :
(Al) When A is cyclic of odd order and n = 2, the condition (AIH) is equivalent to (AIG).
We start a bit more generally. Let p be an absolutely irreducible representation of G into GLn(K) for a field K. For the moment, n is arbitrary. We assume that A is cyclic of order prime to n. We prove that p cannot contain
a character of H as a representation of H, which shows the equivalence when n = 2. Suppose by absurdity that p restricted to H contains a character X. If x is invariant under the conjugate action of A, x can be extended
to a character of G, and it is easy to see in this case, p has to contain an extension of X, and hence reducible. Thus x is not invariant under A. If x is invariant under a subgroup H' D H of G, again by the same argument as above, p gets reducible on H' containing a character x' of H' extending X. Thus we may assume that conjugates of x' under A' = G/H' are all distinct. By Mackey's theorem, the induced representation Ind(X') for H' to G is irreducible. By Frobenius reciprocity or Shapiro's lemma, the induced representation Ind(X') has a unique quotient isomorphic to p. Thus JA'J = n, which contradicts to the assumption that the order of A is prime to n. Of course, one can generalize the above argument for more general A not necessarily cyclic.
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A.2. - Deformation functors of group representations We suppose that G satisfies the following condition (cf. [T)) (pF)
All open subgroup of G has finite p-Frattini quotient.
We fix a representation p : G -+ GL,, (F) satisfying (ZH). In this section, we study various deformation problems of ;5 and relation among the universal rings.
A.2.1. - Full deformations We consider a deformation functor fH : CNL --> SETS given by
FH(A)={p:H-+GLf(A)Ip-pmodmA}/-where ";zz;" is the strict equivalence in GL,,(A), this is, the conjugation by
elements in GL,,(A) = 1 + mAMn(A). The functor.FH is representable ([T) Theorem 3.3) under (ZH). We write (RH, PH) for the universal couple.
Since pa restricted to H is an element in .'H(RH), we have an d-algebra homomorphism a : RH -> Ra such that apH = paI H. We like to determine Ker(a) and Im(a) in terms of A. By choosing a lift
co(a) E GL,, (0) for a E G such that co(a) - p(a) mod mo, we can define for any p E Fc,.(A), p°(g) = p(aga-1) and pl°1(g) = co(a)-1p°(g)co(a) in JI'H(A). In this way, A acts via a ' ----) [a] on .FH and RH. Then as seen in Section 1, we can attach a 2-cocycle b on A with values in G,,,,(A) p in the following way. to any representation p E F'H (A) with pl°1
First choose a lift c(a) of p(a) in GL,,,(A) for each or E G such that p = c(a)-1 p°c(a) and c(hr) = p(h)c(T) for h c H and T E G. Then we know that c(a)c(T) = b(a,T)c(mr) for a 2-cocycle b of A with values in G1b(A). If we change c by c' such that c'(a) = c(a)[; (a) for ((a) E G,,,,(A), we see from
c(a)c(r) = b(a,T)c(aT) that c'(a)c'(T) = Thus the cocycle b' attached to c' is cohomologous to b, and the cohomology class [b] = [p] E H2 (A, (A)) is uniquely determined by p. If [p] = 0, then for a 1-cochain C. We then modify c by ct; and b(a,T) =
by constant so that c(1) = 1. Then c extends the representation p to a representation it of G (Theorem A.1.1). LEMMA A.2.1. - Suppose (ZH) and that n is prime top and p[°1 -- p for
p E .FH(A). If det(p) can be extended to a character of G having values in an A-algebra B containing A, then p can be extended uniquely to a representation it : G -> GL1,(B) whose determinant coincides with the extension to G of (let (p).
Proof : by applying "det" to c and b, we know that[det(p)] = [det(b)] _ [p]n. If n is prime to p, the vanishing of [p]n in H2(A, G,,,,(B)) is equivalent
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTA77ONS
119
to the vanishing of [p]. Thus if det(p) extends to G (that is [p]' = 0), then p extends to a representation 7r of G which has determinant equal to the extension of det(p) prearranged. We now show the uniqueness of 7r. We get, out of 7r, other extensions 7r 0 X E .F'G(B) for X E H1(A, d, (B)) = Hom(A, d,,, (B)). Conversely, if 7r and 7r' are two extensions of p in Yc(B), then for h E H, = 7r(o,)p(h)7r(a)-1 and hence 7r(o,)-17r'(a) 7r'(o,)p(h)7r'(a)-1
commutes with p. Then by Lemma A.1.1, x(v) = 7r(a)7r'(o) is a scalar in x(?T) = 7r(UT)-17f'(aT) = -7r(T)-17r(Q)-17r'(a)7r/(T) = 7r(7-)-'X(01)7r,(T) = X(Or)X(T).
(B)) and 7r' = 7r 0 x, which shows that Thus x is an element in H1 (A, det(ir') is equal to det(7r)xm. If det(7r') = det(ir), then xn = 1. Since x is of p-power order, if n is prime to p, x = 1. Here is a consequence of the proof of the lemma : COROLLARY A. 2. 1. - Let Iro E .FG(B) be an extension of p E FH(A) for an A--algebra B containing A. Then we have
{7ro0XIXEHomo,d,n(B))}_{7rEFc(B) I7fjH=p}. It is easy to see that if H2 (A, F) = 0, then H2 (A,
(A)) = 0 for all A
in CNL. Therefore we see, if H2 (0, IF) = 0,
(*) Ff (A) =H°(A, FH(A)) -FG(A)l0(A) for 0(A) =Hom(A, d,n(A)) .
Here we let x E
act on .FG(A) via it 7r 0 X. Suppose that .FHA is represented by a universal couple (RH,o, pH,&) and [pH,o] = 0 in H2(A,G,n(RH,o)). Then for each p E .FH(A), we have cp : RH,o --+ A p. Then cp*[pH,A] = [p] and therefore, [p] = 0 in such that 'PPH,o H2(A,Gm(A)). This shows again (*). 1
>
Let us show that the functor FHA is representable by applying the Schlessinger criterion (see [Schl and [T] Proposition 2.5). For a Cartesian diagram in CNLO :
A3=A1 XAA2
A1
a2
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we need to check the bijectivity of the natural map
yH : 'FH(A1 XAA2) -' YH(A1)
(A2)
We already know from the representability of JrH that yH :.FH(A1 XA A2) _ .FH(A1) X.FH(A) FH(A2)
Since YHA is a subfunctor of TH, YH(A1) x.FH(A) FH(A2) is a subset of .FH(A1) X Jr,(A) FH (A2), and hence yH is injective. Take an element (PI, P2) of F'H (A1) X(A) .FH (A2). Then alpi = a2P2, that is, there exists
x E GL-.(A) such that xaipix-1 = a2p2. We may assume that a1 is surjective (cf. [T] Proposition 2.5). Then we can lift x to x' E GLn(A1). Then replacing p1 by x'pix'-1, we may assume that alpl = a1p2. Thus p = P1 x A p2 has values in GL7, (A1 x A A2 ). It is easy to see that p is invariant
under A. Thus yH (p) = (p1, p2), and therefore y°H is surjective. Then it is obvious that YHA is represented by RH,A = RH/E,EARH([a] - 1)RH
PROPOSITION A.2. 1. - Suppose (ZH). Then YHA is represented by (RH,o, PH,A) for RH,A = RH/a with a = E'EORH ([o] - 1)RH and PH,A = PH mod a. If either [pH,A] = 0 in H2 (A' G,n(RH,A)) or H2(0, lF) = 0, then we have FG/0 = YHA via 7r +-+ irIH.
We now consider the following subfunctor FG,H of FH given by FG,H (A) = {PI H E
'H(A) I p E FG(B) for a flat A-algebra B in CNL0}.
Here the algebra B may not be unique and depends on A. Let us check that JC*G,H is really a functor. If cp : A ---+ A' is a morphism in CNL and PIH E .'FG,H(A) with p E .FG(B), B being flat over A, then A'®AB
is a flat A'-algebra in CNL. Then (cp ® id)p E .FG(A'®AB) such that cp(PIH) = ((cP 0 i(i)p)IH Thus .FH(cp) takes .FG,H(A) into .FG,H(A'), which shows thatFG,H is a well defined functor. For each p E FG,H (A), we have an extension p E .FG(B). By the universality of (RG, pa), we have cp : RG -+ B such that copG = p. Then PI H = (PPG) I H = 'P(PGI H) = cpapH. This shows that cpa is uniquely determined by PIH E .FG,H (A). Therefore cp restricted
to Im(a) has values in A and is uniquely determined by PIH E FG,H(A) Conversely, supposing that [apH] = 0 in H2(A, d,,,,(B)) for a flat extension B of Im(a) in CNL, for a given cp : Im(a) -+ A which is a morphism in CNL, we shall show that p = coaPH is an element of .FG,H(A). Anyway apH can be extended to G as an element in .F'c(B), and hence apH E .FG,H(Im(ca)). We note that p can be extended to G because [cpcrpH] = cp. [apH] which
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vanishes in H2(A,G,, (B')) for B' = B®1i,,lal,w A. Thus p e .FG,H(A), and FG,H is represented by (Im(a), apH) as long as [apH] = 0 in H2 (0, G,,,(B)) for a flat extension B of Im(a) in CNL. We have the following inclusions of functors : FH D .J' D FG,H D .FG/0, the last inclusion being given by p 1--- pIH. The functor FH is represented by RH/a for a = E.EORH([cr] - 1)RH. Because of the above inclusion, if [apH] = 0 in H2(0,G,,,(B)) for a flat extension B of Im(a) in CNL, the ring Im(a) is a surjective image of RH/a = RH,o. If [pH,A] = 0 (for pH,A = p mod a) in H2(A, G,,,(B')) for a flat extension B' of RH,A in CLN, then PH,o E.FG,H(RH,A) and thus Ff = FG,H PROPOSITION A.2.2. - Assume (ZH) and that [apH] = 0 in H2 (0, (LII,,,, (B))
for a flat extension B of Iin(a) in CNL. Then FG,H is represented by (Im(a), apH). If further [pH,A] = 0 in H2 (A, G,,,, (B')) for a flat extension B' of RH,A, then we have .FG,H = .FH .
The character det(pH) induces an 0-algebra homomorphism 0[[Hab]]
RH for the maximal continuous abelian quotient Hab of H.
We write its image as AH and write simply A for AG. Thus we have a character det(pH) : H --> A. We consider the category CNLAH of complete noetherian local AH-algebras with residue field F. We consider the functor FAH,H : CNLA,_, -> SETS given by
FAH,H(A)={p : H-
I p =-p mod MA and det(p) = det(pH)}/-- .
Pick p : H --> GL,,(A) E FA,,,H(A). Then regarding A as an 0-algebra naturally, we know that p E FH (A). Thus there is a unique morphism cp : RH --+ A such that cPPH p. Then cp(det(pH)) = det(p), and cp is a morphism in CNLA,,. Therefore (RH, PH) represents FAH . Similarly to .FG,H, we consider another functor on CNLA : .
A,G,H(A) = {PI H I P E .FA,G(B) for a flat A-algebra B in CNLA}/
.
Take p E FA,G,H (A) such that p = p'I H for p' E .FA,G(B). Then there exists a unique cp : RG --> B with det(p') = cp(det(pG)). Since the A-algebra structure of B is given by det(p'), cp induces a A-algebra homomorphism of Im(a)A into B for the algebra Im(a)A generated by Im(a) and A. From p = ('PPG)I H = (P(PGI H) = c'aPH, we see that the A-algebra homomorphism
cp restricted Im(a)A is uniquely determined by p. Supposing that [apH] vanishes in H2 (A, d,,, (B)) for a flat extension B of Im(a), we know that [apH] vanishes in H2(A,G,,,(Im(a)A B)). For any morphism cp : Im(a)A -> A in CNLA, [cpapH] = cp,[apH] vanishes in H'2(A,G,n(B')) for
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B' = A ®Im(n) B which is flat over A. Thus we have an extension Tr of p to G having values in B'. Suppose further that n is prime to p. In this case, as already remarked, we can always extend p without extending A and without assuming the vanishing of [apH], because det(p) can be extended to G by cp o det(pc). Thus we know : FA,G,H(A) = {PIH I P E FA,G(A)}/
Since det(p) can be extended to G without changing A, there is a unique extension of it with values in GL,ti(A) such that det(7r) = t o (det(pc)), which implies that it E FA,c(A) and hence irJH E ,TA,c,H(A). Thus .FA,c,H is represented by (Im(a)A, apH) if n is prime to p. We consider the natural transformation : .17A,G -> Jc'A,G,H sending in to irIH. As we have already
remarked, the extension of p E .FA,c,H(A) to it E FA(A) is unique if n is prime to p. Thus in this case, the natural transformation is an isomorphism of functors. Therefore (RG, PC) = (Im(a)A, apH). Thus we get Suppose (Z11) and that either n is prime to p or [apH] vanishes in H2(A, G9z(B)) for a flat extension B of Im(a). Then FA,G,H is representable by (Im(a)AG, apH). Moreover if n is prime to p, we have the Ti-IEOREM A.2.1.
equality Rc = Im(a)Ac.
Since a restricted to Ail coincides with the algebra homomorphism induced by the inclusion H C G, a(AH) C A. We put R' = Im(a) ®AH A. By definition, the character 1®det(pG) of G coincides on Hwith (aodet(pH))®
1 in R'. Thus apH can be extended uniquely to p'c : G -> GL,,,(R') such that det(pG) = 1 ® det(pc) if n is prime to p. Thus we have a natural map t : RG --> R' such that tPG = p'c. Since RG is an algebra over A and Irrr(a), it is an algebra over R'. Thus we have the structural morphism t' : R' -> RG. By Theorem A.2. 1, t' is surjective. By definition, tapH = tpHIH = tp'IH = aPH 0 1 and tdet(pc) = det(p'G) = 10 det(pG).
Thus t'tapH = t'(apH 0 1) = apH and t'tdet(pG) = t'(1 0 det(pc)) _ det(pc). Thus t't is identity on A and Im(a), and hence t't = id. Similarly, tt'p'9 = rpc = p'. This shows that tt'(apH ® 1) = t(apes) = (aPH ® 1) and tt'(1 0 det(pG)) = t((Iet,(pG)) = 1 ® det(pG)
.
Thus tt' is again identity on Im(a) 0 1 and 1 0 A, and tt' = id. Let XP (resp. X(P)) indicate the maximal p-profinite (resp. prime-to-p profinite) quotient of each profinite group X. Write w for the restriction of det(pc) to (Gab)(P) . Define it : Gab -> O[[G,,'']]x by ic(g) = w(g)[gpl for the projection
g, of g into G`";", where [x] denotes the group element of x E GP6 in the
123
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTA7TONS
group algebra. Assuming that F is big enough to contain all g-th roots of unity for the order g of Im(w), we can perform the same argument replacing (AH, AG, det(pc)) by (O[[H,,"]], 0[[G"61], 1 ® rc). Thus we get COROLLARY A.2.2. - Suppose (ZH) and that n is prime top. Then we have (Re, pG) - (Im(a) ®AH AG, aPH ® det(PG)) (Im(a) ®0[[Hpb11 O[[G;G]], aPH ®rc) .
In Particular RG is flat over Im(a).
By Hochschild-Serre spectral sequence, we have an exact sequence
H2 (A, ZI,)
Ho (A, H1(H, ZJ,)) -- f H1(G, ZI,) -f Hi (A, Zp) -' 0 III
III
Hof O
),
III
GJll,
r,
_-
Oan r
-' 1,
where the subscript "p" indicates the maximal p-profinite quotient. Suppose that F is big enough to contain all d°-th roots of unity for the prime-to -p part do of the order d of A. Then the inclusion H C G induces the following commutative diagram :
a:AH
AG.
As seen in Corollary 2.2, this diagram is Cartesian. Thus AG is flat over AH. If H2 (A, 7GI,) = 0 H2 (A, Qr/Z7,) = 0), Spec(Im(a')) - Spec(0[[H;6]])1
and hence Spec(a(AH)) = Spec(AH)°. From the exact sequence, if 0 contains a primitive g-th roots of unity for the order g of Lab, we get
Im(a') = H°(A(G), 0[[G;,' ]])
,
where x E o(0) takes EgEGvi,a(g)[g] to EgEGpba(g)X(g)[g]
A.2.2. - Nearly ordinary deformations
124
H. HIDA
Now we impose the following additional condition to our deformation problem : let S = SG be a finite set of closed subgroups of G. For each D E S, let S(D) be a complete representative set for H-conjugacy classes fl H I g E G}. In the main text (Section 2), the data S is given by of a choice of decomposition subgroups of G = Gal(FE/E) at primes dividing p. For simplicity, we assume that D fl H E S(D) always. Then the disjoint {gDg-1
union SH = ODES S(D) is a finite set, because IS(D)l = IH\G/DI. Let PD be a proper parabolic subgroup of GL(n)lo defined over 0 indexed by D E S. For each D' E S(D) such that D' = H fl gDg-1, we define PD, = a lift c(g) E GL,,,(0) of ;5(g). We assume c(g)PDc(g)-1 for
(NO)
p(D) C PD (IF) for each D E SG S.
Then we consider the following condition : (NOH)
there exists gD E GL,, (A) for each D E SH such that gDp(D)gD' C PD(A) ,
where GL,I(A) = 1 +rnAM,t(A). We define a subfunctor.F, of the functor F?, with various restriction "?" introduced in the previous section, by 17-'(A) = t p E .F? (A) I p satisfies (NOx) } ,
where X denotes either G or H depending on the group concerned. Then by (NO), (NOx) and our choice of PD, F?-'(F) = {pIx} # 0. Let us write gl (resp. PD) for the Lie algebra of GL,,(F) (resp. PD(IF)). Note that D acts on gl and PD by conjugation. We can identify gI with V ®V* = HomF(V, V)
for the representation space V of p, where V* is the contragredient of V. Then PD can be identified with
{gEHomF(V,V) IgFDCFD}, for a filtration FD : {0} = Fo C Fl C C F,. = V. Here gFD C FD implies that gFi C Fi for all i. The filtration FD naturally induces a double filtration FAD on gC and PD. Since this filtration is compatible with that of PD, it induces a filtration of g1/PD, which is stable under the adjoint action Ad(p) of p. As shown in [T) Proposition 6.2, under the following regularity condition for every D E Sx, (RegD)
H°(D, 9(/PD) = 0,
FX ° is representable for X = H or G. We can think of a stronger condition : (RGD)
H°(D, gr(g(/PD)) = 0.
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTATIONS
125
This condition is stronger than the condition (RegD), because on gr(g(/PD), D acts through the Levi-quotient of PD. Writing the representation of D on Fi1Fi-1 as pD,i, (RGD) is equivalent to (RG'D)
HomD(PD,i,PDj) = 0 if i > j .
In the same manner as in the previous section, we can check that A acts on FH °. Take D E S and put D' = D fl H E S(D). Since p is invariant under A and p E F3-'(F), (Inv)
° , i =PD ,i PD
in 9r(V) for alliandaED.
For p E we have 9D E GL,,(A) such that p(D) C gD1PD(A)gD. This implies that V(p) has a filtration FD(p) : {0} = Fo(p) C F1(p) C C F,. (p) = V (p) stable under D such that Fi (p) is a direct A-summand of V (p) for all i and FD(p) ®A ]F = FD. We write PD,i for the representation of D on FF(p)/Fi_1(p). Now suppose p E and [p] = 0 in H2(0,G,,,,(B)) for a flat A-algebra B. Then we find an extension it : G -> GLn (B) of p. Let a E D and D' = Hf1D. Thus ir(a)p(d')ir(a)-1 = p(ad'o,-1) E 9D1PD(A)9D'
for all d' E D' and hence 7r(a)p(d')7r(a)-'p(d')-1 E g31PD(A)gD'. From this and (RegD'), it follows that 7r(a) E 9D1PD(B)9D' for a E D (see [T] Proof
of Proposition 6.2). Thus, taking 9D = gD', we confirm that 7r E Since is stable under the action of &, all the arguments given for.Fx in the previous paragraph are valid for FX°. Writing (RX°, pX°) for the universal couple representing o, we conclude THEOREM A.2.2. - Suppose (ZH), (RegD) for all D E SH and that n is
prime to p. Then we have the equality R° = where n,.o no RH n.o - R; is an 0-algebra homomorphism given by an opIj pn.o and AG° is the image of
H
in RC°. Moreover we have
o, an.oPHo Pco) - (Im(an.°) OA,, An ® det(Pc°)) G
(Im(an.°) ®O[[H;''I[
0[[G',*']],
an.opH0
® x) .
A.2.3. - Ordinary deformations
Fix a normal closed subgroup I = ID of each D E S. For D' _ gDg-1 flH E S(D), we put ID' = gIDg-1 flH. We call p E .. if p satisfies the following conditions : (Ordx)
°(A) ordinary
PD,1 is of rank 1 over A and I C Ker(pD,l) for every D E Sx.
126
H. HIDA
We then consider the following subfunctor Ff d of.., : Fyrd(A) = {p E
'°(A) I p is ordinary}.
It is easy to see that the functor FX d is representable by (RX', pXd)under (RegD) for every D E SX. Let p E F) d(A). Suppose [p] = 0 in H2(A,G,m(B)) for a flat A-algebra B. Then we have at least one extension 7r of p in F, °(B). We consider 1rD,1 : D -> A" for D E S. We suppose one of the following two conditions for each D E S :
(TRD) JID/ID n HI is prime to p; (EXD) Every 1--power order character of ID/ID n H can be extended to a character of A having values in a flat extension B' of B so that it is trivial on IDS for all D' E S different from D.
Under (TRD), as a homomorphism of groups, 1rD,1 restricted to ID factors through TD ,1 which is trivial on I. Thus irD,l is trivial on ID. We note that 7rD,l is of p-power order on ID/H n ID because pD,1 is trivial on ID and pD,1 is trivial on ID nH. Thus we may extend 1rD,1 to a character 77 of A congruent 1 modulo MB'. Then we twists 1r by 77-1, getting an extension 9r' = 7r ® 17-1 such that 1r'D 1 is trivial on ID. Repeating this process for the
D's satisfying (EXD), we find an extension 7r E F `I(B) for a flat extension B of A. We now consider FG H (A) = { pl H E
ji `I (A) I P E Fo d (B) for a flat extension B of A).
