L-Functions and Arithmetic (London Mathematical Society Lecture Note Series)

) the space of functions / * so generated on H(F)\H(f\). What is the relation between this new H(A)-module 0(TT, tp) (the theta-series lifting O/TT), and the H(A)-module 0^(TT) (the Howe lifting of TT)? The answer should be that 0(?r, ij)) realizes ©^(TT) provided @(TT, ip) is not identically zero! The problem is that the non-vanishing of 0(TT,^) is subtle to detect. Prototype example ([W.2]) Fix G = 51(2), and H = SO(V), where V is the 3-dimensional space of trace zero 2 x 2 matrices equipped with the quadratic form q(X) = —det(X). In this case, Howe's correspondence relates representations of SO(V) (which is isomorphic to PGL{2)) with representations of SL(2) (or rather the two-fold cover of SX(2), since MPip(F2 ® V) does not split over SL(2)). In particular, the local correspondence establishes a bijection between the set of irreducible admissible genuine representations of SL(2) which possess a ^-Whittaker model and the set of all irreducible

2 Howe's correspondence and the theory of theta-series liftings

39

admissible representations of PGL(2). Globally, we are dealing with an integral of the form (2.1), where G is SL(2); although both Q(g,h) and ££2) is that 0^(^) is automorphic if and only if £(TT, | ) = 1; here e(ir,s) is the e-factor in the functional equation L(S,TT)

= e(ir,s)L(l-syic)

satisfied by the Hecke-Jacquet-Langlands Z-function of TT. Moreover, even when the condition £(TT, | ) = 1 is satisfied, 0^(TT) will be realizable in 0^(?r, 0) only if this latter space is non-zero; this happens if and only if the stronger condition is satisfied. In the reverse direction, SL2 —> PGL2, Howe's correspondence is equally subtle. Here it turns out that Q^cr) is always automorphic on PGL2 if cr is automorphic, but 0() is non-zero (and then realizes Q^cr)) if and only if the V>th Fourier coefficients

do not vanish identically for fa in the space of a. There is also an intriguing characterization of the non-vanishing of these Fourier coefficients in terms of special values of ^-functions; this we shall describe later after introducing some general zeta-integrals involving theta-series (zeta-integrals of Shimura type). Examples involving functoriality The construction of Shalika-Tanaka relating X and TT(X) was independently obtained in [J-L] with E and GL{2) in place of E1 and SL{2). At this level, it is easy to check that the lifting x —» ^(x) is 'functorial' with respect to the natural X-group morphism \P : (Cx x Cx) x Wp —-> GL2(C) x Wp. Another example involving functoriality is the following. Let G = GSp2, the group of symplectic similitudes of a 4-dimensional symplectic space Wu and let GSO(V3i3) denote the group of orthogonal similitudes of a 6-dimensional orthogonal space H,3 = A2(F4) equipped with the

40

Arthur & Gelbart - Lectures on automorphic L-functions: Part II

inner product (W,W) = WAW'e A4(V) « F. Although we are dealing here with groups of similitudes inside GSp(Wi®V3)3), a suitable modification of the theory of dual reductive pairs leads us to liftings and ?r <-> 0(TT,I/>) between GSp2 and GSO(V). Remarks (1) When dealing with groups of similitudes, it turns out that Howe's lifting (or the theta-series lifting) is independent of ij). (The subscript ij) may therefore be suppressed in these cases, though we shall usually refrain from doing so.) (2) There is a natural injection of GL4/l2 into GSO(V). Therefore, the 0-series correspondence TT —> 0(TT,?/>) may be viewed as a correspondence between representations of GSp2 and GL4 (by restricting functions in 0(TT,VO t° GL4/l2); similarly, Howe's correspondence for the pair (GSp2,GSO(6)) naturally defines a correspondence between representations of GSp2 and H = GL4 which we again denote by ©^(TT). With these remarks in mind, we state the following: Theorem ([J-P-S.3]) (a) Suppose ?r is an automorphic cuspidal representation of G(A); then 0(TT,^) ^ {0} if and only if-K is globally generic, i.e., possesses a standard Whittaker model (on the space of its ^-Fourier coefficients). In this case, 0(TT,^>) realizes the Howe lift 0t/,(?r), and the lifting is compatible with Langlands' functoriality in the following sense: if p : LG —> LH = GL4(C) denotes the standard embedding of GSp2(C) in GL4(C), and {<jv(7r)} is the collection of conjugacy classes in LH determined by TT = ®7rv, then p{av(ir)} coincides with the collection of conjugacy classes in GL4(C) determined by ©^(TT). (b) Suppose n is an automorphic cuspidal representation of H(f\). Then n is the 0-series lift of some (globally generic) TT on G(f\) as in (a) if and only if the degree 6 .//-function