In the same manner as in Section 2, if either n is prime top or [aordpHdl = 0 in H2 (A, d,,, (B)) for a flat extension B of Im(a°rd) in CNL0,we know that .Ford is represented by (Im(a°r"), aordpHd) where cord : RHd RGd is an 0-algebra homomorphism given by a°rdpH( pGdI H. Let p E FGr'H(A) and it be its extension in F d(B) for a flat A-algebra
B in CNLO. The character det(1r) is uniquely determined by p on the subgroup of G","' generated by all ID,,,, because another choice is Tr 0 x for a character x of A and (ir 0 x)D,1 = x on ID,p. If GPr' is generated by the ID,1,'s and H7 det(ir) is uniquely determined by p. Thus assuming that n is prime to p, 1r itself is uniquely determined by p. Therefore the
natural transformation :.F " - F, `H given by p - pI H identifies G d
with a subfunctor ofJ' d , inducing a surjective 0-algebra homomorphism RGd such that PGdl H = /apo d. Since po dI H = apHd Q /3 : Im(aord) _, is the identity on Im(a°r`I) and we conclude that Im(a°r`I) = RGd. This implies
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTATIONS
127
THEOREM A.2.3. - Suppose (ZH), (RegD) for D E SH, either (TRD) or
(EXD) for each D E S and that n is prime to p. Suppose further that the ID,P's for all D E S and HP generate Gab. Then we have Im(aord) = Rod.
In particular, for any deformation p E there is a unique extension A d] yd 7r E (A) such that IrIH = p. If further [pH' = 0 in H2(A, dm(B))
for a flat extension B of RH o, then Ro = Im(aord) = Ro d, where o RH,o = RHd/EaEORtid([a] - 1)RHd. A.2.4. - Deformations with fixed determinant
We take a character x : G - O" such that x - det(p) mod mo. We then define .l'X'?(A) = {p E -'F;c(A) det(p) = xlx}. Supposing the representability of FX , it is easy to check that..X ? is representable. Since the determinant is already fixed and can be extended to G, by the argument in the previous sections shows that if n is prime to p, I
-FH'?'A
_ FG, H =
Fox
Write (Rx'?, pX?) for the universal couple representing .Fx'? and define RH? ax'? : -+ R"? so that ax'?pH PGX'?. Then we have PROPOSITION A.2.3. - Suppose (ZH), (RegD) for D E SH and that n is prime to p. Then we have RH?/E1EDRH?([a]
RG? - 1)RH? = R'? G,H= Im(ax'?) =
where Rx' is either R.x, R:'° or Rte '°r` For each p E F °(A), we decompose det(p) = xl; so that is a p-power order. If n is prime to p, there is a unique character S1/n : H -+ Gm(A) exists. Then we define px = p ® p-'/n, which is an element of FH'"'°(A) Writing fH for the deformation functor FH for xIH in place of pJH, we have a natural transformation :.AEI ° -+ yH"'° x fH given by p --+ (px, det(p)). If (px, det(p)) = (p'x, clet(p')), then
p = px ® ((Iet(p)/x)l1n = x ® (det(P)/x)l/n = p. Thus the transformation is a monomorphism. For a given (pX, det(p)), we can recover p as above. Thus we get .F ° = yH'"'° x fH. Since (O[[H ]], ic) represents fH, we see, if n is prime to p, (RHo' PHo) =
SHo ®Rl/")
128
H. HIDA
Similarly we get
(RH,PH) 5-- (RH[[HPb]],. H ®rl/").
Note that, if n is prime to p, .FH
=.FH'?'oXA'=fc;nxftt and .F6H=.FG'HX-G,H.
Thus a? = ax,? x a' for a' as in the end of the paragraph A.2. 1. This shows
that THEOREM A.2.4. - Suppose (ZH), (RegD) for D E SH and that n is prime to p. Then if 0 contains a primitive I1 -root of unity, we have Im(a?) = Rx'?®o(d
where Rx ? is either Rx or
RGn.°
Manuscrit recu le 23 octobre 1995
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTATIONS
129
Corrections to "On A-adic forms of half integral weight for SL(2)/Q" by Haruzo Hida
in "Number Theory, Paris 1992-93" Lecture Note Series 215, 139-166
p. 145 line 15 : "U(pa) _ {S E S I
U(p")={sEUI p. 145 line 1 from the bottom :
sI, _ (0 1
s1,
I0
) mod p°` }" should read
1
Imodpo
w° ju." should read "for
"for
wl®/2 ® w° IUa, where w° is the dualizing sheaf on
Xa,zp".
p. 146 line 8: "the first horizontal map" should read "the first vertical map".
p. 146 line 10 : "the first row" should read "the second column". p. 146 line 14 : "second row" should read "first column".
p. 146 line 14 : "the vertical maps" should read "the horizontal maps". p. 146 line 15 : "rows" should read "columns" The second diagram in p. 146 should be replaced by the following :
0
0
1
H°(U.y, w(k + )) ®7L/ j,0Z
t
E.
?
H°(Uy, w(k + 2)) 0 Z/1/ Z
1
H°(U.y,w (k+ ) ®7G/p 1Z) 2
En
I H°(U.y,w (k + 1) ® Z/p f3Z)
1
I
H°(U.y, O(D)) 0 Z/p1Z
H°(U.y, 0(D) 0 Z/p Z)
H. HIDA
130
The diagram in p. 147 should be replaced by the following :
0
0
1
H°(UU, w(k + )) ® Z/pAZ
Ea
1
H°(U,,,w(k+ 2) ®7G/p3Z)
2
1
1
H°(UU, w(k + 2)) 0 Z/p' Z
H°(Urx, w(k + 1) 0 Z/ppZ)
1
1
H°(U,,, CA(D)) ® Z/j'Z
H°(U, ,CA(D) 0 Z/p'3Z)
The second formula in (4.1) : "a(n, f I T(q2)) = a(p2n, f) if q I Npa", should read "a(n,.f I T(q2)) = a(g2n,f) if q I
Np°,
p. 149 line 9 from the bottom "Pr(F')-1" should read tar(p)-1"
In the formula of Theorem 3 in p. 153 : "Or(n + m)" should read "V)v (n/ m)".
At several places in pp. 155-157, "Qelil" should read "Qt". In the proof of Lemma 3, (k/2) should read k + (1/2) (thus (k/2) - 1 is replaced by (2k - 1)/2). p. 157 line 5 from the bottom : "it2 # a" should read "/L2 = a".
ON SELMER GROUPS OF ADJOINT MODULAR GALOIS REPRESENTATIONS
131
References (C] H. CARAYOL. - Formes modulaires et representations galoisiennes a
valeurs daps un anneau local compact, Contemporary Math. 165 (1994), 213-237. (CR] C.W. CURTIS and I. REINER. - Methods of representation theory, John Wiley and Sons, New York, 1981.
[DHI] K. Doi, H. HIDA and H. Isi-in. - Discriminant of Hecke fields and the twisted adjoint L-values for GL(2), preprint, 1995. [F] M. FLACI. - Afiniteness theorem for the symmetric square of an elliptic
curve, Inventiones Math. 109 (1992), 307-327. [G] R. GREENBERG. - Iwasawa theory andp-adic deformation of motives,
Proc. Symp. Pure Math. 55 Part 2 (1994), 193-223. [H] H. HIDA. - Elementary Theory of L -functions and Eisenstein series, 1993, Cambridge University Press.
[HI] H. HIDA. - Galois representations into GL2(Z [[X]]) attached to ordinary cusp forms, Inventiones Math. 85 (1986), 545-613. [H2] H. HIDA. - Modules of congruence of Hecke algebras and L -functions associated with cusp forms, Amer. J. Math. 110 (1988), 323-382.
[H3] H. HIDA. - Hecke algebras for GL1 and GL2, Sem. Theorie des Nombres de Paris, 1984-85, 63 (1986), 131-163. [H4] H. HIDA. - On the search of genuine p-adic modular L -functions for GL(n), preprint 1995. [HT] H. HIDA and J. TILOUINE. - On the anticyclotomic main conjecture for
CM fields, Inventiones Math. 117 (1994), 89-147. (M] B. MAZUR. - Deforming Galois representations, in "Galois group over Q", MSRI publications, Springer, New York. (MT] B. MAZUR and J. TILOUINE. - Representations Galoisiennes, diperen-
tielles de Kahler et "conjectures principales", Publ. IHES 71 (1990), 65-103.
132
H. HIDA
[Sch] M. SCHLESSINGER. - Functors ofArtin rings,, Trans. Amer. Math. Soc.
130 (1968), 208-222. [T] J. TILOUINE. - Deformation of Galois representations and Hecke algebras, Lecture notes at MRI (Allahabad, India).
[W] A. WILES. - Modular elliptic curves and Fermat's last theorem, Ann. of Math. 142 (1995), 443-551. [EGA] A. GROTHENDIECK. - Elements de geometrie algebrique N, Publ. IHES,
vol. 20, 1964. Haruzo HIDA
Department of Mathematics, UCLA, Los Angeles,
Ca 90095-1555, U.S.A.
Number Theory Paris 1993-94
Algebres de Hecke et corps locaux proches (une preuve de la conjecture de Howe pour GL(N) en caract@ristique > 0) Bertrand Lemaire
1. - Introduction Enonce de la conjecture. Soient
F un corps local non archimedien de corps residuel fini [rappelons les deux possibilites : ou bien F est une extension finie d'un corps p-adique Q p, ou bien F est isomorphe a un corps de series formelles k((w)) en une indeterminee w sur un corps fini k], OF 1'anneau des entiers de F et PF l'ideal maximal de OF. G le groupe des F-points d'un groupe reductif connexe G defini sur F, muni de la topologie totalement discontinue heritee de F. f(G) l'algebre de Hecke des fonctions G -> C localement constantes a support compact, munie du produit de convolution * induit par une mesure de Haar dg sur G. K un sous-groupe ouvert compact de G, eK = vol(K, dg) -1 iK l'idempotent de 7-1(G) associe a K et 7-1(G, K) = eK * H(G) * eK la sous-algebre des fonctions K-biinvariantes.
Jo C f(G)* 1'espace des distributions AdG-invariantes sur G. S2 une partie compacte modulo conjugaison dans G (i.e. fermee, AdGinvariante et contenue dans AdG(W) pour une partie compacte W de G). JG(Sl) C JG le sous-espace des distributions a support dans Q. CONJECTURE DE HOWE. - Pour tous G, K et Sl comme ci-dessus,
dimd(JG(c )I
'H(c,K)
) < 00 .
Commentaires. Pour F de caracteristique nulle, cette conjecture a ete prouvee par Clozel pour n'importe quel groupe reductif connexe G defini sur F ([Cl 2]).
134
B. LEMAIRE
La conjecture de Howe est le type meme de resultat local de finitude induisant une algebrisation toujours plus profonde de la theorie des representations automorphes (cf. les derniers papiers d'Arthur ou l'ingredient "conjecture de Howe" est presque toujours present). Dans le meme ordre d'idee, it est significatif de constater le glissement progressif d'une analyse harmonique basee sur 1'espace de Schwartz (HarishChandra) vers une analyse harmonique basee sur l'algebre de Hecke l (G) (Bernstein/Kazhdan), meme si 1'espace de Schwartz conserve toute son utilite dans certaines situations (cf. [Cl 3]).
Quid de la caracteristique > 0? Pour F de caracteristique > 0 et G = GL(N), cette conjecture est prouvee dans [Le] Chap. 4; c'est cette preuve que nous presentons ici. Essentiellement deux obstacles s'opposent a la generalisation de la demonstration de Clozel a la caracteristique > 0 :
1. Le nombre de classes de conjugaison de sous-groupes de Cartan, contrairement a la caracteristique nulle, peut etre infini lexemple : si k = Z/27L, it y a une infinite de classes d'isomorphisme d'extensions quadratiques separables de F = k((ra)), chacune fournissant une classe de conjugaison de sous-groupes de Cartan (en l'occurence elliptiques) de GL(2, F)]. On ne peut donc, apres reduction de la conjecture de Howe a une conjecture portant sur les integrales orbitales de G, raisonner comme le fait Clozel en ne s'interessant qu'aux elements appartenant a la trace sZ n r de St sur un sous-groupe de Cartan fixe r de G. 2. L'integrabilite locale des caracteres.
Rappels : on appelle representation lisse de G un triplet Or, G, V) oil V est un C-espace-vectortel (de dimension le plus souvent infinie) et it : G --+ Autc(V) un homomorphisme de groupes tel que pour tout v E V, le stabilisateur G = {g E G, 7r(g) v = v} de v est ouvert dans G. Notant (7r*, G, V*) la representation duale de (7r, G, V) donnee par
(v*EV*, 9EG, vEV), on definit la representation contragrediente (i, G, V) de (7r, G, V) comme la partie lisse de (7r*, G, V*) ; le choix d'un couple (v, v) E V x i7- definit alors un coefficient de 7r w.-
(gEG).
Une representation lisse Or, G, V) est dite admissible si pour tout sousgroupe ouvert compact H de G, V H = {v E V, 7r(h) v = v pour tout h E H} est un sous-espace de dimension finie de V.
ALGEBRES DE HECKE ET CORPS L OCAUX PROCHES
135
Une representation lisse (ir, G, V) induit un homomorphisme d'algebres it : 1-1(G) -> Endc(V) donne par -7r(p) = fG cp(g)7r(g)dg (cp E 1-1(G)). Si (7r, G, V) est admissible, alors l'operateur ir(cp) (cp E 1-1(G)) est a valeur dans
un sous-espace de dimension finie de V, et 1'on peut definir le caracteredistribution O, E JG de it par < O7r, cp > = trace(ir(cp))
(cp E 7-1(G))
.
Pour F de caracteristique nulle, on sait grace a Harish-Chandra ([HC 11) que si (zr, G, V) est admissible et irreductibles (i.e. s'il n'existe aucun sous-
espace propre non trivial de V stable sous l'action de G), alors 0, est une distribution localement integrable au sens ois it existe une fonction 'y,r E Lioc(G) telle que < E), cp > = fG cp(g)-y, (g)dg ((p E 7-1(G)).
Pour F de caracteristique > 0, cette integrabilite locale des caracteres est dans le cas general (c'est-a-dire pour n'importe quel groupe reductif connexe G defini sur F) encore conjecturale. D'ou l'impossibilite d'utiliser la formule d'integration de Weyl, pas plus que le(s) procede(s) de troncature de la trace de Clozel.
Pourquoi se limiter a GL(N)? L'application exponentielle, outil frequemment utilise en caracteristique nulle pour remonter du groupe a son algebre de Lie, West pas d6finie en caracteristique > 0. Mais pour GL(N), on dispose de la miraculeuse application x 1--> 1 + x et des resultats de Bushnell-Kutzko sur 1'entrelacement des strates simples (cf. la section 3 ci-apres). La presence d'elements inseparables (i.e. dont une au moans des com-
posantes irreductibles du polynome minimal a travers un plongement p : G -* GL(d, F) est inseparable sur F), et les phenomenes d'instabilite des orbites a leur voisinage, impliquent de revoir 1'analyse harmonique et tout particulierement la theorie des integrales orbitales - developpee en caracteristique nulle [exemples : degenerescence des orbites inseparables par extension radicielle du corps de base (les orbites inseparables fermees
de G ne sont plus fermees lorsque considerees dans G(F) pour une cloture algebrique F de F) ; absence, pour les elements inseparables, d'une decomposition de Jordan en parties semi-simple2 et unipotente commutant entre elles] ; on a donc tout naturellement commence par nettoyer GL(N), cf. [Le] Chap. 3.
Et meme seulement admissible de longueur finie. 2 On entend par semi-simple un element s E G tel que le polynome minimal de p(s) est produit de composantes irreductibles (non necessairement separables) sur F apparaissant chacune avec une multiplicite 1. 1
136
B. LEMAIRE
Desormais et jusqu'a la fin de 1'expose G = GL(N) (N entier > 2), et soient
KF = G(OF) et KF (m entier > 1) les sous-groupes de congruence modulo PT de KF. BF le sous-groupe d'Iwahori standard de G(F) i.e. l'image reciproque par la projection KF - G(OF/PF) du sous-groupe de Borel de G(OF/PF) forme des matrice triangulaires superieures, et BF (m entier > 1) les sous-groupes de congruence modulo PF de BF.
2. - L'idee : comparer 1'analyse harmonique des groupes G(F) et G(E) pour F de caracteristique > 0 et E de caracteristique nulle. Resultat de Kazhdan sur les corps locaux proches. DEFINITION. - On dit que deux corps locaux non archimediens E et E' sont
m-proches pour un entier m > 1 si les anneaux locaux tronques OE/PE et OE /PE sont isomorphes. TrIEOREME3 (Ma]).
- Soit n un entier > 1. Alors it existe tin entier
m = m(n) > n tel que pour tout corps local E m proche de F, les algebres de Hecke'J-L(G(F), KF) et 7-l(G(E), KE) sont isomorphes. CONJECTURE ([Ka]). - L'entierm(n) = n convient.
Cote galoisien, Deligne, inspire par le resultat de Kazhdan et par la philosophic de Langlands, obtient un resultat du meme genre, quoique beaucoup
plus precis. Notons £(F) la categoric des extensions finies separables de F et £n(F) (irn entier > 1) la sous-categoric pleine de £(F) formee des extensions F'/F telles que le m-ieme groupe de ramification en notation superieure Gal(F"/F)'n de la cloture normale F"/F de F'/F est trivial. `THEOREME' ([De]). - Soit n un entier > 1. Alors la categorie £,,,(F) "ne depend"4 que du (OF/PF, PF/PF+1, EFn), oU.EF,, design le morphisme de OF/PF-modules PF/PF+1 _, OF/PF induitpar l'inclusion PF --> OF par passage au quotient.
Retour ciite automorphe ou 1'on traduit sur les isomorphismes d'algebres de Hecke de Kazhdan la precision obtenue par Deligne cote galoisien. 3 Enonce et montre par Kazhdan pour n'importe quel Z-groupe reductif deploye.
4 Plus precisement, Deligne donne une notion generale de triplet, organise ces triplets en une categoric, puis construit une categorie £(Tr.(F)) des "triplets au-dessus de Trn(F)" et une equivalence de categorie canonique £n(F) -> £(Tr,,,(F)).
ALGEBRES DE HECKE ET CORPS LOCAUX PROCHES
137
Fixons un entier n > 1, un corps local E n-proche de F et un isomorphisme d'anneaux A : OF/PF -i OE/PE. Fixons aussi un jeu d'uniformisantes (tvF, WE) respectivement de F et E, compatible avec A au sens oU A(WF + PD) = WE + P. Via la decomposition de Bruhat-Tits G(F) _ flv, BFwBF ou w parcourt les elements du groupe de Weyl affine Wz(WF) = SN X ZN, on construit aisement un isomorphisme d'espaces vectoriels
C = ((A, WF, WE) : x(G(F), BF) -' x(G(E), BE) PROPOSITION 1. - ( est un isomorphisme d'algebres (et meme d'anneaux).
Preuve : cf. la presentation de 7-l(G(F), BE) par generateurs et relations donnee dans [Ho). L'application ( induisant une bijection entre ('ensemble des generateurs de 1-l(G(F), BF) defini par WF et 1'ensemble des generateurs de H(G(E), BE) defini par WE, it suffit de verifier qu'elle preserve les relations. COROLLAIRE. - Pour tout sous-groupe ouvert compact H de G(F) con-
tenant BF, ( induit par restriction un isomorphisme d'algebres 1-l(G(F), H) - 7-1(G(E), ((H)) ou. ((H) est le sous-groupe ouvert compact de G(E) contenant BE defini par lC(H) = ((1H). Preuve : apres avoir verifie que la partie ((H) ainsi definie est bien un groupe, on conclut grace a 1'egalite H(G(F), H) = eH * 7-l(G(F), BF') * CH-
On a en fait legerement mieux. La donnee du jeu d'uniformisantes (AF, WE) permet de relever l'isomorphisme A en un isomorphisme de groupes A : PF/PF+1 -4 PE/PE+1 defini par A(WFX) = WEA(x) (x E OF/PF ), compatible avec les structures de modules sur OFIPF et OE/PE au sens of A(ax) = A(a)A(x) (a E OF/PF, x E PF/PF+1) et induisant un diagramme commutatif A
0F/P F I
EF, r.
Pr/PF+1
OE II PE I
EE,n.
PEIPE+1
Alors ((A, WF, WE) = ((A, A) Itraduction : la classe d'isomorphisme de 1'algebre de Hecke H(G(F), BF) "ne depend" que de celle du triplet Trn(F)]. Consequence immediate (et seduisante) quant a la theorie des representations : le foncteur (ir, G(F), V) H (7rn, H(G(F), BF), V,,) (ou Vn designe
138
B. LEMAIRE
le sous-espace des vecteurs v E V tels que 7r(b) v = v pour tout b E BF) de la categorie des representations lisses de G(F) dans celle des f(G(F), BF)modules, induit une bijection'7r''--> (('7r') entre 1'ensemble e,a(G(F)) des
classes de representations admissibles irreductibles5 de G(F) ayant un vecteur non nul fixe par BF et 1'ensemble e,, (G(E)) [en fait, on obtient meme
une equivalence entre la categorie des representations lasses de G(F) qua sont engendrees par leurs vecteurs fixes sous l'action de BF et la categorie correspondante pour E ([Le] Chap. 2)]. Comme toute representation lisse de G(F) possede un vecteur non nul fixe par BF pour n suffisamment grand, et comme on peut approcher un corps local de caracteristique > 0 d'aussi pros que l'on veut par un corps local non archimedien de caracteristique nulle Isi F = k((ui)) avec #k = pr, alors toute extension de Qr de degre
residuel r et d'indice de ramification > n, est n-proche de F], on peut resumer la situation en disant que pour.F de caracteristique > 0, la theorie des representations de G(F) est limite des theories des representations de G(E), E extension finie de Q1, telle que CAE/PE =CAF/PF, quand l'indice de ramification absolu e(E/QI,) tend vers l'infini.
3. - Une preuve de la conjecture de Howe pour GL(N) en caracteristique > 0 Le resultat (comme chez Clozel) se montre par induction sur la dimension
des sous-groupes de Levi de G. On procede en trois etapes, les deux premieres (independantes l'une de 1'autre) permettant dans la troisieme de traiter les elements elliptiques, c'est-a-dire ceux intervenant dans le dernier cran de l'induction.
Premiere etape : un relevement uniforme des integrales orbitales elliptiques.
Soient n un entier > 1 et y un element elliptique (i.e. de polynome caracteristique irreductible sur F) de G(F). Notations : soient x E G(F), dg,; une mesure de Haar sur le centralisateur G(F),, de x dans G(F) et cp E 7-l(G(F)). On definit l'integrale orbitale de cp au point x dans G(F) par IG
X (Ir
)
0(g-lxg) dg G(F).\G(F)
'iJx
ou dg est la mesure de Haar sur G(F) arbitrairement normalisee par vol(KF, (1g) = 1. Pour tout x, cette integrale est absolument convergente. Soit G(F)e 1'ensemble (ouvert) des elements elliptiques de G(F).
5 On sait (grace a Jacquet) qu'une representation lisse irreductible est admissible.
ALGEBRES DE HECKE ET CORPS LOCAUX PROCHES
139
Normalisation : si de plus x E G(F)e, soit I G(F) x) l'integrale orbitale au point x def nie par la mesure dg,; sur G(F),; telle que vol(OFIx[, dgy) = 1. THEORGME 1. - It existe un voisinage V = V(y, n) de y daps G(F)e tel
que IG(F) (co x) = I G(F) (p, y) pour toutefonction cp E 7-l(G(F), KF) et tout element x E V.
Quitte a remplacer y par un de ses conjugues, on suppose que y est en position standard, i.e. que le sous-groupe parahorique H de G(F) normalise par F[y] X satisfait la double inclusion BF C H C KF. THEOREME 2. - II existe un entier r = r(y, n) > n tel que pour tout corps
local E r-proche de F et tout isomorphisme d'algebres (: l(G(F), BF) 7-l(G(E), BE) comme daps la Proposition 1, le voisinage V duThEoreme I est BF-btinvariant, la partie t;(V) definie par 1S(v) = t;(1v) est contenue dons G(E)e et I G(E) (((p), x') = I G(F) (p, y) pour toutefonction cp E f-l(G(F), KF) et tout element x' E cp(V).
Principe de la preuve : 1'idee est, grace au Principe de submersion d'Harish-Chandra [Ha 21 applique a I'application partout submersive bp,y : G(F) x P --f G(F), (g, p)
g-1 ygp ou P designe un sous-groupe de Borel de G(F), d'ecrire l'integrale orbitale I G(F) (p, y) d'une fonction cp E 1-1(G(F), Kr) comme combinaison lineaire infinle (indexee par les elements w du groupe de Weyl affine) d'integrales de la forme fJ(F) ap,y(g)cp,,,(g)dg (cp E 1-1(G(F))), puffs de "calculer" cette application ap,y E 11(G(F)) issue du principe de submersion en la cassant en une combinaison lineaire (finie) de fonctions caracteristiques d'ouverts compacts de G(F) (ce qui revient 1
)
a controler l'image des ouverts de G(F) x P par la submersion bp,y); le voisinage V(y, n) du Theoreme 1 comme 1'entier r(y, n) du Theoreme 2, se lisent alors directement sur la formule obtenue. Ces deux resultats "calcul" des integrales orbitales elliptiques et leur transport - constituent la partie la plus delicate de la demonstration, les arguments utilises etant pour 1'essentiel puises dans le travail de Bushnell-Kutzko [BKI, cf. ci-dessous. Soit x un element irreductible (i.e. de polyni me minimal irreductible
sur F) de G(F). Alors F[x] = L est une extension de F contenue dans g = M(N, F) et l'on note b C g le commutant de L dans g. Identifiant de maniere standard g et b avec leur dual de Pontrjagin, on appelle corestriction
moderee daps g relative a L/F un homomorphisme de (b, b)-bimodules s : g -f b realisant la restriction des caracteres6. c'est a-dire tel que OFotr9/F(gb) = 'IL 0trb/L(s(g)b) ((g, b) E g x b) pour des caracteres additifs
6 Pour nous, un caractere est simplement un homomorphisme continu dans C><.
140
B. LEMAIRE
OF : F -> Cx et 'OL : L -> CX de conducteurs respectifs PF et PL. Si C est
un QF-ordre hereditaire dans g normalise par L", alors s(G) = b fl 9 est un OL-ordre hereditaire dans b et la classe x + 9k (k E Z), appelee strate de g et notee [G, -vg(x), -k, x] ou vg designe la "valuation" sur g induite par les puissances du radical de Jacobson de 9, est une strate simple si x minimise le degre des extensions F[g]/F (q E X + gk, irreductible). Dans cette situation, on a une relation explicite entre la corestriction moderee s et l'application adjointe [x, ] : g -> g, relation permettant a Bushnell-Kutzko de calculer l'entrelacement {g E G(F), g-1 (x + Gk)g fl (x + gk) # 0} de la strate [9, -vg(x), -k, x] ; en particulier si x E G(F)e, cet entrelacement est compact modulo le centre et contenu dans le normalisateur de !9 dans G(F). C'est ce calcul de 1'entrelacement des strates simples elliptiques que nous utilisons pour controler l'image des ouverts de G(F) x P par 1'application 5p,, puis pour transporter la formule obtenue de G(F) a G(E).
Deuxieme etape : une majoration du resultat de Clozel independante de la ramification. Soient n un entier > 1, q = pT, El/QT une extension non ramifiee de < aN) degre r, et F une famille finie de suites ordonnees (a) = (al < (ai E Z). Pour chaque extension finie E/El, on definlt la partie XE,.Y = de G(E) on (WE designant une quelconque uniformisante de E) (a) = dlag('G7a E l , ... , WI Li7E EN)
((a) E ZN)
Ti ORI ME 3. - It existe une constante c = c(q, n, F) > 0 telle que pour toute extension fine totalement ramifiee E/El et toute partie 12E de G(E) fermee AdG(E)-invariante et contenue dans XE,,F,
dill/JG(E)(QE)I x(G(E) KE)) < C. Principe de la preuve : on reprend le(s) article(s) de Clozel en controlant
a chaque etape de la demonstration que la majoration quit obtient ne depend pas de ]a ramification. L'idee centrale de la preuve de Clozel consiste a casser 1'expression trace(7r(cp)) (ou Tr est une representation admissible
irreductible de G(E) et cp E f(G(E)). Les deux versions qu'il donne de sa demonstration correspondent a deux troncations differentes de cette expression. Dans la premiere version (IC! 1]), it coupe cette trace en "trace elliptique" et "trace non elliptique" ce qui l'amene a considerer toutes les (classes de) representations elliptiques7 de G(E), ceci bien que la ramification des fonctions considerees (i.e. le niveau n du sous-groupe de congruence KR) soit fixee, et donc a travailler modulo une hypothese conjecturale 7
Une representation admissible irreductible x de G est dite temperee s'il
ALGEBRES DE HECKE ET CORPS LOCAUX PROCHES
141
de finitude des exposants speciaux de la serie discrete8; cette hypothese a ete verifiee par Clozel pour le groupe lineaire ([Cl 11) [notons qu'en theorie des formes automorphes, cette hypothese de finitude correspondrait au fait que les poles des series d'Eisenstein construites a partir des formes cuspidales relativement a un sous-groupe parabolique donne, appartiennent a un ensemble fini, independant de la forme cuspidale induisante ; c'est une conjecture tres forte, pas meme verifiee pour le groupe lineaire]. Dans la deuxieme version ([Cl 21), la trace est coupee suivant des termes indexes par les classes de conjugaison de sous-groupes paraboliques de G(E) et corres-
pondant a la stratification de Deligne-Casselman; Clozel etablit ainsi une formule remarquable exprimant la trace complete trace(7r(cp)) en termes des "traces compactes" des modules de Jacquet de 7r relatifs a un systeme de representants des classes de conjugaison de sous-groupes paraboliques de G(E) (nous utilisons en fait la version duale de cette formule, montree par Clozel dans [Cl 31, exprimant la trace compacte de 7r(cp) en terme des traces completes des modules de Jacquet de 7r relatifs a ces memes sousgroupes paraboliques). Nous raisonnons essentiellement sur la seconde troncature de Clozel, utilisant neanmoins la propriete de finitude des exposants speciaux de la serie discrete puisqu'on est amenes a compter (grace a la parametrisation de Zelevinski) les orbites du groupe des caracteres non ramifies (i.e. les caracteres triviaux sur le groupe G(E)o = {g E G(E), det(g) E OE}) de G(E) agissant par torsion sur ('ensemble des classes de representations elliptiques de G(E) ayant un vecteur non nul fixe par KE.
Troisieme etape. On suppose F de caracteristique > 0. Soient K un sous-groupe ouvert compact de G(F) et fl une partie compacte modulo conjugaison dans G(F).
Soient n un entier > 1 tel que Kr C K et F une famille finie de suites ordonnees (a) = (al < < aN) (ai E Z) telle que S2 C existe un caractere w de G tel que les coefficients de ]a representation tordue w & 7r sont dans 1'espace de Schwartz. Une representation temperee 7r de G est dite elliptique si son caractere-distribution O7 West pas identiquement nul sur l'ouvert des elements elliptiques de G. 8 Une representation admissible irreductible 7r de G est Bite essentieUe-
ment de carre integrable s'il existe un caractere w de G tel que les coefficients de la representation tordue w 0 7r sont de carre integrable modulo le centre de G. La serie discrete de G est ]'ensemble des classes de representations essentiellement de carre integrable de G. Noter que pour GL(N), une representation est essentiellement de carre integrable si et seulement si elle est elliptique (Jacquet). Pour 1'hypothese sur les exposants speciaux, on renvoie a la Definition 1 de [Cl 11.
142
B. LEMAIRE
On raisonne par induction sur la dimension des sous-groupes de Levi de G. On suppose la conjecture de Howe vraie pour tous les sousgroupes de Levi standards M de G (elle est trivialement vraie pour le tore diagonal A0, une partie compacte modulo conjugaison dans A0(F) etant tout simplement compacte). La propriete bien connue de descente des integrales orbitales non-elliptiques, jointe a un argument de descente des parties compactes modulo conjugaison dans G(F), entrainent que les fonctionnelles lineaires cp a--> I G(F) (cp, x, dgx) (x E S2 non elliptique, cp E f(G(F), KF)) engendrent un sous-espace vectoriel de di-
mension finie du dual de ?-l(G(F), KF). Grace aux etapes 1 et 2, on montre (petit raisonnement par 1'absurde) que les fonctionnelles lineaires cp G(F) (p x)dg,) (.x E 1 elliptique, W E f(G(F), KF)) engendrent elles aussi un sous-espace vectoriel de dimension finie du dual de
t-, I
l(G(F),KF). On conclut grace a la propriete de densite des integrales orbitales I G(F) (cp, x, (Ig,) (x E 1) dans 1'espace JG(F) (St), lequel s'obtient grace aux resultats de Gelfand-Kazhdan en rangeant les nappes de Dixmier de G(F) par dimension d'orbites croissante, cf. [Lei Chap. 3.
4. - Conclusion L'integrabilite locale des caracteres en caracteristique > 0, si elle reste conjecturale dans le cas general, est desormais montree pour le groupe lineaire (cf. [lei Chap. 5, a paraitre dans Compositio Math.). On peut des lors suivre d'encore plus pres la demonstration de Clozel et montrer cette conjecture de Howe pour GL(N) en caracteristique > 0 directement, c'est a-dire sans passer par la comparaison des analyses harmoniques [la demonstration, non encore redigee, repose alors sur le theoreme de densite des caracteres-distributions des representations temperees dans 1'espace
JG(F), dont la preuve ne depend pas de la caracteristique du corps de base[. Cette seconde approche se pretant nettement plus facilement que la premiere a une generalisation a n'importe quel groupe reductif connexe, it est naturel de commencer par essayer de montrer 1'integrabilite locale des caracteres dans le cas general. A suivre donc.
Manuscrit recu le 22 mars 1994
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Bibliographie [BK] C.J. BUSHNELL et P.C. KUTZKO. - The admissible dual of GL(N)
via compact open subgroups, Ann. of Math. Studies 129, Princeton U. Press, Princeton, New Jersey, 1993. [CI 1 ]
L. CLOZEL. - Sur une conjecture de Howe I, Compositio Math. 56 (1985), 87-110.
[C12] L. CLOZEL. - Orbital integrals on p-adic groups : a proof of the Howe conjecture, Ann. of Math. 129 (1989), 237-251.
[Cl 3] L. CLOZEL. - The fundamental lemma for stable base change, Duke Math. J. 61 (1990), 225-302.
[De] P. DELIGNE. - Les corps locaux de caracteristique p comme limites de corps locaux de caractenstique 0 dans Representations
des groupes reductifs sur un corps local, Travaux en cours, Hermann, Paris (1984), 119-157. [Ha 11 HAPISH-CHANDRA. - Admissible invariant distributions on reduc-
tive p-adic groups, Queen's Papers in Pure and applied Math. 48 (1978), 377-380. (Ha 21 HAPISH-CHANDRA. - A submersion principle and its application in
Papers dedicated to the memory of V.K. Patodi, Indian Academy of Sciences, Bangalore, and the Tata Institute for Fundamental Research, Bombay (1980), 95-102.
[Ho] R. HOWE. - Harish-Chandra homomorphismfor p-adic groups, CBMS Regional Conf. Series in Math. 59, Amer. Math. Soc., Providence, Rhode Island, 1985. [Ka] D. KAZHDAN. - Representations of groups over close localfields,
J. Analyse Math. 47 (1986), 175-179. [Le] L. LEMAIRE. - These, Univ. de Paris-Sud, 8 fevrier 1994. Bertrand LEMAIRE
Universite Paris-Sud Departement de Mathematiques Batiment 425 91405 ORSAY CEDEX
Number Theory Paris 1993-94
Aspects experimentaux de la conjecture abc Abderrahmane Nitaj
1. - Introduction Si n est un entier non nul, le produit de ses differents facteurs premiers,
note r(n), sera appele radical de n (certains auteurs utilisent le terme support, conducteur ou noyau), avec par convention r(1) = 1. La conjecture
abc de J. Oesterle et D.W. Masser est liee a la notion de radical. Cette conjecture date de 1985 et est devenue maintenant classique. CONJECTURE (abc). - Soit s > 0. II existe une constante c(e) > 0 telle que,
pour tout triplet (a, b, c) d'entiers positifs verifiant a + b = c et (a, b) = 1 on ait : c < c(e)(r(abc))'+E, oil r(abc) est le radical de abc.
La conjecture abc a des consequences tres etonnantes. Elle implique en particulier le theoreme de Fermat asymptotique (i.e. vrai pour les grands exposants), le theoreme de Faltings We Conjecture de Mordell) et s'applique
bien aux equations diophantiennes. On peut trouver certaines de ses applications dans [2,3,5,6, 11,12,13,14,15,16,17,19]. L'inegalite de la conjecture abc implique que, pour tout triplet d'entiers positifs (a, b, c), verifiant a + b = c, (a, b) = 1 etc > 3, on a : log c
log r(abc)
<1+s+
log c(e) log c(s) <1+s+ log r(abc) log 6
Ainsi, le rapport (1.2)
a = a(a, b, c) =
log c
log r(abc)
A. NITAJ
146
est borne. Elle implique de meme que le rapport p = p(a, b, c) =
(1.3)
log abc
log r(abc)
est borne car a(a, b, c) < p(a, b, c) < 3a(a, b, c). Experimentalement, les plus grandes valeurs connues pour les rapports (1.2) et (1.3) proviennent respectivement d'un exemple de E. Reyssat (1987), et d'un exemple de I'auteur (1992) : (1.4)
J 2 + 310.109 = 235 l 13.196 +2 30 .5 = 313.112.31,
a = 1, 62991, p = 4, 41901.
Le meilleur resultat prouve provient d'un travail de C.L. Stewart et K. Yu, qui ont etablit le theoreme suivant : THEOREMS 1.5 (Stewart-Yu). - Soite > 0. It existe une constante cl (e) > 0
telle que, pour tout triplet (a, b, c) d'entiers positifs verifiant a + b = c et
(a,b)=1,onalt: log c < cl (e)
(r(abc))213+e
.
L'inegalite de ce theoreme est cependant assez loin de celle de la conjecture abc.
Dans la partie 2 de cet article, on etudie les deux rapports (1.2) et (1.3), et dans la partie 3, on decrit des algorithmes qui permettent de "tester" numeriquement la conjecture abc. Ces resultats numeriques nous permettront de proposer des conjectures plus faibles que la conjecture abc, mais dans lesquelles les constantes sont explicites.
2. - Etude des rapports (1.2) et (1.3) Les inegalites (1.1) impliquent que les rapports (1.2) et (1.3) verifient : (2. 1)
11I11 Slip, cO a(a, b, c) = 1,
lim SupC_O. p(a, b, c) = 3,
ou les limites sont prises pour des triplets d'entiers positifs verifiant a+b = c et (a, b) = 1. En effet, le theoreme de Mahler assure que
limes,- r(abc) = +oo. En particulier, les limites (2.1) montrent qu'il ne peut y avoir qu'un nombre
fini de triplets d'entiers positifs (a, b, c), verifiant a + b = c, (a, b) = 1 et admettant des rapports (1.2) ou (1.3) respectivement superieurs a 1, 62991 ou a 4,41901, et donc meilleurs que les exemples (1.4). D'autre part, independamment de la conjecture abc, nous demontrons la proposition suivante.
ASPECTS EXPE°RIMENTAUX DE LA CONJECTURE abc
147
PROPOSITION 2.2. - Pour tout reel ao > 0, it n'y a qu'un nombrefini de
triplets (a, b, c) d'entiers positifs uenfiant a + b = c, (a, b) = 1 et tels que a(a, b, c) = ao.
Preuve : soient (xi, yi, zi), i = 1, 2, deux triplets d'entiers positifs differents verifiant pour i = 1, 2, xi + yi = zi, (xi, yi) = 1 et tels que a(xi, yi, zi) = ao. On pose ri = r(xiyizi), i = 1, 2. Si rl = r2, alors zl = z2 et 1'equation x + y = zi n'admet qu'un nombre fini de solutions telles que r(xyzl) = rl, ou rl est fixe. Supposons maintenant que rl 54 r2. Soit (x3i y3i z3) un triplet d'entiers positifs verifiant X3 + y3 = Z3, (x3i y3) = 1 et tel que a(x3, y3, z3) = ao. Soit r3 = r(x3y3z3). Si r3 = ri, i = 1, 2, alors z3 = zi. Supposons donc que r3 # ri, i = 1, 2. Ainsi log z1
log rl
_
log z2
log r2
_
log z3
log r3 = ao,
et ao n'est pas rationnel puisque r(zi) 0 ri, i = 1, 2, 3, en excluant le triplet trivial (1, 1, 2). Alors, le theoreme de Lang (voir [20], p. 51) implique
que log rl, log r2 et log r3 sont Q -lineairement dependants. Soient ai,
i = 1, 2, 3 trois entiers positifs non nuls tels que (al, a2, a3) = 1 et a3 log r3 = al log r1 + a2 log r2. Ainsi r33 = r"r22 r" et ri, i = 1, 2, 3, est sans facteurs carres. Si (ri, r2) = 1 alors a3 = al = a2 et donc r3 = rlr2. Si (rl, r2) 0 1, alors a3 = al + a2, a3 = ai, avec i = 1 ou i = 2, et donc r3 = r2 ou r3 = rl, ce qui est impossible. Ainsi, pour ao donne, it n'existe au plus que trois families, finies, de triplets (a, b, c) d'entiers tels que a(a, b, c) = ao. Chaque famille est representee par le meme radical r(abc).
Remarques : 1) Si la conjecture de Lang, appelee conjecture des quatre exponentielles
(voir [20], p. 59) est vraie, on peut montrer que pour chaque reel ao it ne peut y avoir au plus qu'une seule famille, finie, de triplets (a, b, c) d'entiers, ayant le meme radical et tels que a(a, b, c) = ao. 2) Une famille de triplets interessante est la suivante. Pour un entier n > 2, on pose : an = 2(2n-1 - 1),
bn
= (2n -
1)2,
xn = 2n - 1,
yn = 2n+1(2n-1 - 1).
Alors an + bn = xn + yn, r(anbn) = r(xnyn) et donc a(an, bn, an + bn) _ a(xn, yn, xn + yn)
3) Voici d'autres exemples de triplets qui ne sont pas de la forme cidessus, ayant le meme rapport (1.2), avec :
148
A. NITAJ
- Deux representants : 1 + 2.3 = 22 + 3 = 7; - Trois representants : 3 + 22.5 = 5 + 2.32 = 23 + 3.5 = 23; - Quatre representants : 2 + 3.52 = 2.52 + 33 = 25 + 32.5 = 7.11; - Cinq representants : 25.5.7 + 32 = 23.33.5 + 72 = 22.52 + 3.73 = 210 + 3.5.7 = 1129.
Nous allons maintenant montrer que les triplets (a, b, c), d'entiers positifs verifiant (a, b) = 1, a+b = c et pour lesquels a(a, b, c) > 1 et p(a, b, c) > 3
sont nombreux. Soit (X, Y, Z) un triplet d'entiers verifiant (X, Y) = 1, X# Y, X + Y = Z et Z pair. Ecrivons X = Ax3, Y = By3 et Z = Cz3 Of, A, B et C sont les parties sans facteur cube de X, Y et Z. Definissons les suites (xn), (yn) et (zn) par x0 = X010 = y, zo = z et par les relations de recurrence : d:Cn+1 = xn(Ax3a + 2By3), (2.3)
dy't+1 = -yn(2Ax,3,, + By3),
dzn+1 = z,z(Axn - By3),
ou d designe le pgcd des trois termes de droite. PROPOSITION 2.4. - Soient (xn), (yn) et (zn) les suites defines par (2.3).
Alors pour tout n >_ 0, Axn + By3 = Czn, (Axn,Byn) = 1 et 23lzn. En particulier les trois suites sont infinies.