has a pole at s = 1 for some grossen-character

2 Howe's correspondence and the theory of theta-series liftings

41

Concluding remarks 1. The example just given shows how Conjecture B can be proved using the theory of 0-series liftings, at least for generic TT. One should also be able to establish this lifting (again for generic TT) as a special case of the 'converse theorem' program outlined in Section I.I; for the case at hand, this requires an analysis of the ^-functions L(s,U x r) on GSp2 x GL2 already studied in [PS-So]. In general, for arbitrary TT, one must also eventually be able to establish this lifting using the trace formula. In any case, the functorial identity where p : LG —> LH, already implies that (some twisting of) L(s, II, A2) must have a pole at s = 1 if II = 0^(TT) = /O(TT), since poA2 contains a one-dimensional subrepresentation (and therefore L(S,TT ® %, poA2) will - for some x ~~ contain the Riemann zet a-function as a factor). 2. As the preceding examples confirm, instances of functorial lifting can be established using the trace formula, X-functions, the theory of thetaseries liftings, or any combination thereof. In some cases, such as the Shalika-Tanaka example x —> ^(x)? e a c n o n e °f these methods provides an (alternate) proof; the L-function method was applied in §12 of [J-L] (albeit at the level of GX(2)), whereas the trace formula approach was developed in [L-L], and led to new and provocative results. Zeta-integrals of Shimura type Although of interest in its own right, our lengthy detour through the theory of 0-series was motivated entirely by our interest in the analytic properties of automorphic X-functions. The connection between these subjects is provided by zeta-integrals of Shimura type, themselves modifications of Rankin-Selberg integrals involving 0-series. To explain the general construction, we fix G = Sp n , and we consider zetaintegrals of the type (2.2)

C(*,
Here ip(g) is a cusp form in the subspace of Ll(G(F)/G(f\)) realizing the irreducible cuspidal representation TT of G(A); T is an n x n symmetric nondegenerate matrix, which we confuse with the n-dimensional orthogonal space VT it determines; 0|(#) 1S (th e value at (#, h) = (g, 1) of) the theta-kernel

e?(5,&)=

E

42

Arthur & Gelbart - Lectures on automorphic L-functions: Part II

corresponding to the dual pair (Spn, O(Vr)) in Spn and the choice of SchwartzBruhat function $ on the Lagrangian subspace Fn®VT (viewed as nxn matrix space); finally, E(g,F,s) is the Eisenstein series on G(f\) of the form

where P = {(~

)} in G is the parabolic subgroup whose Levi component

M is isomorphic to GLn, and Fs(g) belongs to Ind^^\detMm(p)\s+n21. For convenience, we assume n is even (and therefore the metaplectic group never need occur here). The zeta-integral ((s,
does not vanish (there S — {X} denotes the F-vector space consisting of symmetric 2 x 2 matrices). Let p denote the standard 5-dimensional representation of the L-group of 5O5(C) of Sp2 and let XT(°>) denote the quadratic character (a, —det T). Then the i-function

has meromorphic continuation to C, with only finitely many poles, and is analytic for Re(s) > 2. Sketch of proof

Everything follows from the basic identity

(2.3)

((s, tp, * , F) = G^Lsis

+ i , 7T ® XT, />),

where Fs(g) is suitably normalized, i.e., multiplied by a suitable product of zeta-functions, such that (1) the identity (2.3) holds and (2) the resulting (normalized) Eisenstein series has only finitely many poles (in this case, at

2 Howe's correspondence and the theory of theta-series liftings

43

certain half-integral or integral points of the interval [—§,§]). The function GQO(3), which depends on the local data $ v and FVttJ is a meromorphic function in C which can be chosen to be non-zero at any one of the possible poles of E(s,g, F). From this it is clear that L(s + | , w ® XT> p) can have a pole at s = s0 only if E(s,g,F) does. Moreover, a pole of E(s,g,F) will produce a pole of L(s\ 7T ® XT> p) only if this pole survives integration against This circumstance brings us full circle back to the theory of 0-series liftings. In particular, we have the following important Corollaries to the proof just sketched: Theorem [Li] The L-function £s(s, TT (g) XT? p) has a pole at s = 2 if and only if the Fourier expansion of any tp in TT has only one 'orbit' of non-zero Fourier coefficients (pT, i.e., TT is T-distinguished in the sense of [Li]. Remark There is always some nondegenerate T such that <pr ^ 0; this was proved earlier by Li in his thesis [Li.3]. But then for T = 'A^TA" 1 , with A in GLn, we have

(2-4)


t°A]g).

Thus PGSp2(C) w 5O5(C) C GZ5(C)). Then Ls(s9ir ®XT,P) has a simple pole at 3 = 2 if and only if either one of the following conditions are satisfied: (1) the theta-series lifting of TT to GO(T) is non-zero, i.e., ©(TT,^) 7^ 0; or (2) TT is a CAP representation with respect to either the Borel subgroup of G, or the parabolic subgroup Q which preserves a line. Remarks 1. The first condition follows immediately from the fact that Ress-3/2E(g, s, F) is independent of g, and therefore L(s\ TT ®XT-,P) has a pole at s' = 2 iff ,^$,F)

=c I G(F)\


44

Arthur & Gelbart - Lectures on automorphic L-functions: Part II i.e., iff 0(TT,^) is not identically zero on GO(T). (Again, some modification of the theory of dual pairs is required for the similitude groups GO(T) and GSp2] in this case, as already mentioned in a recent remark, 0(TT,VO no longer depends on tp.)