Preuve : par definition des suites (xn), (y.,,) et (zn), on a Axo + 3) = 1. Supposons ceci vrai pour les terByo = Cz0 avec (AX3 mes xn, Yn, zn et que 2nlzn. Alors, par les relations (2.3), on a Axn+1 + By3+1 = Czn+1 Puisque (xn+1, yn+1, z3+1) = 1 et C est sans facteur cube, alors (xn,+1, yn+1) = 1. On a de meme (xn+1, zn+1) = (yn+1, zn+1) _
1. Ainsi (Axn+1, By3+1) = (A, y3+1)(x3+1, B). Si p est premier et si pI
(Axn+1,
By3+1). alors pI Cz3+1 et pl (A, y3+1) ou pI (xn+1, B), ce qui n'est
pas possible. Soit d comme clans (2.3). On a By3.))
= (Axn
(xn(Axn + 2B y3), yn(2AX3 +
+
2Byn, 2Ax3n + By3)
= (3Ax3 n, 3By3n)
Alors d = 1 ou d = 3. Comme Axn et By3 sont impairs, alors 2I (Axn - By3) et donc 2n+1Izn+1. La conclusion de la proposition decoule du fait que les
termes de la suite (zn) ne s'annulent jamais. On pose maintenant pour tout n > 0 : an = min (IAx3I, IBy3l, ICz3I) (2.5)
cn = max
(IA-C31,
bn=en - an,
JBy3I, ICz3I)
,
149
ASPECTS EXP$R/MENTAUX DE LA CONJECTURE abc
PROPOSITION 2.6. - Soient (x,,,), (y,) et (zn) les suites defines par (2.3). Alors it existe un entier no tel que pour tout n > no, on ait : p(an,bn,Cn) > 3.
a(an,bn,cn) > 1,
Preuve : pour tout n > 0, les triplets (an, bn, cn) verifient an + bn = cn et (an, bn) = 1. Alors r(anbncn) = r (I ABCII xnynzn13) et
r
(2ABcxvIit)
(IABCIIxnynzn13) =r
< IABCxnynznl/2n_1
Soit no > 1 un entier tel que IABCI/2n°-1 < IABCI1/3. Alors pour tout
n>no,ona:
r(anbncn) < I ABCI1/3IXnynznl = (anbnCn) 1/3 < Cn,
et donc a(an, bit, Cn) > 1 et p(an, bn, cn) > 3.
La constante c(e) de la conjecture abc agit comme un terme correcteur. Nous pouvons utiliser les suites definies par les relations (2.3) pour montrer que la condition e > 0 est obligatoire dans la conjecture abc. PROPOSITION 2.7. - Soit e ----> c(e) une application verifiant la conjecture abc. Alors
1imE.o c(e) = +00.
Preuve : On reprend les suites (xn), (yn) et (zn) definies par les relations (2.3). Soit no un entier tel que IABCI /2no-1 < IABCI113. Appliquons la conjecture abc aux triplets (an, bn, cn) definis par (2.5) avec n > no : 1+E
Cn C c(E)
Alors c(e) >
(r((LbnCn))1+E
c,-,E2(n-n°)(I+E)
< c(s) (2
n°
.
et donc
1lnllnf c(E) > 2"` -no E-+0
ce qui implique que limE,o c(E) = +oo.
3. - Recherche de bons triplets pour la conjecture abc. Nous dirons qu'un triplet (a, b, c) d'entiers positifs verifiant a+b = c, (a, b) = 1 est bon pour la conjecture abc si l'un de ses rapports a(a, b, c) ou
150
A. NITAJ
p(a, b, c) est assez grand par rapport aux valeurs conjecturales 1 et 3. Ceci ne peut se produire que si c ou le produit abc est assez grand par rapport au radical r(abc). Nous resumons dans cette partie une etude amplement detaillee dans [9, 10].
Soit n > 2 un entier et soient A > 0, B
0 et C > 0 des entiers
premiers entre eux deux a deux. Notre recherche de bons triplets pour la conjecture abc va etre basee sur la resolution de 1'equation diophantienne :
Ax'n-By''=Cz,
(3.1)
avec (y, C) = 1. Cette equation a des solutions si et seulement si la congruence
Ate' =- B (mod C),
(3.2)
a une solution t avec 0 < It) < C/2. Les entiers a, b et c dont on calculera les rapports a(a, b, c) et p(a, b, c) seront pris parmi IAxnl, IBynI et ICzi, en les reduisant par leur pgcd et pour les quels I zI est assez petit. La recherche de solutions pour 1'equation (3.1) avec IzI = 1 est basee sur les theoremes suivants. THEOREME 3.3. - Soient B < 0 et n pair. Si (x, y, 1) est une solution de ('equation (3.1) avec (y, C) = 1, alors U existe une solution t de (3.2) auec 0 < Itl < C/2 et une reduite u/y de la fraction continue de t/C teUes que
x = ty - Cu. Supposons maintenant B > 0 ou n impair. On peut alors definir les quantites : b = (B/A)1/n, 2n It. i
=
1/(n-2)
-1 ) Anon(_1
Pour un reel x, [x] designe sa partie entiere.
THEOREME 3.4. - Soient B et n deux entiers tels que B > 0 ou. n est impair. Soit (x, y, ± 1) une solution de l'equation (3.1) et soit 1
E=
cos-(27r [(n -l)/21)
si xS > 0, si xh < 0.
Sin = 2 et AB > 4 ou si n > 3 et y > yo, alors it existe une solution t de (3.2) avec 0 < Iti < C/2 et une reduite u/y de la fraction continue de (t - Eb) /C
telles que x = ty - Cu.
ASPECTS EXPERIMENTAUX DE LA CONJECTURE abc
151
Les theoremes (3.3) et (3.4) nous permettent d'ecrire deux algorithmes
pour chercher de bons triplets pour la conjecture abc. Pour cela, fl faut choisir des entiers A, B et C tels que r(ABC) soit petit par rapport a IABCI. On peut choisir par exemple IABI petit et C de la forme pe oil p est premier et ou e est un entier assez grand. Les deux algorithmes ont le meme principe :
Determiner les solutions t de la congruence (3.2) en utilisant une des differentes methodes connues (voir [4, 22]).
Determiner les reduites u/y de (t - A) IC oii A = 0 si B < 0 et n pair et A = Eb sinon. Poser ao = A (ty - Cu)', b° = -Byn et co = a° + b° et les diviser par leur pgcd. Poser a = min (Iaol, Ibol, lcol), c = max (Iaol, Ibol, Icol) et b = c - a. Calculer a(a, b, c) et p(a, b, c) par (1.2) et (1.3). Si Fun de ces rapports est assez grand, enregistrer le triplet (a, b, c). Nous avons applique ces deux methodes dans les cas suivants :
1) n = 2, 1 < A < CBI < 300, C = pe, ou p < 31 est premier et e est entier verifiant pe _< 2f0 pour B < 0 et pe < 290 pour B > 0.
2) n = 3, 5, 1 < A < B < 200, C = pe, ou p < 31 est premier et e est entier verifiant pe < 240. Cette recherche a donne 86 triplets (a, b, c) verifiant a(a, b, c) > 1, 4 sur 115 connus et 103 autres triplets verifiant p(a, b, c) > 3,8 sur 140 connus.
La table (3.6) liste les triplets connus avec a > 1.49 et la table (3.7) ceux avec p > 4, 00, suivant les ordres decroissants de a et p. On peut utiliser une autre methode pour chercher de bons triplets pour la conjecture abc. Cette methode repose sur la resolution de 1'equation diophantienne lineaire : (3.5)
Ax - By = Cz,
ou A > 0, B # 0, C > 0 sont premiers entre eux et ou Ixyzl est petit par rapport a IABCI. Cela peut se faire en determinant les reduites u/y
de (at - b)/(ac) avec t - ba-1 (mod C), 0 < Itl < C/2 et en prenant x = ty - CU. On peut augmenter les chances de determiner de bons triplets en prenant par exemple A = pi' , B = pZ2 et C = p33 ou pour i = 1, 2, 3, pi est un nombre premier et ou ei > 1 est assez grand.
La recherche de bons triplets pour la conjecture abc, basee sur la determination des petites solutions de 1'equation (3.5) nest pas completement achevee. Elle a permit cependant la decouverte de certains bons triplets (notes "N. (1994)" dans les tables (3.6) et (3.7)).
Dans les deux tables ci-dessous, les auteurs des differents exemples sont designes par leurs initiales :
A. NITAJ
152
B.-B. : J. Browkin et J. Brzezinski, G.. X. Gang, M.-R.: P. Montgomery et H. to Riele, N. R.
A. Nitaj,
E. Reyssat. W. B.M.M. de Weger.
Table 3.6 a
b
c
a
Auteur
2
310.109
235
112
32.56.73
221.23
19.1307
7.292.318
28.322.54
1,62991 1,62599 1,62349
283
511.132
28.38.173
R. (1987) W. (1985) B.-B. (1992) B-B, N. (1992) W. (1985) W. (1985) N. (1994)
1
2.37
54.7
73
310
2".29
72.412.3113
1116.132.79
2.33.523.953
53
20.317.132
115.17.313.137
13.196
230.5
313.112.31
318.23.2269
173.29.318
210.52.715
239
58.173
210.374
52.7937
713
218.37.132
22.11
32.1310.17.151.4423 213.77.9412
59.1396 316.1033.127
73
1,58076 1,56789 1,54708 1,54443 1,53671 1,52700 1,52216 1,50284 1,49762 1,49243 1,49159
R-M. (1994) N. (1992) N. (1994) B.-B., N. (1992) W. (1985) N. (1992) N. (1992)
ASPECTS EXPERIMENTAUX DE LA CONJECTURE abc
153
Table 3.7
a
b
c
13.196
230.5
313.112.31
25.112.199
515.372.47 711
37.711.743 311,53.112
327.1072
515.372.2311
Auteur p 4,41901 N. (1992) 4,26801 N. (1994)
4, 24789 W. (1985) 4,23069 N. (1994) 210.5 2 715 4,22979 N. (1994) 173.29.318 318.23.2269 4,22960 N. (1994) 174.793.211 229.23.292 519 117.372.353 4,22532 N. (1994) 25 .3.7 13 514 19 4, 20094 N. (1992) 72.116.199 2.138.17 321 27.1910.79 4,14883 N. (1994) 32.476.733 518.6359 113.315.101.479 1078 231.34.56.7 4,13000 N. (1994) 311.54 217.173 7.116.43 4,10757 G. (1986) 2'.79.283 4,09700 N. (1994) 36.511.41 1310.53 4,09655 N. (1992) 216.41.71 315.72 197 79.312 29.115.571 4,09647 N. (1992) 312.56 215.52.372 4,09080 N. (1992) 78.19 3.177 312.7.134 242 4,08331 N. (1994) 795.677 224.35 5.195.592 710.167 4,07114 N. (1992) 335.233 2.112.1074.359.20947 53 7s 2911 4,07038 N. (1994) 19.47.716 321.1932 27.512.1272 4,06347 N. (1994) 2310 36.1573.283 230.52.112.13 4,05990 B.-B., N. (1992) 232.32 78.173 11.135.232.31 4,05301 N. (1994) 213 313 113 13.29.436.673 4,04710 N. (1992) 520.17 217.193.23 513.13 317.283 4,04498 N. (1992) 37.514.72 251.112 295.73.4192.1039 4,03039 N. (1994) 229.13 32.57.79 117.192 4,02943 N. (1992) 25.37 3 36.1912 .594. 19603 717.112.71 4,02904 N. (1994) 241 972 738.2347 32.55.1110.43.61 4,01847 N. (1994) 2.59 314 75.11.47 4,01342 N. (1992) 237 320.853 52.76.29.1912 4,01312 N. (1994) 111f.132.79 72.412.3113 2.33.523.953 4,00968 N. (1994) 336.19 22.73.895.347.997 74.2311 4,00764 N. (1994) 222.11.43 32.177 72.415 4,00751 N. (1994) 210 1910 56.134.295 320.4425749 4,00292 N. (1992) 211.114.132.23 325 73.295.137 4,00238 N. (1994) 317.72 225.241 53.114.312 4,00087 N. (1992) 219.13.103 235.72.172.19
A. NITAJ
154
Plusieurs autres methodes ont ete mises au point pour la recherche de bons triplets pour la conjecture abc, avec plus ou moins de succes. - En 1985, de Weger [21], a donne dans sa these une premiere table, en resolvant 1'equation x + y = z avec r(xyz) fixe. - En 1992, Brzezinski et Browkin [1] ont utilise une methode basee sur la determination des reduites des nombres dl/n pour d et n donnes, en prenant les entiers a, b, c parmi IxnI, lyndl et IX'n - dyne. - En 1992, Elkies et Kanapka ont teete tous les triplets (a, b, c) avec
c < 232 et tentent de mener leur recherche en poussant la borne a 240 (communication privee).
- En 1994, Montgomery et to Riele ont cherche de bons triplets en utilisant une variante de 1'algorithme LLL (Communication privee). Malgre toutes ces recherches, les exemples (1.4) restent les meilleurs pour les rapports (1.2) et (1.3). Il est a noter que les exemples (1.4) cor-
respondent a des triplets (a, b, c) dans lesquels c est relativement petit a l'interieur du domaine explore. D'autre part, le theoreme de Mason (voir [8]) empeche 1'existence de bonnes families infinies pour la conjecture abc, qui
soient parametrables polynomialement. Ceci nous amene tout naturellement a proposer: CONJECTURE 3.8. - Soit (a, b, c) un triplet d'entiers
vdnfiant
a+b=cet(a,b) = 1.Alors: C<
r(abc)1,63
abc <
r(abc)4,42
Les tables (3.6) et (3.7) nous renseignent aussi sur 1'expression de la constante c(e) de la conjecture abc. Pour en donner une estimation, it faut tenir compte de la proposition (2.7), mais aussi du theoreme suivant (voir. [7]).
ThEOREME 3.9 (Masser). - Pour tout S > 0, it existe un triplet (a, b, c) d'entiers posittfs venfiant a + b = c, (a, b) = 1 et tel que abc, > r3exp{ (12 - 6) (log r) 112 (log log r) -1 },
od r = r(abc). Ce theoreme implique en particulier que 1'inegalite de la conjecture abc ne peut pas etre de la forme c < kir(abc) log r(abc)k2 oil k1 et k2 sont des constantes. Par consequent, la constante c(e) ne peut pas etre de la forme c(e) = (1/e)k3 oU k3 est une constante.
Manuscrit recu le 20 decembre 1994
ASPECTS EXPERIMEN"TAUX DE LA CONJECTURE (ahe
155
BIBLIOGRAPHIE
[1] J. BROWKIN and J. BRZEZINSKI. - Some remarks on the abc-conjecture,
Math. Comp. 62 (1994), 931-939. 12] N.D. ELKIES. - ABC implies Mordell, Intern. Math. Res. Notices 7 (1991),
99-109. [31 G. FREY. - Links between elliptic curves and solutions of A - B = C, Number Theory, Ulm 1987, Lect. Notes in Math. 1380. [41 K. HARDY, J.B. MUSKAT and K.S. WILuAMS. - A deterministic algorithm
for solving n = fca2 + gv2 in coprime integers u and v, Math. Comp. 55 (1990), 327-343. [5] S. LANG. - Old and new conjectured diophantine inequalities, Bull. Amer.
Math. Soc. 23 (1990), 37-75. [61 M. LANGEVIN. - Cas d'egalite pour le theoreme de Mason et applications de la conjecture (abc), C. R. Acad. Sci. Paris, t. 317 (1993), 441-444. [71 D.W. MASSER. - Note on a conjecture of Szpiro, Asterisques 183 (1990), 19-23. [8] R.C. MASON. - Diophantine equations over Function Fields, LMS Lecture Notes 96, Cambridge University Press 1984. [91 A. NITAJ. - An algorithm for finding good abc-examples, C. R. Acad. Sci.
Paris, t. 317 (1993), 811-815. [10] A. NITm. - A lgorithms for finding good examples for the abc and the Szpiro
conjectures, Experimental Math. 2 (1993), 223-230. 1 111 A. NITAJ. - La conjecture abc, L'Ens. Math. A paraitre.
[121 J. OESTERLE. - Nouvelles approches du theor@me de Fermat, Seminaire Bourbaki, 1987-88, no 694. (Asterisque, vols. 161-162, 165-186) Paris, Soc. Math. Fr. 1988.
1 131 M. OVERHOLT. - The diophantine equation n! + 1 = m2, Bull. London Math. Soc. 25 (1993), 104. [ 141 D. RICHARD. - Equivalence of some questions in mathematical logic with some conjectures in Number Theory, Number Theory and Applications, R. A. Mollin ed. , NATO-ASI Ser. (C-265), Kluwer, (1989), 529-545.
156
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[151 W.M. SCHMIDT. - Diophantine Approximations and Diophantine Equations, Lect. Notes in Math. 1467, Springer-Verlag 1991. [161 J.H. SILVERMAN Wieferich's criterion and the abc-conjecture, J. Number
Theory 30 (1988), 226-237. [171 C.L. STEWART and R. T[JDEMAN, On the Oesterle-Masser conjecture, Monatsh. Math. 102 (1986), 251-257. [ 18] C. L. STEWART and K. Yu. - On the abc-conjecture, Math Ann. 291 (1991),
225-230. 1191 P. Vo.ITA. - Diophantine Approximation and Value Distribution Theory, Lect. Notes in Math. 1239, Springer-Verlag 1987. [201 M. WALDSCHMIDT Nombres Transcendants, Lect. Notes in Math. 402, Springer-Verlag 1974.
[211 B.M.M. de WEGER. - Algorithms for diophantine equations, CWI Tract, Centr. Math. Comput. Sci. , Amsterdam 1989. 1221 K.S. WILLIAMS and K. HARDY. - A refinement of H.C. Williams'qth root
algorithm, Math. Comp. 61 (1993), 475-483. Abderrahmane Nitaj Universite de Caen Departement de Mathematiques 14032 Caen Cedex (email : [email protected])
Number Theory Paris 1993-94
Heights of points on subvarieties of G,',;, Wolfgang M. Schmidt*
1. - Introduction Beginning with the deep 1992 work of Shouwu Zhang [ 14[, much work has recently been done on the height of algebraic points on Zariski-closed
subsets V of G n, where G,n = G,n(C) is the multiplicative group of the complex numbers. Let h(x) be the absolute logarithmic height (whose
definition will be recalled in Section 5) of an algebraic number x. Let A,,,, = Grn(Q) be the multiplicative group of algebraic numbers. A very special case of Zhang's results is that when (x, y) E A2, is on the line (1.1)
x + y = 1,
but is not of the type (w, w2) where w is a sixth root of 1, then (1.2)
h(x)+h(y)>_c1>0
with an absolute constant cl. Then Zagier [13[ gave an elegant proof of this
inequality and determined the best possible constant : cl = 2 1og(2 (1 + /5)). Next, Schlickewei and Wirsing [ 11 [ used a method inspired by Zagier's to show that the algebraic solutions to (1.1) are in some sense well spaced with respect to the height function, and this in turn [9[, [10[ led to bounds c2 (r) depending only on r for the number of solutions of (1.1) with (x, y) in a subgroup r of Gin of rank r. From now on, a point x will be understood to lie in G;,, but whenever heights are involved, it will be understood to lie in A; ,. Given a point
x = (xi.... , xn), set n,
hy(x) _
h(xi). i=1
Supported in part by DMS-9401426.
W. M. SCHMIDT
158
Given two points x, y set b(x, y) = h.(xy-1),
with the product and inverse taken in A,'. Then (i) 6(x, y) > 0, with equality precisely when xy-1 E U', where U is the group of roots of 1, (ii) b(x, y) = 6(y, x), (iii) b(x, z) b(x, y) + b(y, z). Therefore b is a semidistance on A , and it induces a distance on A; L/Un. Given E > 0, call x, y neighbors of distance < e if b(x, y) < e. Carrying the ideas of Zagier, and of Schlickewei and Wirsing further, the author [12] showed that for certain curves V C Qv 2,,, there are constants
q > 0, e > 0 depending on V in a rather simple way, such that a point
x E V has at most q neighbors of distance < E in V. Very soon afterwards, Bombieri and Zannier [ 1 ] went even further. They gave an elementary proof of theorems of Zhang [ 141, [15] on heights of points on varieties V C G; , with bounds depending only on n and the degree of V. They showed in particular that for a certain subvariety Va of V, there are numbers q > 0,
E > 0, such that a point x E G has at most q neighbors of distance < e in V\Va. Here q = q(n, d) and E = E(n, d) when V is defined by polynomial
equations of degree 5 d. Zagier [13] and the author [12] had used the involution x H x-1 but Bombieri and Zannier [1], inspired by work of Dobrowolski [4], used the isogeny x --* xr in their arguments, where p is a sufficiently large prime number.
Whereas the constants q(n, d) and E(n, d) are explicitly computable by the method of [1], Bombieri and Zannier did not carry out such a computation, and indeed some estimates obtained by their method would involve n exponentiations. Our goal here will be to obtain explicit and representable (although still quite large or quite small) values for all the constants, and to make some other improvements. In order to achieve this, we will avoid as much as possible automorphisms of G; , resultants and induction on n. In contrast to [1], we will work both with the partial degrees, as well as the total degree of a polynomial. A polynomial in n variables of total degree d has up to (1.3)
N(d)
(n+ dl
< (2n)r1
nonzero coefficients.
An algebraic subgroup of G Tit is a subgroup which is an algebraic variety, i.e., it is Zariski-closed. A torus is an algebraic subgroup which is
HEIGHTS OF POINTS ON SUB VARIETIES OF G.vn,
159
(absolutely) irreducible as a Zariski-closed set. By coset we will understand a coset gH where H is an algebraic subgroup. A torus coset will be a coset
gH where H is a torus. A torsion coset will be a coset uH where u is a torsion point of G , i.e., u E Un. Now let V C G 6 be an algebraic variety. A coset gH contained in V will be called a maximal coset in V if there is no
coset gH' with gH # gH' C V. Let V" (H) be the union of all cosets gH contained in V, and Vl (H) the union of all maximal cosets gH contained in V. Similarly define V" (H) and VV` (H) as the union of all torsion cosets uH contained in V and the union of the maximal torsion cosets uH contained in V. Finally set
V" =
VU(H)
I
V' = UV'A (H), H
H
dim H>0
the unions being over algebraic subgroups H of G"'. We will suppose throughout that
n?2. THEOREM 1. - For given d, there are algebraic subgroups H1, ... , depending only on n and d, each a union of at most (2d)n tort, and with (11d)n2,
m <
(1.4)
having the following property. When V is defined by polynomial equations of total degree <_ d, then In
(1.5)
V12 =
912
V"(Hi) = U i=1
dim H,>0
712
(1.6)
U i=1
V1 (Hi),
dim,Hi>O
In
V,1 = U V92(Hi) = U Vi (HH) i_1
i=1
THEOREM 2. - Let V C G; be defined by polynomial equations of total degree < d. Then (i) Each V"(H) is Zariski-closed and defined by polynomial equations of degree <_ d.
(ii) When V is defined by equations with rational coefficients, then each V"(H) is defined by equations of degree <_ d with rational coefcients.
160
W. M. SCHMIDT
NO Each VV` (H) is the union of fewer than
exp(4N(d)!)
(1.7)
torsion cosets uH. (iv) When V is defined by equations with rational coefficients, then each V1 (H) is the union of fewer than exp(3N(d)3/2 log N(d))
(1.8)
torsion cosets.
In Laurent [6] and again in Bombieri-Zannier [1] it had been shown that V(H) is the union of at most c(d, n, [K : Q], M) cosets uH when the defining polynomials of V had coefficients in a number field K and had heights < M. In Section 5 we will define a certain height H'(f) for nonzero polynomials f with algebraic coefficients. THEOREM 3. - Suppose V C G u is defined by polynomial equations ft = 0 (2 = 1, ... , t) with total degree <_ d, with coefficients in a normal number field K of degree k, and with heights H'(fl) < M (Q = 1, ... , t). Then every x E V \V" has
h.(x) > 1/(dkN(dk). e3"r')
(1.9)
where
M1 = k! MO.
In particular, M1 = M when K = Q. We will also point out how Sieve Methods can be used to replace (1.9) by the alternative estimates hs(x) > co(N(dk))M1 3N(dk)-10
(1.91)
where co(N(dk)) depends only on N(dk) (hence only on n, d, k), and (1.9")
It, (x) > exp(-3000N(dk) log2 N(dk))Mj 300oN(dk)
THEOREM 4. - Suppose V C G,,, is defined by polynomial equations fe = 0 t) of total degree < d. Then every x E V\Va with at most q := exp((4n)2dN(d))
(1.10) exceptions,
(1.11)
has It,, (X) >
HEIGH75 OF POEMS ON SUB VARIETIES of G,n,,
161
COROLLARY. - A pointy E G n has at most q neighbors of distance < e in
V\V°.
For when x E V\V then z=xy-1 E y-1V\y-1V y-1V\(y-1V) and by Theorem 4 applied to y-1V, all but at most q points z in y-1V\(y-1V)" have hs(z) > e. THEOREM 5. - Let V be as above. Let r C An be a group containing at most r multiplicatively independent elements. Then given C >_ 1, there are at most q(gC)r
(1.12)
points x E r fl V\V" with h,,(x) < C. Of particular interest is the case (I = 1. The variety V then is of the type n
(1.13)
E aeYxi = 0
(?. = i, ... , t)
i=O
where we have set xo = 1, and where x = (X1, ... , xn) E G; When P is a partition of {0, 1, ... , n}, let Hp consist of the points x having xi = xj for any i, j lying in the same subset A of T. Then Hp is a torus, and it turns out (see Section 3) that in Theorem 1 we may replace {H1,... , H,n} by the set of groups Hp where P is any partition. There are <_ (n + 1)n+l such groups. A coset 9H,,j) consists of points x having xi = gix), (for 1 <_ i <_ n, i E A and A a set of P) with arbitrary xa (A in 9)), except that xa = 1 for the set A containing 0. Further V"(Hp) consists of points x having
E aeixi. = (l
V = 1, ... , t)
iEA
for every set A of the partition. Since dim Hp > 0 except when 9) is the trivial partition with the only set A = {0, ... , n}, we see that V\V° consists of x E V such that there is no subset A 0 of {1, ... , n} such that
2a,ixi=0 iEA
(P.=1,...,t).
Solutions X E V\V" are usually called non-degenerate. When (1.13) is a single non-homogeneous equation, Schlickewei [8) proved that there are at most 24(n+1)? non-degenerate solutions x E U.
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W. M. SCHMIDT
In the present situation N(d) = n + 1, and since n ? 2, Theorems 4, 5 hold with q = 4o(n)
exp((4n)3n).
2. - Lattices and Algebraic Groups In our usual pedantic manner we begin by recalling basic facts about algebraic groups H C G G. When X = (X,,. .., X9L) is a variable vector and
i = (ill ... , in E Z ,set
X''=X1'...Xn. Similarly define xi when x is a point. Given i, the map x '- xi is a character X on Gn,. Let el = ( 1 , 0, ... , (l), ... , e9,, _ (0, ... , 0,1) be the standard basis
vectors. When T E GL(n, Z), say Te, _ (all, ... ,
ai91), let TT be
the map
Gn1. -' GM defined by vT (x)
(x
T(el) ,
... , x T(e,,) ) = (:C1all
aln 1 ... 1 and 1
C71
xnann)
When also o E GL(n,Z), then cp,T(x) = cpT W, (x). It is now clear that TT
is an automorphism of G. When V C Gnn is Zariski-closed, then so is cpiV, and when V is irreducible (as a Zariski-closed set), then so is cpTV. For algebraic subgroups Hl, H2 of Gn, write Hl ^, H2 if H2 = cpTHl for some r E GL(n, Z). When a = (al, ... , (1"71) E.Zn, then T(a) = al r(el) +
+ a91T(e9L), so
that
xr(a) = ( T(e1))a1 ... (cPT((x))a. Applying this with T-1, T(a) in place of T, a, we obtain (2.1)
xa = XT 'T(a) = (7'T-1(x))T(a).
By lattice we will understand a subgroup of Z. A full lattice is a subgroup of rank n. When A is a lattice, let S(A) C IRn be the space spanned by A, and set A = S(A) fl Z. Then A is a lattice containing A, and the index p(A) of A in A is finite. The lattice is called primitive if A = A, i.e., if p(A) = 1. When A is a lattice, let HA consist of x E Gn with xa = 1 for every a E A. Then HA is an algebraic variety defined over Q. In fact it is an algebraic subgroup of G; , since (xy)a = xa ya. In view of (2.1) we have (2.2)
7'T-1HA = HT(A).
HEIGHTS OF POINTS ON SUF3VARIETIES OF G ;u
163
LEMMA 1. - Suppose A is a lattice of rank r. Then
HA=FxM'-r,
(2.3)
where the product x of groups is direct, F is finite of order p(A), and
MI-1 -
Hn_T where Hn-' consists of x with x1 = = xr = 1. Therefore HA has p(A) irreducible components, and in particular HA is a torus precisely when A is primitive. When B is a lattice of rank r containing A, we have (2.4)
HB = F' X Mn
where F' is of order p(B), and is contained in F. Remark. F is not uniquely determined by A, but Mn_T is, consisting of elements m of HA which are "divisible" by any integer q # 0, i.e., in = m1 where (mi E HA.
Proof : pick rr E GL(n,, Z) such that r(A) consists of the vectors (b', 0) where b' E Z' and 0 is the origin in Z,-r. Then T(A) = (A', 0), where A' is a full lattice in Z` with p(A') = p(A). Thus HT(A) consists of
(x', x") E G;t x G,-- = ((D;;;,, with x' E HA, and x" arbitrary in Hn-' Since A' is full in Zr, the group HA, is easily seen to be finite of order .
p(A') = p(A). We have HT(A) = HA, X Hn-T. Therefore by (2.2), HA is of the type (2.3). The other assertions of the lemma follow easily.
LEMMA 2. - The map A H HA sets up a byection between lattices and algebraic subgroups of G;;,,.
Proof : we begin by showing that the map is injective. Suppose HA = HB. Then when x E HA, we have xa = 1 for every a E A, further x6 = 1 for b E B, therefore x° = 1 for every c E A + B. Therefore HA C HA+B. Since the reverse inclusion is obvious, HA = HA+B. It now follows from Lemma 1 that rank A = rank (A + B) and p(A) = p(A + B). Therefore A = A + B, so that B C A. By symmetry A = B. It remains to be shown that every algebraic subgroup H equals HA for some lattice A. Since H is Zariski-closed, this will follow from Lemma 4 in the next section. LEMMA 3. - Let A, B be lattices. Then (2.5)
HAHB = HAnB
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W. M. SCIIMIDT
Proof : we first will show that a product HH' of algebraic groups H, H' in G; is an algebraic group. Since HH' is obviously a group, we need only show that it is Zariski-closed. Since each group H or H' is a finite union of torus cosets, it will suffice to show that a product of two such cosets is closed, hence suffice to show that a product of two tori is closed. In fact we will show that the product of two tori H, H' is a torus. By Lemma 1,
H = r(H") with r E GL(n, 7G) and some b, so that H consists of points (yb. , ... , with fixed b1, ... , b,, E Zb, and where y runs through G. Similarly, H' consists of points (z`1, ... , z'-) where z runs through G,`,.L, say. Therefore HH' consists of points (y61 Z'', ... , ybnz°n). It will suffice to show that when g > 0 and dl,. . . , d,, E Z9 are arbitrary, the set T of points
(dl w
n w d") E Girt
as w runs through G9,, is a torus. Let A consist of vectors a = (ai,... , a,,.) E Zn having aldi+ +andn = 0. Then A is a primitive lattice, and T C HA. We claim that T = HA. Without
loss of generality, dl,.
. .
,
d, are independent, and we have relations
adw=a;,ld1+...+(ti,d,. (r 0. As w runs through G9,, then z = (wd1, ... , Wd,.) runs through G;,,. Now T is "divisible" by every a > 0, so that it consists of elements x" with x E T, i.e., of elements (2.6)
a (zl,...
z,.a,z
ar+1.1
y(4,.+1.,. ... -7, ,...,ziand ... Zram )
with z E G n. The vectors
a lattice B C A with b = A. Since T contains the points (2.6), we have HB C T. Therefore HA C HB C T, where p = p(B) is the index of B in A. Since HA, being a torus, is divisible by p, indeed HA C T. The proof of the lemma is now completed as follows. We have already shown that HAHB is an algebraic group. It is the smallest group containing both HA and HB. The 1-1 correspondence between lattices and algebraic
groups reverses inclusion relations. Since A fl B is the largest lattice contained in both A and B, the assertion (2.5) is established.
HEIGHTS OF POINTS ON SUB VARIETIES OF G; ,
165
3. - Proof of Theorem 1 Let V C G;n be Zariski-closed and defined by polynomials fl V _ 1, ... , t). We may write (a1jX i
ft =
(2
= 1, ... , t),
iE I
where I is a finite set of n-tuples i. Let D(I) be the difference set, consisting
of points i - j with i, j E I. An algebraic subgroup H C V is called a maximal subgroup contained in V if there is no subgroup H' with H C H' C V. LEMMA 4. - Let H be a maximal algebraic group contained in V. Then H = HA where A is generated by vectors in D(I).
Proof : we follow Laurent (6] and Bombieri-Zannier (1]. Given i, the restriction of the map x --f x' to H is a character Xi on H. When x is any character on H, let Ix be the set
Ix={iEI with Xi=X}. On H we have the relations (Lei
X = 0 (?. = 1, ... , t).
x
By Artin's Theorem on linear relations on characters, these must be trivial relations, so that ati = 0
(3.1)
(.Q = 1 , ... , t; a n y
).
iEIx
For X E H, (3.2)
x' = x3 if i, j E Ix for some X.
Therefore H C HA, where A is the lattice generated by the vectors i - j with i, j E Ix for some X. When X E HA, then (3.2) implies that xi has a common value for every i E Ix, say the value xx, and then
fe(x) = E E afi X
iEIx
xx = 0 (Q = 1, ... , t)
166
W. M. SCHMIDT
by (3.1). Therefore HA C V. Since H was maximal, H = HA. When V is defined by equations of total degree <_ d, we may take
I = 1(d), where I(d) consists of n-tuples of nonnegative integers whose sum is < d. Therefore I C dI where I consists of the points >; E R' with L1-norm < 1.
Then D(I) C dD(I) = 2d1. A lattice A with generators in 2d1 does not necessarily have a basis in 2d1. However, when rank A = r, say, then there
is a basis al, ... , ar with aj E max(1, j/2) - 2d1 (j = 1, ... , r) ; see, e.g., Lemma 8 in 12, Ch. V]. So when ,n > 1 and A is of any rank, it is generated by n vectors in ndl. The number of integer points in ndl with nonnegative components is N(nd). Considering the 2n quadrants, the number of integer points in ndl is < 2nN(nd) = 2n
nd +n n
< 2n(2dn,)n/n! < (4ed)n < (11d)n,
since n! > (n/e)n. The number of choices for A then is < (11d)n2. In view of Lemma 4, every subgroup HA obtained in this way is in a set {H1, ... , H.} where m satisfies (1.4). Further when A of rank r has r linearly independent vectors in 2(I, these vectors will have Euclidean norm <- 2d, hence will span a parallelepiped of r-dimensional volume <- (2d)r < (2d)n. Therefore (3.3)
p(A) S (2d)n,
and HA is the union of at most (2(l)n tori. LEMMA 5. - Every algebraic subgroup H C V is contained in a maximal algebraic subgroup contained in V.
Proof : the argument for Lemma 4 shows that H C HA C V where A is generated by vectors in D(I). Since D(I) is finite, there will be a group HA which is maximal in V. The proof of Theorem 1 follows immediately : every coset xH C V has H C x-1 V, hence by Lemma 5 is contained in a coset xH' C V where H' is maximal in x-1V. Thus H' is among the groups H1,. .. , H. constructed above.
When V is defined by linear equations, I consists of the n + 1 vectors eo := 0, el,... , en, hence D(I) of the differences eti - ej. Given a lattice A generated by some vectors from D(I), write i - j if ei - ej E A. Then HA consists of x =...,xxn) ,,(x1, having xi = xj when i - j, where we have set xO = 1. The equivalence relation - defines a partition P of {0, 1,
. .
. , n},
HEIGHTS OF POINTS ON SUB VARIETIES OF G-
167
and HA = Hp consists of x E G;;,, having xi = xj when i, j lie in the same set of the partition. This verifies a claim made in the Introduction.
4. - Proof of Theorem 2 (i) By Lemma 1, H = F x r(H't-') with r E GL(n, Z), so that H consists of elements vr(y) where v E F, y E H". When V is defined by polynomial equations ft, = 0 (f = 1,... , t), then xH C V precisely if (4.1)
(P = 1, ... , t)
.ft(xvT(y)) = 0
f o r every v E F and every y = (1, ... , 1, y,.+1, .... yn) E H". Each expression on the left hand side is a polynomial in y,.+1 i ..., yn. The coefficients of
these polynomials are certain polynomials g* in xv (s = 1,... , t'), and we need that (s = 1, ... , t') g* (xv) = 0 for every v E F. But this means that x has to satisfy certain polynomial equations (4.2)
g+,,,(x) = 0
(rn = 1,
. . . ,
t").
V a (H) consists precisely of the points x satisfying the equations (4.2), which are clearly of total degree < d. (ii) Now the polynomials fI defining V and the polynomials gs introduced in (i) have rational coefficients. Consider a particular such g* = g*, say, and the system of equations (4.3)
g*(xv) = 0
(v E F).
When V E F is fixed, (4.3) implies the system g*(vk'x) = 0
(k E Z).
The components of v are powers of some root of 1, say powers of where ( is a primitive rn-th root of 1. The automorphisms of Q(() are the Q-linear maps aj with ak;(c) = (k where (k, m) = 1. Applying these automorphisms to the coefficients of f (X) := g* (vX) we obtain the polynomials f (k) (X) = g* (vk`X ). Therefore (4.3) implies that x not only
satisfies f (x) = 0, but also the q(ni) conjugate equations f (k) (x) = 0. These equations are equivalent to a system of gy(m) equations with rational coefficients. Clearly all these equations have the same degree as g*.
Before commencing with the proof of parts (iii), (iv), we need to insert the following.
W. M. SCI-IMIDT
168
LEMMA 6. - Consider a system of linear equations n
(2 = 1'...'0.
aP.iui = 0
(4.4)
i=1
A solution u = ( u 1 , . .. , un) will be called non-degenerate if there is no subset
Iof{1,...,n}with 0
aeiui = 0 iEI
Then up to proportionality, (4.4) has at most 16(n) :
24n'
non-degenerate solutions whose components are roots of 1. In the special case when the coefficients in (4.4) are rational, O(n) may be replaced by r. (n)
exp(2n312 log n).
Proof : let us first deal with the case of general coefficients. The case t = 1 is due to Schlickewei 181 (note that Schlickewei considers an inhomogeneous equation in n variables, which corresponds to a homogeneous equation in n + 1 variables). The general case is easily deduced from the case t = 1 . For let U 1 ,- .. , UN be distinct non-degenerate solutions. For
each Iwith 0
has E (tpi'(tnci iEI
0.
We therefore may pick c1, ... , ct such that
t=1
iEI
for every I with 0 < III < n and every m, 1 <_ m <_ N. Then ul, ... , uN are non-degenerate solutions of the single equation lliui = 0 i.-1
HEIGHTS OF POINTS ON SUBVARIETTES OFQv-
169
with bi = E_1 ceaei. By Schlickewei's result N V) (n). Now consider the case of rational coefficients at. By the argument given above it will suffice to deal with the situation when t = 1. By Conway and Jones [3], up to a factor of proportionality, the ui will be m-th roots of 1 where m is of the type m =
p1p2...P1
with distinct primes pi,... , pt having e
1:(pi-2)
(4.5) i=1
Now when pl <
< pe, then pi > 2i - 1, so that 1
e,
(4.6)
J:(pi i=1
e2 - 2t, 1:(2i - 3) = i=1
- 2) >_
and (4.5) yields P. < /+ 1 < 2 Vn-. Since each pi S n, we have m < exp(2n1/2log n) [by the prime number theorem, m < exp((l+E)(n log n) 1/2) when n > no (e), but we are interested in a simple and explicit estimate]. Given m, our equation has up to proportionality at most mom'-1 solutions (ul, ... , 71n) - Therefore the number of non-degenerate solutions is < exp(2n3/2 log n,) = rc(n).
Let P be a partition of {1, . (4.49))
. .
E (pi'ai=0
,
n}. Consider the system of equations
(f=1,...,t; AET),
iEA
where A E P signifies that A is a set of the partition 9). Let 8(P) be the set of solutions u E U'n of (4.4) which satisfy (4.4 P), but not (4.4 Q) for any refinement Q of T. Every solution of (4.4) lies in some (not necessarily unique) set S(O).
Consider the equations (4.4 P) for fixed A E P and i? = 1,...J. By Lemma 6, there are up to proportionality at most b(JAI) solutions in roots of unity ui (i E A) for which the sums over a set A' C A with 0 < (AEI < JAI do not all vanish. Denote these solutions by }iEA, where 1 < pa < q,\, with 0 <_ qa < ''(CAI). Thus, since a factor ua of proportionality is allowed, the solutions are {uaui1'' }iEA.
170
W M. SC!!MIDT
This can be done for every A E T. The solutions in 8(P) therefore are of the type (4.7)
when i E A and A E P,
ui =
where ua (A E P) are arbitrary and 1 <_ pa < qa (A E P). Let 8(P, {pa}) for a given system {pa}aEp be the solutions of (4.4) of the type (4.7) with the given values of pa. The number of systems {pa } is V)(n),
f (la < 11,O(JAI) AE'P
AE`.P
since Vi(a)V)(b) < 0(a + b). The number of partitions P is < nn. Every solution u E Un of (4.4) lies in one of the sets 8(P, {pa}), and there are (4.8)
< n'''(,I,l) = n'
2411! < 257d
< exp(4n!)
such sets. We now are ready to continue our proof of Theorem 2. (iii) We have to show that for given H, there are fewer than (1.7) torsion cosets uH which are maximal in V. Now uH C V precisely when u satisfies the equations (4.2). Write
9?n(X) =
b11e.iXi
(In = 1, ... t )>
iEI (d)
so that uH C V precisely when (4.9)
E b1,t.iu' = 0
(rn = 1, . .
.
,
tit).
iEI(d)
Set ui. = ui; then {ui}iEI(d) satisfies the system of linear equations (4.10)
L b7nvai = 0
(in = 1, ... , t").
iEI((I)
This is exactly like the system (4.4), except that the set { 1, ... , n} has been replaced by I(d), and the subscripts i. by i. This time, by (4.8), the solutions fall into strictly less than (4.11)
exp (411(d) j!)
I-IEIGIITS OF POINTS ON SUBVARIETIES OFGr,
sets 8(P, {2)a}). Let
171
and O,a}A p be given; solutions in this set are
7i z
when i E A and A E
,
where the uA (A E P) are arbitrary. We have to go back from (4.10) to (4.9). We have to find u E U' such
that u' _ via i E A and A E P, where the ua are arbitrary. Possibly there is no solution for a given P and {pa}AEP. But if there is a solution u, the general solution will be uh where
h' for i E A depends only on A. In other words, hi = hi if i Z j, where P is the equivalence relation induced by T. Thus h E HB, where B is the lattice generated by the vectors i - j with i j. Therefore if for given P, {pa} there is a solution u at all, the general solution will be
uu, with u' E HB n U". But in fact for any h E HB, the point uh will be a zero of the polynomials so that uhH C V. But then uHBH C
V, where HBH is an algebraic group by Lemma 3. Since we demanded that uH C V be maximal, HB C H. Therefore all the cosets uu'H with u' E HB n U11 will be the same. Therefore for given P. {pa}, there is at most one maximal torsion coset uH. The number of maximal torsion cosets uH C V" therefore is bounded by (4.11). Since II(d)j = N(il), (1.7) follows.
(iv) By part (ii), uH C V precisely if u satisfies certain equations g,,,(u) = 0 where each g,,,, has rational coefficients and is of total degree < d. We proceed as for (iii), but with it in place of zli, so that (4.8) is replaced by (4.12)
< n",t(n) = exp(n log n + 2n312 log n) < exp(3n312 log n),
and (4.11) is replaced by (4.13)
exp(31I((1)1312 log II ((_I) 1).