2. We shall come back to CAP representations at the end of Lecture 4. Suffice it now to give the definition: a cuspidal ir is C(uspidal), Associated to a) P(arabolic) if there exists a proper parabolic subgroup P = M£/, and a cuspidal automorphic representation r of Af(A), such that for almost all w, TTV is a constituent of indPvrv. Put more colorfully, TT is in the 'shadow' of an Eisenstein series! 3. A similar characterization of those IT such that L(s, TT® %T> p) might have a pole at a different singularity of E(s,g,F) depends (at least) on the possibility of identifying the remaining residues of E(s,g,F) via some kind of Siegel-Weil formula. For example, for the point s — ^, such a Siegel-Weil formula is developed in [Ku-Ra-So], and applied to give a characterization of the pole of L(s, ir ® XT, p) at s = 1 (analogous to the theorem above). Comments on the Shimura zeta-integral (1) Theorem [P-R.3] above (and also Li's theorem) generalizes to Spn with n even. It gives the finiteness of poles result missing from the authors' earlier work on L(s, TT, p) via the Godement-Jacquet type integral of [PR.l]. (Recall that for the latter zeta-integral, the required non-vanishing of the bad local integrals has yet to be established in full generality.) This result also improves on Shahidi's theory, which gives the finiteness of poles result only for generic TT. (2) The analysis of the poles of the Eisenstein series E(g,syF) is a delicate business involving intertwining operators for the induced representation 7nd|det|* ; this is the subject matter of §4 of [P-R.2] as well as §3 of [P-R.l]. A still more difficult problem is the analogous analysis of intertwining operators and Eisenstein series for the induced (from cuspidal!) representations Ind^r\det\s' on, say, G — SO2n+u until this problem is resolved, the finiteness of poles result for the Rankin-Selberg L-iunctions on G x GL(n) will remain incomplete. For a discussion of what must be proved, see Chapters II and III of [Ge-Sh].

2 Howe's correspondence and the theory of theta-series liftings

45

(3) For Spn, with n odd, these methods must be modified to involve Eisenstein series on the metaplectic group. Indeed 0 T ( # ) is now a genuine function on the metaplectic cover of Spn, and hence must be multiplied by an Eisenstein series of the same 'genuine' type (so that the resulting integrand in the Shimura integral (2.2) will be naturally defined on Spn). For arbitrary n, the theory has not yet been worked out. However, for n = 1 we encounter the zeta-integral

interpolating the degree 3 L-function for SL(2) corresponding to the L-group homomorphism p = Ad: PGL2{C) —» SO2}1(C) C GL3(t). In this case, Gelbart and Jacquet have shown that L(S,TT ® X>^0 is entire for all twists x> unless the theta-series lifting of TT to an isotropic Gf = £0(2) is non-trivial. If this is so, there can be a pole at s = 1, again for reasons of Langlands functoriality. Indeed, such a w is then the Shalika-Tanaka lift of some TT' on 50(2) coming from an i-group homomorphism \I> : LG' —> PGL2(C) with the property that tyop contains the identity representation, i.e., L(s,7c,p) = L(s,7r',/?o\I>) has a degree 1 factor producing a pole at s = 1. In any event, an application of the converse theorem for GL(3) yields the Gelbart-Jacquet lifting from GL(2) to GL(3). (4) A further modification of the zeta-integral (2.2) comes from replacing

46

Arthur & Gelbart - Lectures on automorphic L-functions: Part II

Waldspurger's work (on the non-vanishing of Fourier coefficients and special values of L-functions) and counterexamples to Ramanujan's Conjecture Consider again the dual pair (SL2,PGL2). Recall that in this case, the theory of theta-series liftings gives a bijection between cuspidal a on SL2 with nonvanishing ^>th Fourier coefficients and cuspidal TT on PGL2 with L(ir, | ) -=/=• 0. But the question remains: how can we characterize the non-vanishing of this Fourier coefficient intrinsically in terms of
Here ip (x) = tp(ax) is any character such that 0 ( < J , ^ ° ) ^ {0}, and \a is the corresponding quadratic character x —• (a, x). In [W.2] it is shown that Shtp(a) depends only on 0, and not on the choice of a in F*. From the bijective properties of the correspondence 0(-,^) recalled above, it then also follows that, for any automorphic cuspidal representation ?r of PGL2, 7T = Sh^ia) for some cuspidal a on SL2 if and only if H2, * ® Xa) 7^ 0 for some a in Fx. It is interesting to note that this same characterization of the image of Sh^ has been sketched recently by Jacquet using the 'relative' trace formula (see [Ja]). However, the following remarkable result seems to lie deeper, and is (thus far) proved only in [W.3]. Theorem Suppose a = ®av is an automorphic cuspidal representation of SX2? and / is any cusp form in the space of a. Then / admits a non-zero ^th Fourier coefficient if and only if: (i) for all v, crv has a ^-Whittaker model; and

(ii) Ls{\,Sh+(a))?0.