172
W. M. SC!IMIDT
5. - Heights When K is a number field, let MK be its set of places, and for v E MK let Iv be the absolute value belonging to v, normalized such that it extends the
ordinary or a p-adic absolute value of Q. Further set lixliv = xlk-,k where k is the degree of K and k the local degree associated with v (caution : our II
'
II is denoted by I . I in [1]). The product formula
j1
IIaIIz, =1
v E A-iK
holds for a E K'. Given x E K', set
H(x) = jI max(1, Ilxii,,) vEMK
This height, the absolute multiplicative height of x, is independent of the field K in which x is embedded. The absolute logarithmic height of x is h(x) = log H(x). By the product formula, (5.1)
h(x) =
max(0, log llxll,,) = I log IIxIIvI 1: 2 vEMK vEMK
One has h(x) > 0, with equality precisely when x E U, one has h(1/x) =
h(x) and h(xq) < h(x) + h(y). For points x, y with nonzero algebraic coordinates set again it
It., (X) _ L h(xi),
[-1
6(x, Y) = h (xy-1). Then 6 clearly has the properties (i), (ii), (iii) stated in the Introduction. We will now introduce a slightly different height, and we apologize to the reader for the inconvenience! When X E K', set milx(I:1' I
Ixi,, = I
I, .
)
when v is finite, when v is infinite,
1:1,11. IV)
Ix1I,,+ "' + Ix,,I,,
and Ilv = Ixi "/k . Further when x 4 0, set a11X
He(x) = fi lixliv. .,,E MK
HEIGHTS OF POINTS ON SUBVARIETIES of G ,
173
Then H(x) is independent of the field in which the coordinates of x are
embedded. By the product formula, H(ax) = H'(x) when a # 0 is : x,L) in projective algebraic, so that H' depends only on the point (x1 : space F"-1. When f is a nonzero polynomial with coefficients in K, define If 1,,, If 11. and H'(f) in terms of the coefficient vector. When we substitute x into f , where f has partial degrees :: d, we obtain 12
If I" fl MaX(1, IX, 1")d.
If (X) I",
(5.2)
i=1
6. - Auxiliary Lemmas When 0) is a partition of a set, let ated with T.
.P
be the equivalence relation associ-
LEMMA 7. - Suppose x1, ... , x,, are nonzero. Suppose they satisfy a system of equations 7L
a&x'=0 for 0<_j
(6.1) i=1
Then there is a partition 9) of { 1, ... , n } such that
xi=xj
if
P i^3
and
Eati=0 for AEP, 1<_B<_t.
(6.2)
iEX
Proof : let 9) be the partition with i - j precisely when xi = xj. For A E P. write xa for the common value of xi with i E A. The equations (6.1) become (6.3)
> Ixj =0 for 05jSn-1, 1
iEA
Sin ce the xA with A E P are distinct, the Vandermonde determinant IX 2\I
with A E P and 0 < j <- API - 1, where ITI is the number of sets A E P,
is nonzero. Then (6.3), applied with 0 < j < ITI - 1, shows that each "coefficient" is zero, i.e., that (6.2) holds.
174
W. M. SCHMIDT
LEMMA 8. - Let f f # 0 be a polynomial with coefficients in Q, with partial
degrees <_ d, and of height H' (, f) . Let P > H' (f) be a prime number. Then every algebraic point x with
f (x) = 0,
f (xI) r 0
has (6.4)
1 log(p/H'(f)) - pd
h., (x) >_
Proof : this is essentially Lemma 1 of [ 1 ]. For completeness we will give a proof. We may suppose that f has coefficients in Z, and these coefficients have no common factor > 1. By Fermat's Theorem,
f'' (x) = .f (x") + pg(x), where g(X) E Z[X ] has partial degrees 5 pd. Since f (x) = 0 by hypothesis,
f(x") = -pg(x)
(6.5)
Set a = f (x"). When v 11) we have
In Ialy <
Pd
Hmax(1,lxi1,,)
If Iv
i=1
by (6.5) and since If I, = 1 in view of our convention on the coefficients of f. For the other v's we have it,
Ial1, < J Inax(1, lx,) '" I l.f I, i=1
by (5.2). If we raise these inequalities to the exponent kv/k, take the product over v E MK, observe that or = f (xI') 54 0 satisfies the product formula,
and that Ro llplly = p-1, we obtain pd
7b
1
1
(J-JH) (xi) a
H'(f),
1
therefore
hs(x) _
h(xi,) i.=1
pd1log(p/H'(f))
HEIGHTS OF POINTS ON SUBVARIETIES OF G-
175
LLEMMA 9. - Let f be as above. Let m be an integer all of whose prime
factors are
> eH'(f). Let x be a point with (6.6)
f(xm)#0.
f(x)=0,
Then
hs(x) > 1/(md). Proof : clearly m > 1. We will use induction on the number of prime factors of m. When m is a prime, the result follows from (6.4). When m is not a prime, write m = prrn' where 1, is a prime. Now if f (x"`) # 0, our induction
gives hs(x) > 1/(m'd) > 1/(md). But if f (x"L) = 0, we apply Lemma 8 to x", obtaining hs(x'n) > 1/(pd), so that hs(x) = (1/m')hs(xm') > 1/(md). We will need arithmetic progressions of natural numbers (6.7)
a, (L + b,..., (,t + (n - 1)b
whose members have no prime factor below some quantity r, and are small in size. When a = 1,
b = fir, p
there are indeed no prime factors p < r in (6.7). Moreover, b = exp(L9(r)) where 19 is Chebyshev's function, which satisfies 19(r) < 1.017r (see, e.g., [7)). Therefore b <- ei.oirr We therefore obtain the almost trivial LEMMA 10. - There is an arithmetic progression of length n of natural numbers less than (6.8)
n e1.0 7r
all of whose prime factors exceed r. Deeper results may be obtained by Sieve Methods. Professor Halberstam kindly pointed out to me that (6.8) may be replaced by (6.8')
C3(n)r3''
H10
On the other hand, following a suggestion by Professor P.D.T.A. Elliott, I
could use the explicit sieve estimates in Chapter II of his treatise [5) to obtain the explicit bound (6.8")
exp(90n log2 2n) r3ooo"
176
W. M. SCHMIDT
No doubt this can be much improved. No proofs of (6.8') or (6.8") will be given here.
7. - Proof of Theorem 3 when K = Q Let V C G' be defined by polynomial equations ft = 0 (1 = 1,. .. , t) with rational coefficients, of partial degrees < d and heights H'(fe) < M (t = t) Set r = [eM]. By Lemma 10, but with n replaced by N(d), there is an arithmetic progression
a,a+b,...,a+ (N(d) - 1)b whose members are positive integers < e1.o17rN(d), all of whose prime factors exceed eM. We may suppose that gcd (a, b) = 1. Let X E V, so that ft(x) = 0 (P. = I,, , t.). Case (i) fe(xa+it,)00
for some f in 1 S P <-- t and some j in 0 <-_ j <- N(d) - 1. Since the prime factors of a + jb exceed eM, and since a + jb < e1.o17' N(d) < e3MN(d), Lemma 9 yields h, (x) > 1/(d' e3MN(d)), so that (1.9) holds with k = 1.
If instead of (6.8) we had used (6.8') or (6.8"), the members of our arithmetic progression would have been bounded respectively by C3(N(d))7,3N(d)+10
< C4(N(d) )M3N(d)+10
or (since N(d) > n + 1 ? 3) by exp(90N(d) log2(2N((L)))r3000N(d) < exp(2900N(d) log2 N(d))M3oooN(d),
giving rise to (1.9') or (1.9") with k = 1. Case (ii) .fe(x"+a')=0
for
1 <-P.
So if ft = Ei aeiX', say, we have Eaei(xi)(s+j"=0
i
(1 S?.
177
I-IEIGHTS OF POINTS ON SUBVARIETIES OF Qua-
which is the same as
beix =0 with
(1
ia b..i = ati(x) ,
xi = (X") b The sums are over i E I(d). Lemma 7, applied with N(d) in place of n, gives a partition P of I(d) such that
(x")" when i Z j,
(7.2)
and
bei =0 (A E P, 1 < P < t).
aLi(x'')a
(7.3) iEA
iEA
By (7.2) we have
x' = x,\ui for i E A, A E P, where xa depends only on A, and u' = 1, so that ui is a b-th root of 1. It lies in the field U,, generated by the b-th roots of 1. Now (7.3) may be rewritten
as (7.4)
>
0
E T, 1
iEA
Since gcd (a, b) = 1, there is an automorphism of Ub,/Q which sends b-th roots of 1 to their a-th power. Since the coefficients aIi in (7.4) lie in Q, we obtain
Eati'(6i=0
(AEP, 1
iEA
(this step is not needed in the construction of Lemma 10, for there we had a = 1). We may infer that (7.5)
>aIixi=O
(A E P, 1<-P<_t).
iEA
-
Let B be the lattice in Z generated by the vectors i - j with i j, and A the lattice generated by the vectors bi - bj with i Z j. Then A = bB C B and HB C HA. By (7.2), x E HA. Expressing HA, HB as in (2.3), (2.4), we see that (7.6)
x E uHB
178
W. M. SCHMIDT
for some u E F C U'. Suppose Y E uHB, so that y = xh with h E HB. There are numbers hA(A E P) such that hi = h,, when i E A. Then
Eany' =ha1: iEX
ca,ix''=0
(A E P, 1<_$<_t)
iEA
by (7.5). We may conclude that fP(y) = 0 V = 1, ... , t), so that y E V. Therefore uHB C V, and by (7.6),
xEVu This completes the proof of Theorem 3 when K = Q.
8. - Proof of Theorem 3 in general Let W be an algebraic manifold defined by equations, f¢ = 0 (t = 1.... , t) with coefficients in a normal number field K of degree k. Let a1i... , ak be the embeddings of K into C, and let ,fQZI be obtained from ft by applying ai to the coefficients of fi. Further let W0 (i = 1, . . . , k) be the manifold given by ft(') (x) = 0 (f = 1, ... , t), so that W(I),..., W (k) are the "conjugates" of W.
Lemma 11. - Let W as above be defined by equations ft = 0 of partial degree < d and of height H'(fr) < M (t' = 1, ... , t). Then k:
W* := U W(i)
(8.1)
of partial degree <_ dk with
is defined by equations g,,,, = 0 (Ira coefficients in Q and with
H'(y, )
k ! Al'
Proof : set
F(T,X) = f
.
X)where
)( (±Tefr)
T1,. .. , T, are indeterminates. Expand F as a polynomial in Ti,..., Ti ; the coefficients will be certain polynomials g,,(X ), and W* is the zero set of these polynomials. They are of partial degree <_ dk and have coefficients in Q. Say, without loss of generality, that is the coefficient of T11 ...TJq
HEIGHTS OF POIN1S ON SUBVARiETIES OF G',`n
with positive j1, ... , j,, and with j1 + polynomials (8.2)
+ j,, = k. Then gxn is the sum of f(i(q,1))
f12(1 1)) ...
179
q
... f(i(9,7v)) 9
where i(1, 1), ... , i(q, jq) is a permutation of 1, ... , k.
We may suppose that each of the given polynomials ft has some coefficient equal to 1, and then If f0Ix, > 1 for every e, i and every v E MK. Then the polynomial (8.2) has absolute value (see the definition in Section 5!)
...I, C
I.111)I'V...I.f1A)I'V....1gl)Ix,...If(k)q1".
Write ex, = 0 when v is finite and ex, = 1 when v is infinite. Since gm is a sum of at most k! polynomials (8.2), kk.E"If(1)Ix,...I.fjA)I.,,...I.f,l)Iv...If(k)Iv. Raising this to the exponent kk1,,/k where kx, is the local degree, and taking the product over v E MK, we obtain H'(gx,,,)
k!M`Ik < kk!Mk2
Remark : the lemma holds whether K is normal or not. So now let V be Zariski-closed as in Theorem 3, defined by polynomial equations of degree <_ d, with coefficients in a normal number field K, and with heights <_ M. Let V = V('), ... , Vlk) be the conjugates of V. Let x be a point in V. Without loss of generality we may suppose that (8.3)
xEV(1),...,V(`s), but xVV(`I+1),...,V(k).
Set
W is again defined by polynomial equations with coefficients in K, of degree <_ d, and of height < M. The variety W* defined by (8.1), in view of Lemma 11, is defined by equations with coefficients in Q, of degree < dk,
and with heights <_ M1 = k!Mk2. By the case K = Q of Theorem 3 we have (1.9) (and also (1.9'), (1.9")), unless x E W*", so that x E uH where u E U', H is a torus, and uH C W*. Since uH is irreducible, uH C W(`) for some i. Here WO = 01-1) n ... n v() with distinct ri i ... , rq. But in view of (8.3), the set {r1, ... , r,,} must be {1, ... , q}, so that WO = W and uH C W C V(1) = V, therefore x E V".
180
W. M. SCIIMIDT
9. - A Variation
Suppose V is defined by equations fe = 0 (f = 1,... , t) where ft = >iEI atiX'. Then it is obvious from the proofs that N(d) may be replaced by III = cardl in Theorem 2 (iii), (iv) and in the case k = 1 of Theorem 3. Now suppose that W C G',, is defined by polynomial equations ft = 0 (f = 1, ... , t) where ft = f (X 1, . . . , Xs) has rational coefficients and is of total degree <_ d in each block of variables X. = (X,. .. , Xtin). Then in Theorem 1 we have to replace n, d by ns, ds, and in particular W" is the union of <
(9.1)
(11ds)+L2y2
sets WI"(Hi). The monomials occurring in ft are Xi' ... Xse with x I(d) = I, say, where (i1,...,is) E 1(d) x III = N((I)s.
(9.2)
Therefore, when the coefficients are rational, we may conclude from what we have said above that each Wj`(H) is the union of fewer than exp(3III312 log III) < exp(3III2) = exp(3N(d)2s)
torsion cosets. Combining this with (9.1) we see that W" is the union of (9.3)
< (llds)"2''2 exp(3N((1)2s)
torsion cosets. Ifs satisfies
s<_N(d), then log(llds) <-- log(11N2((1)) < 2N(d) (since N(d) >_ n+1 > 3). and (9.3) becomes < exp(n,2N2((1) 2N(d) + (9.4)
3N((I,)2N(d))
< exp(2N((1)5 + 3N((1)2N(`I)) <
exp(5N(d)2N(`I))
exp(5 (2n)2IN(d))
Assume furthermore that each defining polynomial has H'(ft) < s! < N(d)!. Then by Theorem 3 with k = 1, with n, d replaced by ns <_ nN(d), ds <_ dN(d) respectively, and N(d) replaced by III = N(d)s < N(d)N(d), we
may infer that every(x1, .. , xs) E W\W" has hs(xl, ... xs) > 1/(d N((l)N((l)N((I). e3N(")!) (9.5)
> 1/(N((1)2N(,I) exp(3N(d)!))
> exp(-4N ((I)N(d))
> exp(-4(2n) IN(d))
HEIGH75 OF POINTS ON SUB VARIETIES OF Gm
181
10. - Proof of Theorem 4 Let £(d) be the set of lattices which have a set of generators which lie in D(d) := D(I(d)). We had seen in (3.3) that every A EC(d) has p(A) < (2d)'LEMMA 12. - Let A C Al C . . . C A, be lattices in G(d). Then
v S 2n2d.
Proof : initially consider only chains of lattices which all have the same rank, say r. Then each lattice of the chain is contained in A, and the index of A in A is p(A). Since each index in our chain is > 2, we obtain v5 (log p(A))/log 2 <_ n(log 2d)/log 2 <_ nd.
In general, the rank in the chain is between 0 and n, and can only increase. Only n such increases are possible. Since {0} is the only lattice of rank 0, we obtain
For any A E £(d), let v = v(A) be largest such that there is a chain as in the lemma. Then v(Z") = 0, and v(A) < 2n2d in general. PROPOSITION. - Let V be a variety defined by polynomial equations of
total degree < d. Suppose V C gHA where A E £(d). Then every point x E V\Va, with at most exp(v(A) 5(2n)2dN(d))
h,,(x) > 3e = 3 exp(- (4n)2dN(d) ).
When A = {0}, so that gHA = gGn, = G', the condition V C 9HA is no restriction. Since v({0}) < 2n2d, we see that for any V defined by equations of total degree at most d, all but at most exp(2n2d 5(2n)2dN(d)) < exp((4n)2dN(d)) = q
points x E V \V a have (10.2). Therefore the Proposition implies Theorem 4.
W. M. SCIIMIDT
182
When v(A) = 0, then A = Z", therefore HA = {(1, ... ,1)}, and gHA = {g} consists of a single element. The Proposition is true in this case. We will prove it in general by induction on v(A). So suppose V C gHA where v(A) > 0. Now A is generated by certain
vectors i - j with i, j E I(d), so that HA is defined by certain equations
x' = x3 with i, j E I(d). We obtain a partition P of 1(d) such that i j precisely if xi = xj is a valid relation in HA, i.e., if the characters are the same on HA. X: x '-p and Xj:x The coset gHA consists of points x having
x'=xag' when iEA, AET, with arbitrary xa (A E P). Let I1 C I(() consist of a set of representatives of the sets A E 9), so that 11 contains exactly one element i(A) E A for every A E 3). By Artin's Theorem on characters, the monomials xi with i E Il are linearly independent in HA.
When V = gHA where dim HA > 0, then V" = V and V\V° = 0. When V = gHA where dirnHA = 0, then IVI = IHAI = p(A) < (2d)'. We therefore may suppose that V C gHA. Since V is defined by equations of total degree <- d, the monomials xi with i E I, are linearly dependent in V. Let E be the set of points x E V\V`° which violate (10.2). With X E Ewe associate the vector v(x) with components xi (i E I1). Thus v(x) lies in the space CII' ( of dimension 1111. Let T C C1''1 1 be the subspace of CII' l spanned
by the vectors v(x) with x E E. Since the monomials x' (i E I1) are linearly dependent on V, diinT < I. Say dimT = s - 1 where 1 <-- s c IIii. A matrix with rows v(x1), ... , v(x,) where x1i ... , x, lie in E is an (s x Iii)matrix of rank < s -1. This means precisely that certain s x s-determinants A1, ... AQ depending on x1, ... , x, vanish. The equations
x,) = 0
(P, = 1,...,Q)
define a Zariski-closed set W C (G;;,)-'. The polynomials At have H'(O1) _ s!, and they are of total degree d in each block of n variables. Thus W is of the type considered in Section 9. Therefore every point (x1 i .... x8) E W\W" satisfies (9.5). But this is impossible when x1 i .... x, lie in E, for
then (10.3)
h,(x1i...,xy) _
hY(x.i) <_ 3sexp(-(4n)2dN(d)) J=1
< exp(-4(2,n,) IN(d))
Therefore when x1, ... , x lie in E, we have
xr,...,x,) EW
.
HEIGHTS OF POINTS ON SUBVARIETTES OF Gnz
183
Fix xi, ... , x3_1 in E with v(xi), ... , v(xy_1) linearly independent. Let us restrict to x E E such that (10.4)
(xl,...,x8_1,x) E uH,
where uH is a fixed maximal torsion coset in W". Here k = HB, where by Lemma 4, B is a lattice in Z"' generated by certain vectors b = (b1,... , b5) where each bj E D(d) = D(I(d)); this holds since each At has degree < d in each of x1, . . . , xs. The coset uH is defined by certain equations
xl61 ... x4bb =h,
(10.5)
for certain vectors (b1,. .. , b,,) E D(d) x x D(d). More precisely, since we x D(I1). restricted to exponents i E Il, we have (b1,. .. , b,) E D(I1) x We claim that b. 54 0 for some of the relations (10.5). For otherwise, the relation (10.4) would be independent of x. Thus if it holds for any x at all, it holds for every x, so that
(xi,...
_1, x) E uH C W
for every x, and therefore v(xi), ... , v(x3_1), v(x) would be linearly dependent for every x, contradicting the fact that even if we restrict to x E HA, the vectors v(x) span a space of dimension 11, 1 > s. Consider a particular relation (10.5) with bg 0. Applied to (10.4) we obtain
xbl ... 1
x:b..-ix. 1
s
=t
so that, since xi, ... ,x3_1 are fixed, xb. = b'. Now if there is some xo at all with xo E gHA, xo' = S', the general x will be x = x0h where ha = 1 for a E A, and hbs = 1, so that h E HC where C is the lattice generated by A and by b3. Here b, E D(I1), say b, = i - j with i # j in I1, so that i x j. Therefore xi = xi is not a valid relation in HA, and bs = i - j A. Therefore A C C. Since C is generated by A and b,,, we have C E £(d) and v(C) < v(A) - 1. We have x E xoHC, so that z E V where
f, = V n x(,HC. Here V and HC, therefore V, is defined by polynomial equations of degree 5 d, and V C xoHC. Further x V" yields x Va. By induction, such x, with at most exi)((I/(A) - 1) 5(9l)2(IN(d))
184
W. M. SCHMIDT
exceptions, have (10.2).
In order to deal with general x E V\V`°, we have to multiply by the number of cosets uH making up W". This number is bounded by (9.4), so that indeed the number of exceptions to (10.2) is bounded by (10.1). 11. Proof of Theorem 5. We proceed as in [121. Suppose initially that
r is finitely generated and of rank r. There are x1, ... , x, such that the elements of r are (11.1)
x`T
where u E U9L and i = (i1, ... i,.) runs through Z". Here r, hence the components of xl,... , x,. lie in a number field K. For V E MK put aijv = log IIxijII,
(1 < i < r, 1 < j < n),
tt aij"Si,
ajvW _ i=1
E R'. Set
where F-
4Z
G() = 2
Iajv(&)I. I,EMK j=1
When i E Z' , formula (5.1) yields n
(11.2)
O(i) _
n
h(xl,j=1
h(xj) = h,, (x)
j=1
where x is given by (11.1). We have (a) iG(F-) >>_ 0.
for a E R.
(b) V)(a&) = (c)
5 0() + 'q (rl)
(d) ii(i) ? c(k) > 0
for every i E ZT\{0}, where k = (leg K. The last of these assertions holds since x given by (11.1) does not lie in U'L unless i = 0, and by a well known theorem of Dobrowolski 141.
LEMMA 13. - Suppose,,/) satisfies (a), (b), (c), (d). Let I C 118' be a set of
points such that
,,/,(i-j)>_E>0
HEIGH'T'S OF POINTS ON SUBVARIETTES OF Gnn
for i
185
in I. Then the number of i E I with Vi(i) < C
is <_ ((2C/e) + 1)r.
Proof : this easy result is Lemma 4 of [ 12].
Define q, E by (1.10), (1.11). In view of the Corollary to Theorem 4, an
element x E V\Va has at most q neighbors in V\Va of distance < E. Let 8 be the set of points to be estimated in Theorem 5. Consider subsets 8' of 8 no two of whose elements are neighbors of distance < E. Let 81 be a maximal set with this property, i.e., a set such that no set S" D 81 has this property. It is easily seen that ISI
When x # y lie in 8', we have It,, (xy-1) = 5(x, y) > e. When x is given by (11.1),andy=u'x1' a then (11.3)
Vi(i - j) = h.(xy-1) >_ e.
Let I be the set of exponents i E Zr such that (11.1) holds for some x E 8'. Then
I8'I=III, and (11.3) holds for i
j in T. By Lemma 13, III < ((2C/e) + 1)'' < (3C/E)T,
therefore 181 < gl8'l = qIII < q(3C/e)r.
Note that by (10.2), Theorem 4 holds with 3e in place of e. Therefore with q = E-1+ 181 < q(gC)r.
In general, r is a union of finitely generated groups r1 C r2 C rank r. Our estimate holds for each rti (i = 1, 2, ...), hence also for r.
of
Manuscrit requ le 30 janvier 1996
186
W. M. SCHMIDT
References. [ 11 E. BoMBIEiui and U. ZANNIER. - Algebraic Points on Subuarieties of Gi,
International Mathematics Research Notices 7 (1995), 333-347.
[2] J. W. S. CASSELS. - An Introduction to the Geometry of Numbers, Springer Grundlehren 99 (1959). [31 J. H. CONWAY and A. J. JONES. - Trigonometric diophantine equations
(On vanishing sums of roots of unity), Acta Arith. 30 (1976), 229-240. [41 E. DOBROWOLSKI. - On a question of Lehmer and the number of irre-
ducible factors of a polynomial, Acta Arith. 34 (1979), 391-401. 151 P.D.T.A. ELLIOTT. -Probabilistic Number Theory I, Springer Grundleh-
ren 240 (1980). [61 M. LAURENT. - Equations diophantiennes exponentielles, Invent. Math.
78 (1984), 299-327. 171 J. B. ROSSER and L. SCHOENFELD. - Approximate formulas for some
functions of prime numbers, Illinois J. of Math. 6 (1962), 64-94. 181 H. P. SCHLICKEWEI. - Equations in roots of unity, Acta Arith. (to appear).
[91 H. P. SCHLIcKEwri. - Equations nx + by = 1, Annals of Math. (to appear). [ 101 H. P. SCHLICKEWEI and W. M. SCIIMiDT. - Linear equations in variables
which lie in a multiplicative group, preprint. [11] H. P. SCHLICKEWEI and E. WIRSING. - Lower bounds for the heights of
solutions of linear equations, Invent. Math. (to appear).
[121 W. M. SCHMIDT. - Heights of Algebraic Points Lying on Curves or Hypersurfaces, Proc. A.M.S. (to appear).
HEIGHTS OF POINTS ON SUBVARIETTES OF Gvf1t
187
(13) D. ZAGIER. - Algebraic numbers close to both 0 and 1, Math. Computation 61 (1993), 485-491.
(14) S. ZHANG. - Positive line bundles on arithmetic surfaces, Annals of Math. 136 (1992), 569-587.
(15) S. ZHANG. - Positive line bundles on arithmetic varieties, Journal A.M.S. 8 (1995), 187-22 1. Wolfgang M. Schmidt
Department of Mathematics University of Colorado, Boulder Boulder, Colorado 80309-0395 U.S.A.
Number Theory Paris 1993-94
Some applications of Diophantine
approximations to Diophantine equations T.N. Shorey
1. - In 1975, Erdos and Selfridge [5) confirmed an old conjecture by proving that a product of two or more consecutive postive integers is never a power. In other words, the equation
inintegersk>2,e>2,m>O,y>O
(1)
has no solution. We shall consider a more general equation than (1). For introducing this equation, I need some notation. For an integer v > 1, we
write P(v) for the greatest prime factor of v and we put P(1) = 1. Let b > 1, k > 2, f > 3, m > 0, t > 2 and y > 0 be integers such that P(b) < k. Let dl, ... , dt be distinct integers in the interval [1, k]. We observe that
{dl,...,dt} = {1,2,...,k} ift = k. We shall consider the equation (2)
(m+dl)... (rn+dt) =bye.
Equation (2) with t = k and b = 1 is equation (1). If P(y) < k, equation (2) asks for all the solutions of the inequality
P((m+dl)... (m+dt)) < k and we do not intend to consider this question in this talk. Therefore we make the assumption that P = P(y) > k throughout section 1. Then there is unique i with 1 < i < t such that P I (m + di). In fact PL I (m + di). Thus
m + k > m+(1,,> Pt > (k+ 1)e
190
T.N. SHOREY
which implies the following relation
rn>k1. Before I state the results of this section, I wish to point out that I have taken 2 > 3 in equation (2). In fact equation (2) with £ = 2 was the topic of my talk in this seminar in 1992. The first result on equation (2) is due to ErdOs in 1955. THEOREM 1. ([4]). - Let e > 0. Equation (2) with t>k-(1-e)kloglogk
log k
implies that k is bounded by an effectively computable number depending only on e.
Theorem 1 has been improved considerably by Shorey in 1986-87. THEOREM 2. ([8], (9]). - (a) Equation (2) with
P.>3,t>
1(1+
4,z -88+7 2(t - 1)(2f2 - 5.t? + 4)
2
k=vlk
implies that k is bounded by an effectively computable number depending only on 2.
(b) There exist effectively computable absolute constants C1 and C2 such that equation (2) with
Q> C1,t> kQ-1111+7r(k)+2 implies that k < C2.
We observe that 4, > 2 for f > 3 and
v3
47
56'
135 ' ___ v1 < 3 for f > 5. 192'
v4
Theorem 2 has recently been sharpened for $ > 7 by Nesterenko and Shorey. For stating this result, we define 1 1212 -1601+29
v1 =
2813-7611212-12 1+29
01+11
28i -1881+129
if
1 (mod 2)
if e = 0 (mod 2)
APPLICATIONS OF DIOPHANIINE APPROXIMATIONS TO DI0PHAN77NE EQUATIONS
191
For 2 > 7, we observe that
I (l - (.875)1) ve
e(
1
and
- 1.41 21 )
v7
.4832, v8
< .4556, vg
v11
.3243,V12
.3076,V13
f
1 ( mo d 2)
if f
0 ( mo d 2)
if
< .3878, vlo
< .3664,
< .2787,V14 < .2655.
THEOREM 3. (1131). - (a) Equation (2) with
f>7, t>v1k implies that k is bounded by an effectively computable number depending only .E.
(b) Let e > 0. There exist effectively computable numbers C3 and C4 depending only one such that equation (2) with
£> C3i t> kt?-'+" +ir(k)+2 implies that k < C4.
The proofs of Theorem 2(b) and 3(b) depend on Baker's sharpening [31 in the theory of linear forms in logarithms. Linear forms in logarithms with
as close to one appear in the proofs and the best possible estimates of Shorey 18, Lemma 2], namely replacing log A for log Al log An with A = max Ai, for these linear forms in logarithms are required. For Theo1
rem 3(b), we utilise a refinement due to Loxton, Mignotte, van der Poorten and Waldschmidt [61 of the preceding result of Shorey. The proofs of Theorems 2(a) and 3(a) depend on the method of Roth and Halberstam on difference between consecutive v-free integers and approximations of algebraic numbers by rationals proved by Pade approximations. For Theorem 3(a), we require the following result which is a consequence, as proved in 1131, of a result of Baker [2]. LEMMA 1. - Let A, B, K and n be positive integers such that A > B, K <
n,n>3andw=(B/A)1/n¢ Q. For 0 < 0 < 1, put iS=1+2-0
K ILI = 40n(K+1)(s+1)/(Ks-1), Ell 1 = K2K+s+140n(K+1).
192
T.N. SHOREY
A(A - B)-btij' > 1. L2
w-q>
Agx(s+l)
for all integers p and q with q > 0.
Proof : we may assume that 1
L'
Observe that
fi> 1+K, and
(K+1)s
(K+1)6
(K+1)fi
K6 -1+¢ K+2-0-1+¢ =fi
Ks-1 Put:
Al = 40n(x+1)A A2 = 40n(x+1)(A -
B)x+1A-x
Observe that A2 < 1
if
40n(1+')(A- B)'+* A-' < 1 if
A(A - B)-(1+*)40-n(1+'x) > 1
which is the case by (3). Put
A
logA1
= log
A2
Observe that
-A<s
i.e.
Al < (A2 1)s i.e.
40n(x+1)(s+l) < (A
-
i.e.
A(A - B)-6/L1' > 1,
B)-(x+1)sAxs-1
APPLICATIONS OF DIOPHANTINE APPROXIMATIONS 70 DIOPHAM7NE EQUA77ONS
193
which is (3). For integers r > 0, p and q > 0, Baker [2) proved in 1967 that there exists Pr(X) E Z[XI satisfying
(i) deg Pr < K (ii)
H(Pr)
P,. (Z) # 0
I Pr(w) I< )2.
(lii)
(iv)
Define r as the smallest positive integer such that
a2<
219K.
Then : ,\2-1 > 21K q
Observe that :
(
,2
All = (.\2)^ < 2qK
) ^ _ A 12-^q-^K < A12'q'K
We have: eK
<-I P'(-q)1
Thus
Pr(') - P,(w) I>
21
On the other hand :
Prq)
P, (w)1 < K2K.1i 1 w -
Thus :
Iw--1_> q
1 <- a1K2K+8g8K 1 w - q
2
AgK(8+1)
This completes the proof of Lemma 1.
We shall be applying Lemma 1 with n = e > 7 and
K
((P. - 3)/2 (l
(Q - 4)/2
if £ - 1(mod 2) if t - 0(mod 2).
194
T.N. SHOREY
Let us look at the exponent of irrationality
K(s+ 1) = l Kb +K =
Ki
2--0
+K
=K(110+1)+1 =(K+1)(2--0)
-o
1> 2(K+1)>f-2.
Thus the exponent of irrationality is not far off from the trivial exponent E. In fact, the precision in the exponent of irrationality is not important for the application. Baker [ l I proved earlier in 1964 irrationality measures with the assumption b = 2. The important feature of Lemma 1 is that we can take b < 2. The assumption h = 2 corresponds to Theorem 2(a). Finally, we close this section by giving a sketch of the proof of Theorem 1. Proof of Theorem 1 : we may assume that k exceeds a sufficiently large effectively computable number depending only on E. Recall that m > k. A prime divisor greater than k of the left hand side of equation (2) divides precisely one factor and therefore its exponent has to be congruent to zero mod e. Thus m + di = aixti, 1 < i < t, where P(ai) < k, (xi, [I p) = 1. p
We show that (4)
For i # j, let We may assume that xi > x3 and
k > di - d; = ai(xi - x) > aix -1 > (aix)( -1)/t >
m('-')/t
> k1-1.
For every prime p < k, we take an f (p) E S such that p does not appear to a higher power in the factorisation of any other element of S. Put
S1=S-{f(p)Ip
APPLICATIONS OF DIOPHANTINE APPROXIMATIONS TO DIOPHAN7TNE EQUATIONS
195
Thus
I S1 I>- t - Ir(k).
It is easy to see that
R ati < a,ES1
TT
If-1+[yzl+...
= k! < kk.
p
Then there exists a subset S2 of S1 such that : I S2 I >- ek/2
and
a1 < k(log k)1-' if ati E S2. Further, as in the proof of (4), we show that a;aj with ati E S2i aj E S2
are distinct. The primes satisfy this property and what Erdos has shown is that the estimate for the number of numbers satisfying the above property is like that of primes. Erdos showed that S2 I-< 3k(log k)-c/2.
Hence k is bounded.
2. - In this section,we mention another application of the theory of linear forms in logarithms and Lemma 1. Let b, d, k, B, m and y be positive integers. satisfying P(b) < k, ged(m, d) = 1, k > 2, .P, > 2 and (5)
rn(nn + d)...(in + (k - 1)d) = lry1.
The first result on equation (5) is due to Fermat that there are no four squares in arithmetical progression. In view of the results of 1, we assume in this section that (1 > 2. Shorey and Tijdeman [14] showed that either (m, d, k) = (2, 7, 3) or the left hand side of equation (5) is divisible by a prime
exceeding k. Erdos conjectured that equation (5) implies that k is bounded by an effectively computable absolute constant. Marszalek [7] confirmed this conjecture in the case b = 1 and (I fixed, Shorey [ 101 in the case 2 > 2 and d composed of fixed primes and Shorey and Tijdeman [15] in the case $ fixed and d composed of fixed number of prime divisors. Further Shorey and
196
T.N. SHOREY
Tijdeman [ 16] sharpened the result of Marszalek by showing that equation (5) implies that : (6)
d > kca log log k
where C5 > 0 is an effectively computable absolute constant. Further Shorey [11] proved that there exist effectively computable absolute constants C6 and C7 such that equation (5) with k > C6 implies that : (7)
m>
dl-C7Oe
where De = p-1 (log p)2log log (p+ 1).
The proof depends on Baker's sharpening [3] and the best possible estimates of Shorey mentioned in 1 on linear forms in logarithms with as close to one. The estimate (7) is trivial whenever the exponent 1- C70e is negative which is the case only if p is bounded. For bounded p., Shorey [12] derived from
Lemma 1 that given e > 0 there exists an effectively computable number C8 depending only on a and p such that equation (5) with k > Cg implies
that (8)
in > dl-(6+E)e-'
Now we combine (7), (8) and (6) for deriving that equation (5) with £ > 7 implies that m kC4 log log k where C9 > 0 is an effectively computable absolute constant. In particular, equation (5) with p > 7 implies that k is bounded by an effectively computable number depending only on rn. In other words, the conjecture of ErdOs mentioned above is confirmed whenever f > 7 and m fixed. Manuscrit recu le 5 juillet 1995
APPLICATIONS OF DIOPFIANTINE APPROXIMA77ONS TO DIOPI-IAN77NE EQUATIONS
197
REFERENCES [11 A. BAKER. - Rational approximations to 3 2 and other algebraic numbers, Quart. J. Math. Oxford (2) 15 (1964), 375-383.
[2] A. BAKER. - Simultaneous rational approximations to certain algebraic numbers, Proc. Cambridge Philos. Soc. 63 (1967), 693-702.
[3] A. BAKER. - The theory of linear forms in logarithms, Transcendence Theory: Advances and Applications, ed. by A. Baker and D.W. Masser, Academic Press (1977), 1-27. [4] P. ERDOs. - On the product of consecutive integers III, Indag. Math. 17 (1955), 85-90. [5] P. ERDOs and J.L. SELFRIDGE. - The product of consecutive integers is
never a power Illinois J. Math. 19 (1975), 202-30 1. [6] J. H. LOXTON, M. MIGNOTTE, A.J. van der POORTEN and M. WALDSCHMIDT.
- A lower bound for linearforms in the logarithms of algebraic numbers, C.R. Math. Report Acad. Sci. Canada 11 (1987), 119-124. [7] R. MARSZALEK. - On the product of consecutive elements of an arithmetic
progression, Monatsh. Mr. Math. 100 (1985), 215-222.
[8] T.N. SHOREY. - Perfect powers in values of certain polynomials at integer points, Math. Proc. Camb. Phil. Soc. 99 (1986), 195-207. [9] T.N. SHOREY. - Perfect powers in products of integers from a block of consecutive integers, Acta Arith. 49 (1987), 71-79.
[10] T.N. SHOREY. - Some exponential Diophantine equations, New advances in Transcendence Theory, ed. by A. Baker, Cambridge University Press (1988), 352-365. [I 1] T.N. SHOREY. - Some exponential Diophantine equations (II), Number
Theory and Related Topics, ed. by S. Raghavan, Tata Institute of Fundamental Research (1988), 217-229. [12] T.N. SHOREY. - Perfect Powers in products of arithmetical progressions with fixed initial term, to appear.
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T.N. SHOREY
[13] T.N. SHOREY and Yu.V. NESPERENKO. - Perfect powers in products of integers from a block of consecutive integers (II), to appear.
[ 141 T.N. SHOREY and R. TIJDEMAN. - On the greatest prime factor of an arithmetical progression, A Tribute to Paul Erdtis, ed. by A. Baker, B. Bollobas and A. Hajnal, Cambridge University Press (1990). 385-389. [15] T.N. SHOREY and R. TIJDEMAN. - Perfect powers in products of terms in
an arithmetical progression, Compositio Math. 75 (1990), 307-344. [16] T.N. SHOREY and R. TIJDEMAN. - Perfect powers in products of terms in
an arithmetical progression (II), Compositio Math. 82 (1992), 119-136. T.N. Shorey School of Mathematics
Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400 005 India
Number Theory Paris 1993-94
Fields containing values of algebraic functions and related questions Umberto Zannier
1. - Introduction The problems discussed below originate in the celebrated Hilbert Irreducibility Theorem (from now on H.I.T.) the simplest case of which states Let F E Q[x, y] be an irreducible polynomial. Then, for infinitely many natural numbers n, F(n, y) is irreducible in Q[y]. (Many proofs and generalizations of such result in various directions are known. We quote for instance [De2], [De3], [La], [Sch], [Se], [SZ] for different points of view.)
We can restate H.I.T. as the assertion that the roots of F(n, y) = 0 have degree d over Q for infinitely many n E N or even, letting 6 be an algebraic function solution of F(x, 6(x)) = 0, we may say that the degree over Q of the values of 6 at infinitely many rational integers n equals the degree of 6 over Q(x).(1)
In such a context the following problem seems natural to us : to estimate the degree obtained adjoining to Q many values of type 6(n) . In other words we ask how independent are the values of an algebraic function at rational integral arguments.
The problem is clearly capable of generalizations. However we shall discuss here only the basic case just mentioned.
We shall sketch the proof of some results concerning the order of magnitude, for n -> oo, of the degree over Q of fields of type K*(n) = Q(61i ... , 6n) for a prescribed sequence {6j} satisfying F(j, 6;) = 0. We define, for given F (1) (1)
D*(h) := mi11[K*(n) : Q]
Of course the value of an algebraic function at n is not well defined, but in our context it turns out that the choice of the "branch" makes no difference.
200
U. ZANNIER
where the minimum is taken over all sequences 10j) as above. Let D deg,y F.
Assuming that F is an absolutely irreducible polynomial with D > 1 (as we shall do in the sequel) it may be easily proved, using H.I.T. over general number fields, that D* (n) --> oo. In fact, under such assumptions, there exists, given any n, a natural number n' > n such that F(n', y) is irreducible over K*(n). Then K*(n') contains K*(n) properly. It is in fact possible to improve substantially on such qualitative information. For notational convenience we shall work from now on with the logarithm of D* (n). The obvious upper bound n log D for log D* (n) is not always attained, even as an order of magnitude, as the most simple examples F(x, y) = YD -x show : in such cases K* (n) is contained in the compositum of the fields generated over Q(e ) by the D-th roots of the prime numbers up to n, whence log D* (n) << nn , by Chebyshev's estimate. Such upper bound is anyway meaningful since it generally holds as a lower bound too, as far as the order of magnitude is taken into account. In fact we have THEOREM 1. - Let F E Q[x, y] be an absolutely irreducible polynomial(2) with degy F > 1. Then log D* (n) >> ,',L
In some cases we can actually prove better bounds. Before stating them we introduce a bit of notation, which shall be useful throughout. Let F be absolutely irreducible and define E as the splitting field of F over Q(x). Let 9 be the Galois group of kE over k(x), k being some number field containing
the algebraic closure of Q in E. The function field kE corresponds to a nonsingular curve C over k. Let S(x) be a squarefree polynomial whose roots are the finite ramification points of x (viewed as function on C). In view of our assumptions S is non constant (otherwise C would be a nonramified covering of the Riemann sphere, whence the number of sheets, and so D, would be 1. The Hurwitz genus formula gives an even more precise information). Also, since F has coefficients in Q, the same may be assumed to hold for S. The roots of the polynomial S are precisely the roots of the discriminant G(x)(3), say, of the integral closure of Q[x] in E.
Also, define E(rn) to be the splitting field over Q of the polynomial F(m, y), and denote by D(n) the absolute degree of the composite K(m) of the fields E(rn) for 1 < in < n. It will be seen that D* (n) and D(n) have (2)
This could be replaced with the necessary condition that the absolutely irreducible factors of F have degrees > 2 in y. The proof does not present new difficulties, however, and we restrict to the special case. (3) defined up to multiplication by the square of a rational number
FIELDS CONTAINING VALUES OFALGEBRAIC FUNC77ONS AND RELATED QUESTIONS 201
logarithms of the same order of magnitude, so we shall consider D(n). We have
THEOREM 2. - Assumptions being as in Theorem. 1, let S(x) have an irreducible factor of degree 2 or 3. Then log D(n) >> n.
In both cases the implied constants may depend on F only. Both results were obtained by R. Dvornicich and myself already in 1983 (see [DZ1] or [De4]), but for a number of reasons their publication has been delayed, so only very recently they appeared in [DZ2].
The proofs show that the condition degree 2 or 3 in Theorem 2 could be replaced by degree > 2 provided natural (and classical) conjectures on the distribution of squarefree values of polynomials were true; the connection with such questions (answered at the moment only in quite special cases, see e.g. [Ho]) is somewhat intrinsic to the problem, as the examples F(x, y) = y2 - f (X) show. Concerning the exact order of log D(n), A. Schinzel has proposed the following general conjecture. SCHINZEUS CONJECTURE. - We have either log D(n) >> n or (0 S(x) splits
into linear factors over Q and (ii) G is abelian. By applying Kummer Theory it is easy to see that, conversely, conditions
(i) and (ii) imply that E is contained in a composite of a finite number of fields of type L( '` x + b), where L is a suitable number field and b E Q, so practically we are reduced to the basic examples mentioned above, and in
such cases log D(n)
THEOREM 3. - Let condition (i) in Schinzel's conjecture hold Assume moreover that there exists a prime pl 1 #9 such that 9 has no normal subgroups
of index p. Then log D (n) > -'2n+ o(n). (Contrary to the previous theorems, the constant in the lower bound is now absolute for large n). The result includes all nonabelian groups of squarefree order as well as the nonabelian S. and An (use Bertrand's postulate). We also remark that it suffices if the assumption is verified by some quotient of 9 rather than by 9 itself.