Sketch of proof The only if direction is easy, since the non-vanishing of the ^-Fourier coefficient immediately implies both the existence of a global (and hence local) Whittaker model, and the non-vanishing of the theta-series lift 6(a,V) = * = Sh+i*) (hence L(A,») = L($,Sh,( 0(cr, 0), it will suffice to show that 0(<J, tp) ^ {0} (since this is equivalent

2 Howe's correspondence and the theory of theta-series liftings

47

to the non-vanishing of the ^th Fourier coefficients of / in <J); i.e., we must prove that for some choice of <£>, $, and g, C(g) =

V(h)e*{gt h)dh ± 0

/

(see (2.1)).

Note that for any

{\xa + x') = i(\xa)<j)2(x')(x' G X'a), so that for t in the stabilizer T of xa in PGL2j we have

i.e., for (t, h) G T x SL2, the Weil representation (or theta-kernel) decomposes in this simple way. Let us now assume ((t) = 0 and show this leads to a contradiction of our hypotheses. Let K = F(y/a), and view T as the anisotropic form of norm 1 elements of Kx. Since Tp\T/\ is compact, we can integrate the integral expression for ((t) with respect to T(F)\T(A), and interchange the order of integration to obtain

o= J


e**(t,h)dt\ dh.

The Siegel-Weil type formula proved by Waldspurger asserts that the integral in parentheses above equals the value at s = | of the Eisenstein series

(Here f,(h) = L(s + ix«)|a*l'"*K(&)*2)(0), where wj denotes the Weil representation associated to the dual pair SO(K) x SL2 C Sp4, and h in SL2 has the Iwasawa decomposition h = ( aJ 8

an

* 2 Jn, with u G K&.) Since

/, belongs to IndB^ \<Xh\ Xai d its value at h = e is essentially L(s + 2 ^,Xa)? E* {h,s) is a familiar normalized Eisenstein series on S ^ - F° r s

48

Arthur & Gelbart - Lectures on automorphic L-functions: Part II

sufficiently large, this Eisenstein series converges absolutely, and hence we can write 0 = value (* (s), where

C(s) = J To complete the proof, we note that (*(s)ls precisely the kind of Shimura-type zeta integral introduced in [GePS] to interpolate the L-function L(s, Sh^(
Here (*{s) is a local zeta-integral which does not vanish identically at s = ^ if and only if crv has a ij)v- Whit taker model. Thus the theorem follows. Remarks concerning the statement and proof of Waldspurger's Theorem (a) The idea of the proof of Waldspurger's theorem can be summarized in one sentence: in manipulating the expression for a theta-series lifting, one runs smack up against a (special value of a) particular zeta-integral; thus the non-vanishing of the lifting must indeed be related to the nonvanishing of a special value of the automorphic L-function interpolated by this zeta-integral. An attempt to generalize this phenomenon was made in 1982 by Rallis, who considered higher dimensional orthogonal groups in place of PGL2, and computed the L2-norm of the resulting lifting in terms of a (then) new kind of zeta-integral of Rankin-Selberg type; see [Ra.2]. It was precisely these calculations which eventually gave rise to the general Rankin-Selberg-Godement-Jacquet zeta-integrals used by Piatetski-Shapiro and Rallis in their treatment of the standard automorphic L-functions attached to the simple classical groups (Theorem [PR.l] of §11.1). It is therefore clear that the theory of theta-series liftings is inextricably linked with the theory of automorphic L-iunctions. (b) Let us call a cuspidal representation ifi — globally generic if (0 is a nontrivial character of the maximal unipotent homogeneous space U(F)\U(f\) and) the space of ^th Fourier coefficients of cusp forms in the space of a does not vanish identically. According to the statement of Waldspurger's theorem, a need not be globally ^-generic even though its factors o~v are locally ^-generic. In fact, explicit examples have been given (by Gelbart and Soudry) of cuspidal representations of SL2(f\) which are everywhere

2 Howe's correspondence and the theory of theta-series liftings

49

locally t/vgeneric and hence possess 'abstract' ^-Whittaker models globally, but are not globally ifp-generic. Clearly it will be interesting to resolve the following: Problem. If TT is a cuspidal representation of an algebraic reductive group G, and each 7rv is ^-generic, prove that TT is globally ^-generic. A special instance of this problem for GSp2 arises in the work of [Bl-Ra]. Some progress towards an affirmative solution has recently been made in the work of [Ku-Ra-So]; here G — 5p 2 , and it is shown that local ^-generic implies global ^-generic provided the degree five i-function of 7T is non-vanishing at Re(s) = 1. (c) The statement (and proof) of Waldspurger's theorem is not exactly true as stated since a few special cusp forms on the metaplectic group fail to lift to cusp forms on PGL2. Examples of such cusp forms include the 'elementary theta-functions' on SL2 arising from the dual pair (SL2,0(1)). The explanation for this phenomenon is a part of Rallis' theory of 'towers of ©-series liftings', which we now briefly describe in this special context. Consider the following sequence of orthogonal groups paired dually with SL2: 50(3,2) SL2

->

50(2,1)

^

0(1)