202
U. ZANNIER
In a somewhat different direction P.Debes has pointed out that, if one considers the fields K(t, n) := Jt c2n where c > 0 depends only on D (Debes gives an explicit value for c; also, he works with polynomials over general number fields, and then c depends on the corresponding degree as well).
We shall sketch below the proofs of theorems 1,2,3. Some analytic number theory is used in both Theorems 1 and 2, while a simple sieve method, combined with Kummer theory and a somewhat striking result in field theory (Lemma 4 in [DZ31) is crucial for the proof of Theorem 3. But the common and most essential ingredient in all such proofs is Lemma 2 below which, roughly speaking, constructs prime numbers ramified in E(m) as certain prime divisors of G(m), thus comparing (together with Lemma 1) the discriminant of the specialized field with the specialization of the discriminant. We give in [DZ 1,21 a simple direct proof of such result, though it may be derived from the celebrated Weil's Decomposition Theorem (see (Wel], [We21 Ch.l, or [La], p.263).
We have found further applications of such a principle. A first one allows to obtain, via Baker's fundamental theorems in effective diophantine
analysis (see e.g. (Ba], Ch.4), an effective form of the classical Siegel's theorem on integral points on curves ((Lal, Ch.8 or [Se], Ch.8) in the special case the projection on the x-line induces a Galois cover : this corresponds
to an effective answer to the diophantine question of finding, given a polynomial f E Q[x, y], the integers n such that all roots of f (n, y) = 0 are rational. We have the following theorem, first proved however by Yuri Bilu in 1988: THEOREM 4 (Bilu). - If E has positive genus then the set of m E Z such that F(m, y) has only rational roots is finite and effectively computable.
In [Bill an equivalent result with a different statement is derived (over general number fields). In a more recent approach Bilu proves his statement as a special case of a very interesting criterion of him which implies effective
versions of Siegel's theorem, provided certain supplementary conditions are satisfied (these concern finding covers of a curve, unramified outside a given set A, and such that the A-units have rank > 2) ; see (B13]. Also, Bilu computed explicitly some bounds in (Bi21. We sketch below our point of view about such question, outlining how a proof of Theorem 4 follows from the above ramification criteria (this also appears as an appendix to (DZ2]).
We remark that the methods apply to general number fields.
FIELDS CONTAINING VALUES OFALGEBRAIC FUNCTIONS AND RELATED QUESTIONS 203
A second application, given in [Zal, is also concerned with the distribution of the fields arising from the solution of F(m, y) = 0; it allows to estimate the number of positive integers m < T such that some root of such equation generates a field containing a given number field E (the question originally arose from certain possible applications to analytic number theory, due to M. Nair and A. Perelli). Without uniformity requirements such a problem may be easily solved by means of the known quantitative versions of H.I.T.; made exception for finitely many fields E depending on F, the estimate takes the form O(AT). However, use of the present methods allows an estimate of type 0(c1 T vT), (c = c(F)), uniformly in the field E, with much better bounds as soon as certain mild supplementary assumptions on F are verified. We note that such a result implies a version of H.I.T., uniform over number fields in the following sense :
THEOREM 5. - Let f E Q[:r, y] be irreducible over a number field L, let i(L) be the number of subfields of L and let T > 10. Then the set of positive integers in < T such that F(m, y) is reducible over L contains at most i (L) cg3' /T elements, where c depends only on f . In fact a classical procedure (see e.g. [Sch]) reduces the problem of the reducibility of f (rn, y) over L to the verification whether F(rn, y) = 0 has a root in L, F being a certain polynomial associated to f. Now, if F(m, y) = 0
has a root in L, this root generates a subfield as in the statement. Made exception for finitely many fields depending on f, we can apply the above mentioned estimates, which are completely uniform with respect to the field. For each of the remaining finitely many possibilities, the ordinary version of H.I.T. gives an estimate O(v/T) for the number of relevant integers up to T, and the conclusion follows.
Acknowledgement. It is a pleasure to thank D. Bertrand and P. Debes for helpful conversations, for the interest they have shown towards the above questions and for their kind encouragement.
2. - Proofs We begin with a couple of lemmas, crucial for the proof of all the above theorems. Preliminary to the arguments we choose, once and for all, points P,,,, E C, m E N such that x(P,,,) = in. In the sequel we shall specialize at
P. algebraic functions ' E E, tacitly assuming that in is sufficiently large for the specialization to be defined, in which case E E(m). We also denote by A,,,, the absolute discriminant of E(rn). LEMMA 1. - There exists p0 such that if 1) > p0 is a prime which ramifies in E(m), then 11IG(rn).
204
U. ZANNIER
(This corresponds to Lemma 3.1 in [DZ2]. The result is connected with the Chevalley-Well Theorem.) The proof runs as follows. Choose an integral basis el,..., el, for the integers of E over Q[x]. It may be easily checked that, since the ei span E over Q(x), the algebraic numbers ej(P...)
span E(rn) over Q. We may assume them to be algebraic integers, by multiplying if necessary by a nonzero rational integer independent of m. Let s = [E(m) : Q] and renumber indices to assume that el e, are linearly independent over Q. Let c run over Gal (E/Q(x)) and let r run over Gal(E(m)/Q). We may renumber indices so that T-(et(P,,,)) = ((7jet)(P,,,,) for i _< s, t < It. Let E,,, be the s x It matrix with such entries. Then every s x s minor is divisible by namely the quotient is an algebraic integer. We
may multiply G by a rational number to assume that G(x) = det2(a et). Specialize now at P,,,. The left side gives G(rn), while the first s rows of the matrix on the right side give E.,,,,. Expanding the determinant along the upper s x s minors we obtain that G(rn,) is divisible by A,, i.e. the Lemma.
LEMMA 2. - Let L be a number field, let g E L[.x] be a factor of G, irreducible over L and let e be the ramification index of g in LEf41. Let p be a prime ideal of L, of sufficiently large norm, such that ordg(m) 0 O(mod e). Then p ramifies in LE(m). (This corresponds to Remark 2 after Lemma 3.2 in [DZ2]. As remarked in the Introduction and in [DZ2], it may be derived from Weil's Decomposition Theorem, starting from the observation that the divisor of zeros of g, viewed
as a function on C is divisible by e). To outline a direct proof, let B be the integral closure of L[x] in LE and write (2)
g(x)B = Ste
,
say, normalized to take algebraic integer values at P,,,. Let B(m) be the ring of integers in LE(m) and define 11(m.) as the ideal in B(nn) generated by the wi(P,,,,). From (2) we get:
for some ideal Il E B, generated by
R(I)wl'...w,r
g(x) = (i)
where (i) runs over the r-tuples of natural numbers with sum e and where R(2) E B. We may assume that aR(.j) (P..) is an algebraic integer, for a suitable rational integer a # 0 independent of rn. Specializing at P,,,, we (4)
namely the common ramification index of the prime ideals above g(x) in the integral closure of L[x] in LE
FIELDS CONTAINING VALUES OF ALGEI3RAIC FUNCTIONS AND RELATED QUESTIONS 205
obtain that ng(rn) C (SZ(Tn))e. To obtain an opposite inclusion observe that (2) implies, for each (i) as above S(i)g(x),
for some S(s) E B. As before we get, after specializing, b(fl(m))e C g(m)B(m), for suitable nonzero b E N. Let now Q be a prime ideal of B(m) lying above p, and write v(A) for the order at Q of an ideal A C B(m). We have ev(S2(rn)) < v((ag(m))) and v((g(m))) < v((b)) + ev(1 (m)). We may suppose that p does not lie above primes dividing ab, so v((a)) = v((b)) = 0 and v((g(m))) = ev(1l(rn)) is a multiple of e. On the other hand g(m) E L so, if p were unramified in B(m), then ord,,g(m) = v((g(m))) would also be divisible by e. This completes the proof. It will be noted that the normality of LE over L(x) is crucial to deduce that the ideal generated by q(x) becomes a perfect power in B. In the general
case we would have prime ideal factors with different exponents and a corresponding factorisation would hold after specialization, in accordance with Weil Decomposition. This provides information on the factorisation of g(m), which is nontrivial provided we know the size of some factor. This is the case for instance when the g.c.d. of the exponents is > 1, in which case g(m) will be nearly a perfect power. Otherwise one can use a rather delicate theorem of Bombieri [Bol which supplements Weil's theorem with
an estimate of the size of each factor. An equivalent result, formulated however differently, was obtained by Debes in [Del] and [De2l. Combining the principle of Lemma 2 with such tools, and using Belyi's theorem (see e.g. [Se]), one gets a version of Elkies' argument [Ell deriving an effective Faltings's Theorem from the abc-conjecture (see [Schm l 1, p. 205 for the
case of the rational field) : given a nonsingular curve C defined over a number field, one first uses Belyi's Theorem to obtain a cover x : C --> P1 of degree n, say, unramified outside 0, 1, oo. We can write the divisor of
x (resp. 1 - x) in the form e1P1 + ... e,.P, - giRi - ... - gtRt (resp. f1Qi + ... + f;,Q;, - g1Ri -... - gtRt), where ei, fj, gi, are positive integers and the Pi, Qj, R,, are distinct points on C. The Hurwitz genus formula easily gives, for the genus -y of C, 2-y - 2 + r + s + t = n. Let P E C be a rational point and consider the value x(P). Weil's Decomposition Theorem implies a factorisation x(P) = co" ...1lit 9` (and similarly for 1 - x(P)), where c is a factor of bounded height and the 0i,1lij are algebraic integers (this step roughly corresponds to Lemma 2 above, applied to the irreducible factors x, 1 - .r of the discriminant.) The above mentioned theorem of Bombieri allows to estimate the height of the factors in terms of the height of x(P) (for instance, when x(P) does not approach 0 or oo - in the archimedean sense - each factor has height-: h(x(P))/n). Introducing such
206
U. ZANNIER
Informations into the abc-type equation 1 = x(P) + (1 - x(P)) and using the abc conjecture one obtains h(x(P)) < (1 + f)(' t)h(x(P)) +OE(1) < (1 +e)(n+ 2ry)1i(x(P)) +O(1) and a bound for h(x(P)) follows as soon as Before giving the applications to Theorems 1, 2, 3, let us briefly show that log D* (n) >> log D(n) + O(/), providing the comparison announced in the Introduction. The inequality D*(n) _< D(n) is trivial. To go in the
C K*(m - 1). Then the opposite direction, let m be such that is contained in the normal closure of normal closure (over Q) of K*(m - 1), which is the composite of the E(j) for j < m - 1. Now, if E(j). Plainly there F(m, y) is irreducible, we then have E(m) C are >> log D(n) values of in < n for which such inclusion does not hold. So,
if I(n) denotes the number of in < n such that F(m, y) is reducible, there are >> log D(n) + O(I(n)) values of in < n such that the original inclusion does not hold. At each such in we have D* (m) > 2D*(m - 1). Combining with the bound I(n) << n1/2 given by the quantitative form of H.I.T. we get the claim (even weaker results would do for our purposes).
So it will suffice to estimate D(n). We shall do that by finding many
integers j < n such that some prime p ramifies in E(j) but not in E(i) for i < j : this will prove that D(j) > 2D(j - 1). Let T be very large and consider the set of primes 1) E [T/4, T/2] such that p divides some value G(m). Certainly p will divide S(m) and, using the fact that S is a squarefree polynomial and that 1, is large, it may be easily shown that p will occur with
the first power in S(nt'), say, where nt' = m or m' = m + p. By Lemma 2 (with L = Q) p will ramify in E(m). We then may consider the minimum positive integer r such that p ramifies in E(r), denoting it by j(p). Certainly j (p) < 2p < T. On the other hand if such ap ramifies in E(j) then pIG(j) by Lemma 1. If j < T is given, (and G(j) # 0), this cannot hold for more than O(1) such primes. So there are O(1) primes such that j(p) is a given integer < T, whence the number of the distinct integers among the j (p)'s will be >> than the number of the above primes. But it is known from analytic number theory that the number of primes in [T/4, T/21 which divide some value on N of a given nonconstant polynomial is >> DoT , and this concludes the proof of Theorem 1. The proof of Theorem 2 is similar, but a little more delicate. Let g be the irreducible factor mentioned in the statement. An old result of Nagell (referred to in (DZ21) implies that #{p > T, plg(rn) for some m < T} >> T. Now either an easy argument, if (legg = 2, or a deep theorem of Hooley
on squarefree values of cubic polynomialsi4l, if deg g = 3. implies that (4)
actually a slightly different version of Hooley's Theorem, as given in IHol,
FIELDS CONTAINING VALUES OF ALGEBRAIC FUNCTIONS AND RELATED QUES77ONS 207
#, {p > T, pllg(rn) for some rn < T} >> T. This inequality allows to derive Theorem 2 by the same arguments as above.
We shall now sketch the proof of Theorem 3 in the very special case when F is cubic in y and Q = S3 (naturally the prime p in the statement is now 3); this simple case is sufficiently illustrative of the general method but rather less complicated in detail.
In such cubic case we may assume for our purposes that F(x, y) _ y3 - 3b(x)y - 2c(x), where b,c E Q(x). We now let k = Q(e f'). Also, set a(x) = c2 (x) - b3 (x) ; a is not a square in C(x) in view of our assumption on
Write a = ala2 with squarefree nonconstant al E Q[x]. All the zeros of al appear among the zeros of S(x), assumed to be rational. Also, Cardano's formulae show that kE = k(x, (c + a 1). By combining a simple sieve method(5) with the quantitative version of H.I.T., already used in the proof of Thm. 1, we may easily construct a sequence M = {mj } such that (i) the Galois group of kE(mj) over k is S3, (ii) m, << j and (iii) a1 (any) = cj sj where cj has only prime factors bounded by po and the sj are pairwise distinct squarefree numbers all of whose prime factors are > po. Put a := c + a, and choose once and for all, in an arbitrary manner,
determinations for the values a(rni) := ai, say, and for the cube roots of these numbers. Also, define FI,. = k(al,..., alt). We have kE(mh) C
Fh(ahl) . Assume that It is such that I1-i (3)
kE(mh) C rl kE(mi). i-I
The field on the right is contained in F,, (ai"3, ..., applied with Fl, as base field we have
(4)
al, =
e;
i.I
cxi
so by Kummer theory
o3
for suitable integers ei and ¢ E Fh. Let N denote the norm from Fh to k(ah),
and consider N(ai) for i < It. Certainly ai ¢ k(ah) since, by condition (iii) in the construction of M, the numbers a(mi) and a(mh) have distinct squarefree parts, both > 1 by condition (i). So N(ai) is a power of the is needed (5) it is here that condition (1) of Schinzel's conjecture proves useful
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U. ZANNIER
absolute norm of ai, which equals (c(mi) + a(mi))(c(mi) - a(mi)) _ b3(mi). In particular N(ai) is the cube of a rational number. So, taking the norm N of (4) we deduce that ahF`` `la'`)l would be the cube of an element of
k(ah), and the same would hold for ah, since the absolute degree of Fh is plainly a power of 2. But then kE(mh) would be quadratic over k, violating condition (i) for the sequence M. We have proved that (3) cannot hold, whence D(mh) > 2D(mh_1), and a bound log D(n) >> n follows from condition (ii) for M. To obtain the more precise result with the absolute constant in the lower bound one
has to refine the construction of M. Assume, to start with, deg al > 1. First of all, one can easily prove that, forgetting the requirement about the distinctness of the sj, and making Vo bigger, the density of M may be gotten as near to 1 as we want. Now an elementary argument involving diophantine
approximation shows that for only a few pairs i, j we may have si = s,. Throwing away the corresponding mi's the remaining sequence will have the same density as before and will satify all required conditions. The case of linear al is easier and may be reduced essentially to the distribution of squarefree numbers in an arithmetic progression rm + s with coprime r, s. The value of the constant comes out from this case. For details see Remark 3 in [DZ3].
In the general case both the choices for a and for the determinations for the values at ml, of the algebraic functions involved are more delicate, and the last ones are no longer arbitrary. The analogue of the crucial fact that the norm of c + a is the cube of the rational function is played by Lemma 4 of [DZ3], mentioned in the Introduction. The role of Lemmas 1, 2, invisible in the cubic case, but otherwise fundamental, appears in the need to show, through comparison of ramification, that the specializations at mi of a certain subfield of kE (namely k(x, a) in the cubic case) are pairwise distinct. Let us now discuss the proof of Theorem 4. We may suppose that k is so large to contain all finite ramification points of x. Denote them by il, ...,, . Let ej > 2 be the common value of the ramification indices of points above i;j in C and apply Lemma 2 with L = k, g(x) = x - E;, e = ej. Assume that all roots of F(m, ;y) = 0 are rational. Then kE(m) = k, so no ideal p as in the statement of that lemma may ramify. It follows that the fractional ideal is an ej-th power in k, made exception for a factor having generated by finitely many possibilities. A classical argument involving the finiteness of the class group of k and Dirichlet's unit theorem allows to write, for each j, ff7
ej
n6 - S = - p7j"7
where the pj have finitely many possible values, which can be effectively computed, and where the /j., are algebraic integers in k. So we may suppose
FIELDS CONTAINING VALUES OFALGEBRAIC FUNCTIONS AND RELATED QUESTIONS 209
the p3 to be fixed and investigate the solutions of the above system in algebraic integers m, µl, ..., µ, in k. If r > 2 and el > 3, say, then eliminating m from the first two equations gives an equation tt
tc
P2/,22 = PI U'I' + (Sl - S2)
whose solutions may be found by Baker's method.
If r > 3 and ej = 2 for all j multiplication of the first three equations gives an elliptic equation
PA2 = (m-Sl)(7n-b2)(m-S3) which again can be solved effectively.
In conclusion the solutions of each such system may be found except
when either r = 1 or r = el = e2 = 2. We show that these cases cannot occur. In fact applying Hurwitz genus formula to the field extension kE/k(x) of degree d, say, we get for the positive genus g of kE,
2g-2=-2d+Dei-1)-ei +>(e(P)-1), i=1
P100
the last sum running over the points of C above the infinite point of k(x). This term is clearly bounded by d- 1, whence 2 < 2g < 1+(r-1-E 1 Qi )d. The right hand side is < 1 in both the above cases, a contradiction. We observe that, for the proof to work, the full force of the normality of kE over k has not been used. In fact the method allows to find the integral points on algebraic curves of positive genus provided only certain conditions on ramification (which hold automatically in the present case in view of Hurwitz's theorem) are satisfied. We also remark that all the procedures involved before the application of Baker's theorems (namely the
computation of bases for ideals in function fields, of generators for class groups and unit groups etc.) are known to be effective, thanks to results by Coates and Schmidt (see e.g. (Schm2)).
We conclude by sketching how to obtain the announced estimate for the number of rn < T such that some root of F(m, y) = 0 generates a field containing a given number field E. As observed in the Introduction, such bounds lead to a proof of Theorem 5. It will be sufficient to replace E with its normal closure and to prove the following inequality, excluding the finitely many fields contained in k; namely, if E V- k we have S(T)
#{rn < T : E C kE(m)} << c f-
'
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U. ZANNIER
uniformly in E, where c = c(F). First one lets m run through a sequence such that the lattice of fields between kE(m) and Q is the same as the lattice between kE and Q(x). This can be done by H.I.T. and moreover the excluded sequence will contain << VT elements up to T and thus will not affect the final result. Now, if such an m is counted in S(T), the field E will equal some field r(m), where r is a normal extension of Q(x), contained in kE. Moreover r will not be a constant field extension of Q(x), since E 0 k. r has finitely many possibilities whence it may be supposed to be fixed. Replacing then kE with r leads to a similar problem, so we may in fact bound the number of times E = kE(m). Let gi run through the irreducible (over Q) factors of G, and let 0 be the discriminant of E. Also, let g* _ fl gi. By Lemma 2 we have, for each such in, i, (5)
gi(rn) = tizi',
ei > 2,
where ti are ei-th power free integers, the zi are integers and where the
prime factors of ti are either small or divide A. Let s be the greatest squarefree divisor of A. If s is small, say s < T, then each of the ti has few possibilities. For each such possibility the number of solutions of
each equation (5) with in < T, may be estimated as being << /T--logT uniformly in ti; this can be done either by appealing to certain deep theorems of Evertse and Silverman (see [Schm 11, p.142) or, more simply, by
considering the reduction of the equation modulo many primes congruent to 1 (mod ei) (and not dividing ti); for each such prime one may apply Weil's estimates for the number of solutions to deduce that the reduction of m has few possibilities. Finally a large sieve collects all the informations and gives the required estimate (this approach was used by S.D.Cohen in connection with H.I.T., see [Cohl or [Sel, § 12). When, on the contrary,
s > T, we use Lemma 1 to show that s divides, except possibly for a bounded factor, the value g* (rn). If pl < ... < pl, are (large) primes dividing s and pi...p1,,_i < T < P1.... , then we consider the congruence g* (m) =- 0 (mod p1...p!L). This has << c1 solutions. But h << loglogT. This completes the proof. In fact in [Zal details are carried out in the more general case (needed for the applications of Nair and Perelli) where m runs through an arithmetic
progression qn + r, n < T. The results remain uniform in r and in q too, provided q has, say, << logy T prime factors (complete uniformity in q is perhaps true but leads to difficult problems already in basic cases). This case gives additional combinatorial complications. Also, a uniform version (with respect to r, q) of H.I.T. is now needed already at the beginning of the proof, and it may be accomplished by Cohen's method, which is based
FIELDS CONTAINING VALUES OFALGEBRAIC FUNCTIONS AND RELATED QUESTIONS 211
on reduction to various moduli, and is so well suited for investigating the behaviour in progressions. As mentioned in the Introduction much better estimates may be obtained in some cases. Assuming for instance that every field IF # Q(x) C
r C kE has positive genus, and assuming there is at least one irrational ramification point (analogously as in Thm. 2), the factor VIT disappears. The proof of this result given in [Zal is similar to the above one, but now the use of the Evertse-Silverman estimates is crucial.
Manuscrit recu le 21 decembre 1994
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Umberto Zannier, Ist.Univ.Arch. D.C.A. S.Croce, 191 30135 Venezia (Italy) [email protected]