For each j = 0,1,2, let Ij denote the subspace of (genuine) cusp forms on SL2(f\) whose ©-series lifts to 0(n + l,n) are zero for n < j , but not for n = j . According to [Ra.3]: (i) 70 © /i © I2 exhausts the space of genuine cusp forms on SL2, and (ii) if a C Ij, then the theta-series lift of a from SL2 to its dual pair partner 0(j + 1, j>) is automatically cuspidal (and non-zero); however, the thetalift of this same a to any 'larger' O(n + l,n) is non-cuspidal (and nonzero). In particular, for Waldspurger's dual pair (SL2,PGL2), the ip theta-lift of a will be non-cuspidal precisely when the theta-lift of this same a to 0(1) is non-zero. Such cuspidal a have non-vanishing ^-Fourier coefficients and generalize the classical theta-series Gx(z) = E X (n)n'e 2 "-" 2 '

50

Arthur & Gelbart - Lectures on automorphic L-functions: Part II

where v = 0 or 1, x(""l) = (~^Y\ a n d ©x 1S a c l a s s i c a l cusp form of weight | + v. It is precisely these cuspidal a which spoil the Waldspurger bijection between certain cusp forms on SL2 and PGL2, and hence must be removed. To sum up: the correct space of cusp forms for Waldspurger's bijective correspondence is precisely Rallis' space Ix. What about the space / 2 ? It turns out that J2 is just the space of genuine cusp forms on SL2 with vanishing ^-Fourier coefficients. For example, if tpf(x) = i/>t(x) = i/>(tx), with t $. (Fx)2, then the cusp forms on SL2 which are ^'-theta lifts from 0(1) will have vanishing ^-Fourier coefficients. (Classically, these lifts correspond to theta-series of the form Hx{n)np'e2xin u.) The space of all cusp forms in I2 is the space which Piatetski-Shapiro isolated for study in [PS.2] and showed to be of such great interest in connection with Ramanujan's conjecture. Indeed, let TT on 50(3,2) « PGSp4 be the ^-theta-series lift of a cuspidal space a in I2. From Rallis' theory of towers, it follows that TT is automatically cuspidal. What Piatetski-Shapiro proves in [P-S.2] is that each such 7r also 'satisfies' the following unusual properties: (1) IT provides a counterexample to the generalized Ramanujan conjecture; in fact, such TT'S contain the counterexamples of [Ku] if a does not come from any theta-series attached to a quadratic form in 1-variable, and the counterexamples of [H-P] otherwise; (2) 7T is a CAP {cuspidal associated to a parabolic) representation; (3) the standard (degree 4) .^-function of TT is not entire, and in fact has a pole to the right of the line Re(s) = 1; (4) 7T is not globally ^-generic for any 0; and (5) 7T has a 'unipotent' component in the sense of §1.3. It is this last property which seems to be at the root of the problem of extending Ramanujan's conjecture to groups beyond the context of GL(n) (where properties (2)-(5) are never satisfied by cuspidal representations, and hence one still believes in the truth of the Conjecture). Note that the theory of towers works in the opposite direction as well. For example, corresponding to the diagram 0(2) -» SP2 dpi

=

SL2

one concludes that 'cusp forms' on 0(2) which lift to zero on Spi, i.e., do not play a role in the Shalika-Tanaka construction of cusp forms on SL2 from 0(2), are precisely the forms which lift to (non-zero) cusp forms on Sp2. (It

References

51

is crucial now that we deal with 0(2) in place of S0(2).) Locally, at a place where the quadratic form is anisotropic, we saw earlier that there is just one representation of 0 2 missing from the pairing with SL2, namely the nontrivial character which is trivial on SO(2). The resulting lifted cusp forms on Sp2 are precisely the [H-P] counterexamples to Ramanujan's conjecture mentioned above, and in §3 of Part I.

REFERENCES FOR PART I [A.I] J. Arthur, Automorphic representations and number theory, Canad. Math. Soc. Conf. Proc. 1 (1981), pp. 3-51. [A.2]

, Unipotent automorphic representations: Conjectures, to appear in Asterisque.

[A.3]

, Unipotent automorphic representations: Global motivation. to appear in Perspectives in Mathematics, Academic Press.

[A-C] J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Math. Studies 120, Princeton Univ. Press, 1989. [B] A. Borel, Automorphic L-functions, in Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. 33, Part II, A.M.S., Providence, 1979, pp. 27-61. [B-W] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups and Representations of Reductive Groups, Annals of Math. Studies 94, Princeton Univ. Press, 1980. [F] D. Flath, Decomposition of representations into tensor products, in Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. 33, Part I, A.M.S., Providence, 1979, pp. 179-84. [G] S. Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. 10 (1984), pp. 177-219. [G-J] R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Math. 260, Springer-Verlag, 1972.

52

Arthur & Gelbart - Lectures on automorphic L-functions: Part II [H-P] R. Howe and I. Piatetskii-Shapiro, A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups, in Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. 33, Part I, A.M.S., Providence, 1979, pp. 315-22. [J-L] H. Jacquet and R. Langlands, Automorphic Forms on GL(2), Lecture Notes in Math. 114, Springer-Verlag, 1970.

[J-P-S] H Jacquet, I. Piatetskii-Shapiro, and J. Shalika, Rankin-Selberg convolutions, Amer. J. Math., 105 (1983), pp. 367-464. [J-S] H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations II, Amer. J. Math., 103 (1981), pp. 777-815. [K.I] R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), pp. 611-50. [K.2]

, On the X-adic representations associated to some simple Shimura varieties, to appear in Perspectives in Mathematics, Academic Press.

[K-M] P. Kutzko and A. Moy, On the local Langlands conjecture in prime dimension, Ann. of Math. 121 (1985), pp. 495-516. [Ku] N. Kurokawa, Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two, Invent. Math. 49 (1978), pp. 149-65. [L-L] J.-P. Labesse and R. Langlands, L-indistinguishability for SL(2), Canad. J. Math. 31 (1979), pp. 726-85. [L.I] R. Langlands, Problems in the theory of automorphic forms, in Lectures in Modern Analysis and Applications, Lecture Notes in Math. 170, Springer-Verlag 1970, pp. 18-86. [L.2]

, Representations of abelian algebraic groups, Yale Univ. 1968 (preprint).

[L.3]

, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math. 544, Springer-Verlag, 1976.

References

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[L.4]

, On the notion of an automorphic representation, in Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. 33, Part I, A.M.S., Providence, 1979, pp. 203-7.

[L.5]

,Automorphic representations, Shimura varieties and motives. Ein Mdrchen, in Automorphic Forms, Representations and L- functions, Proc. Sympos. Pure Math. 33 Part II, A.M.S., Providence, 1979, pp. 205-46.

[L.6]

, Base Change for GL[2), Annals of Math. Princeton Univ. Press, 1980.

[L.7]

, On the classification of irreducible representations of real reductive groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Math. Surveys and Monographs, 31, A.M.S., Providence, 1989, pp. 101-70.

Studies 96,

[M-W] C. Moeglin and J.-L. Waldspurger, Le spectre residuel de GL(n), pre- print. [Mg] C. Moeglin, Orbites unipotentes et spectre discret non ramifie, preprint [My] A. Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 108 (1986), 863-929. [P-R] I. Piatetskii-Shapiro and S. Rallis, L-functions for the classical groups, notes prepared by J. Cog dell, in Explicit Constructions of Automorphic L-functions, Lecture Notes in Math., 1254, Springer-Verlag, 1987. [S] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. [Sh] J. Shalika, The multiplicity one theorem for GL(n), Annals of Math. 100 (1974), pp. 171-93.

54

Arthur & Gelbart - Lectures on automorphic L-functions:

Part II

[S-T] J. Shalika and S. Tanaka, On an explicit construction of a certain class of automorphic forms, Amer. J. Math. 91 (1969), pp. 104976. [T.I] J. Tate, Fourier analysis in number fields and Hecke's zeta function, in Algebraic Number Theory, Academic Press, New York, 1968, pp. 305-47. [T.2]

, Number theoretic background, in Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. 33, Part II, A.M.S., Providence, 1979, pp. 3-26.

[Tu] J. Tunnell, Artinys conjecture for representations of octahedral type, Bull. Amer. Math. Soc. 5 (1981), pp. 173-5. R E F E R E N C E S F O R P A R T II [A] J. Arthur, 'Automorphic Representations and Number Theory'. In Canadian Math. Society Conference Proceedings, 1, Providence, R.I., 1981, pp. 3-51. [A-C] J. Arthur and L. Clozel, 'Simple Algebras, Base Change and the Advanced Theory of the Trace Formula'. Annals of Math. Studies 120, Princeton Univ. Press, 1989. [BDKV] J. Bernstein, P. Deligne, D. Kazhdan, M.-F. Vigneras, Representations des groupes reductifs sur un corps local Hermann, Paris, 1984. [Bl-Ra] D. Blasius and D. Ramakrishnan, 'Maass forms and Galois representations'. Preprint 1988. [Bo] S. Bocherer, 'Siegel modular forms and theta series'. In Proceedings of Symposia in Pure Math., 49, A.M.S., Providence, R.I. 1989. [B] A. Borel, 'Automorphic L-functions'. In Proc. Sympos. Math. 3 3 , Part II, A.M.S., Providence, 1979, pp. 27-61.

Pure

[B-J] A. Borel and H. Jacquet, 'Automorphic forms and automorphic representations'. In Proceedings of Symposia in Pure Math., 3 3 , Part 1, A.M.S., Providence, R.I. 1979, pp. 189-202.

References

55

[Bu] D. Bump, 'The Rankin-Selberg Method: A Survey'. In Number Theory, Trace Formulas, and Discrete Groups, Symposium in honor of Atle Selberg, Oslo 1987, Academic Press, 1989. [CL] L. Clozel, 'Motifs et formes automorphes: applications du principe de functorialite'. To appear in Proceedings of the 1988 Summer Conference on Automorphic Forms and Shimura Varieties, Michigan, Academic Press. [Ga] P. Garrett, 'Decomposition of Eisenstein series: Rankin triple products'. Annals of Math., 125 (1987), pp. 209-35. [GePS] S. Gelbart and I. Piatetski-Shapiro, 'On Shimura's correspondence for modular forms of half-integral weight'. In Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979), Tata Institute of Fundamental Research Studies in Math., 10, Bombay, 1981, pp. 1-39; see also 'Some remarks on metaplectic cusp forms and the correspondences of Shimura and Waldspurger', Israel J. Math., 44, No. 2, 1983, pp. 97-126. [GeRo] S. Gelbart and J. Rogawski, 'On the Fourier-Jacobi coefficients of automorphic forms on [/(3)'. In preparation. [GeSh] S. Gelbart and F. Shahidi, 'The Analytic Properties of Automorphic L-functions'. Perspectives in Math. 6, Academic Press, 1988. [Gi] D. Ginsburg, 'L-functions for SOn x GLk\ und angewandte Mathematik. To appear.

Journal fur die reine

[G-J] R. Godement and H. Jacquet, 'Zeta Functions of Simple Algebras'. Lecture Notes in Math. 260, Springer-Verlag, 1972. [Ho] R. Howe, '©-series and invariant theory'. In Proc. Symp. Pure Math., 33, Part 1, A.M.S., 1979, pp. 275-86. [Ho.2]

, 'L 2-duality in the stable range', preprint, Yale University.

56

Arthur & Gelbart - Lectures on automorphic L-functions: Part II [H-P] R. Howe and I. Piatetski-Shapiro, 'A counterexample to the 'generalized Ramanujan conjecture' for (quasi-) split groups'. In Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. 33, Part I, A.M.S., Providence, 1979, pp. 315-22. [Ja]

, 'On the vanishing of some L-functions'. Proceedings Indian Acad. Sciences (Ramanujan Centenary Volume), 97 (Special), Dec. 1987, No.1-3, pp. 117-55.

[J-L] H. Jacquet and R. Langlands, 'Automorphic forms on GL(2)\ Lecture Notes in Math. 114, Springer-Verlag, 1970. [J-P-S.l] H. Jacquet, I. Piatetski-Shapiro and J. Shalika, 'Rankin-Selberg convolutions'. Amer. J. Math. 109 (1983), pp. 367-464. [J-P-S.2]

, 'Relevement cubique nonnormal'. C.R. Acad. Sci. Paris, Ser. I Math. 292 (1981), pp. 13-18.

[J-P-S.3]

, 'The ©-correspondence from GSp(4) to GL{4)\ tion.

In prepara-

[J-S] H. Jacquet and J. Shalika, 'On Euler products and the classification of automorphic representations, I and IF. Amer. J. Math. 103 (1981), pp. 499-558 and 777-815. [KuRaSo] S. Kudla, S. Rallis and D. Soudry, 'On the second pole of the Lfunction for Sp2y. Preprint. [Ku] N. Kurokawa, 'Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two'. Invent. Math. 49 (1978), pp. 149-65. [L-L] J.-P. Labesse and R. Langlands, 'L-indistinguishability for SL(2)\ Canad. J. Math. 31 (1979), pp. 726-85. [L.I] R. Langlands, 'Problems in the theory of automorphic forms'. In Lectures in Modern Analysis and Applications, Lecture Notes in Math. 170, Springer-Verlag, 1970, pp. 18-86. [L.2]

, 'On the functional equations satisfied by Eisenstein series'. Lecture Notes in Math. 544, Springer-Verlag, 1976.

References [L.3]

57

, 'Eisenstein series, the trace formula and the modern theory of automorphic forms'. In Number Theory, Trace Formulas and Discrete Groups, Symposium in honor of A. Selberg, Oslo (1987), Academic Press, 1989.

[Li] J.-S. Li, 'Distinguished cusp forms are theta-series'. Preprint, 1987. [Li.2]

, 'Singular unitary representations of classical groups'. Inventiones Math., 97 (1989), pp. 237-55.

[Li.3]

, 'Theta-series and distinguished representations of symplectic groups'. Thesis, Yale University, 1987.

[MVW] C. Moeglin, M.-F. Vigneras and J.L. Waldspurger, 'Correspondences de Howe sur un corps p-adique'. Lecture Notes in Mathematics 1291, Springer-Verlag, N.Y.1987. [M-W] C. Moeglin and J.-L. Waldspurger, 'Le spectre residuel de GL(n)\ Preprint. [P-S.l] I. Piatetski-Shapiro, 'Zeta-functions of GL(n)\ Preprint, Univ. of Maryland, Technical Report TR 76-46, November, 1976. [P-S.2]

, 'On the Saito-Kurokawa lifting'. Inventiones Math. 71(1983), pp. 309-38; see also 'Cuspidal automorphic representations associated to parabolic subgroups and Ramanujan's conjecture', in Number Theory Related to Fermat's Last Theorem, Neal Koblitz, Editor, Progress in Mathematics, 26, Birkhauser, Boston, 1982.

[P-R.l] I. Piatetski-Shapiro and S. Rallis, 'L-functions for the classical groups'. Notes prepared by J. Cogdell, in Explicit Constructions of Automorphic L- functions, Lecture Notes in Math. 1254, SpringerVerlag, 1987. [P-R.2]

, 'Rankin triple ^-functions'. Compositio Math. 64 (1987), pp. 31-115.

[P-R.3]

, 'A new way to get Euler products'. Journal fur die reine und angewandte Mathematik 392 (1988), pp. 110-24.

58

Arthur & Gelbart - Lectures on automorphic L-functions: Part II

[PS-R-S] I. Piatetski-Shapiro, S. Rallis and G. Schiffmann, '//-functions for G2'. To appear. [PS-So] I. Piatetski-Shapiro and D. Soudry, 'L and e functions for GSp(4) x GL(2)\ Proc. Nat Acad. Set., U.S.A., 82, June 1984, pp. 3924-27. [Ra.l] S. Rallis, 'Langlands functoriality and the Weil representation'. Amer. J. Math. 104 (1982), pp. 469-515. [Ra.2]

, 'L-functions and the oscillator representation'. Lecture Notes in Mathematics 1245, Springer-Verlag, 1988.

[Ra.3]

, 'On the Howe duality conjecture'. Comp. Math. 51 (1984), pp. 333-99.

[Rao] R. Ranga Rao, 'On some explicit formulas in the theory of the Weil representation'. Preprint, 1977-78. [Ro.l] J. Rogawski, 'Representations of GL(n) and division algebras over a p-adic field'. Duke Math. J. 50 (1983), pp. 161-96. [Ro.2]

, 'Automorphic representations of unitary groups in three variables'. To appear Annals of Math. Studies, Princeton U. Press.

[S] A. Selberg, 'Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with application to Dirichlet series'. J. Indian Math. Soc. 20 (1956), 47-87. [Shah] F. Shahidi, 'On the Ramanujan conjecture and finiteness of poles for certain L-functions'. Annals of Math. 127 (1988), pp. 547-84. [S-T] J. Shalika and S. Tanaka, 'On an explicit construction of a certain class of automorphic forms'. Amer. J. Math. XCI, No. 4, 1969, pp. 1049-76. [Shim] G. Shimura, 'On modular forms of half-integral weight'. Annals of Math. 97 (1973), pp. 440-81. [So] D. Soudry, 'The CAP representations of GSp(4,f\)\ angew. Math. 383 (1988), pp. 87-108.

J. reine

References [So.2]

, 'Explicit Howe duality in the stable range'. Math. 396 (1989), pp. 70-86.

59 J. reine angew.

[T.I] J. Tate, 'Fourier analysis in number fields and Hecke zeta functions'. In Algebraic Number Theory, Academic Press, New York, 1968, pp. 305-47. [Tu] J. Tunnell, 'Artin's conjecture for representations of octahedral type'. Bull. Amer. Math. Soc. 5 (1981), pp. 173-5. [W.I] J.-L. Waldspurger, 'Poles des fonctions L de paires pour GL(N)\ Preprint, 1989. [W.2]

, 'Correspondance de Shimura'. J. Math. Pures Appl. 59 (1980), pp. 1-133.

[W.3]

, 'Correspondance de Shimura et quaternions'. Preprint, 1982.

[W.4]

, 'Demonstration d'une conjecture de dualite de Howe dans le cas p-adique, p ^ 2. Preprint, 1988.

Gauss sums and local constants for GL(N) COLIN J. BUSHNELL

Let F be a non-Archimedean local field, G = GLN(F) for some integer N > 1, and 7T a smooth irreducible representation of G on some complex vector space V. (Thus, if N = 1, we have V = C and we may view ?r as a quasicharacter of Fx = GLi(F).) In [6], Godement and Jacquet define zeta-integrals, an X-function £(TT,S) and a local constant £(?r, s,^) attached to TT, completely generalising Tate's treatment [11] of the case N = 1 except in one respect. Tate gives an 'explicit formula' for e(?r, s, tp) in terms of a Gauss sum T(TT,^) essentially identical to a classical Gauss sum. The object of this article is to give an analogous formula in the general case. Apart from a few remarks, there is nothing really new here: all material beyond [6] has already appeared in [2], [1] or [8]. However, this unified account may slightly ease the lot of any potential user. Notation We use the following notation throughout. F = a non-Archimedean local field; oF = the discrete valuation ring in F\ PF = the maximal ideal of o^; qF = (oF • PF) (group index). || • \\F = the absolute value on F such that ||*|| F = (oF:xoF)-\

x€F\

(We frequently omit the subscript F here, as F is fixed for long periods.) tpF = a nontrivial continuous character of the additive group of F. TV > 1 is an integer; A = MN(F), G = GLN(F); ij) = if>A = %l)F o tiAfF, where tr denotes the matrix trace.

62

Bushnell - Gauss sums and local constants for GL(N)

1 THE GODEMENT-JACQUET FUNCTIONAL EQUATION We start with a very brief summary of the main definitions and results of [6] pertaining to the non-Archimedean local case. The survey article [7] is also a convenient reference for this material. We write S(A) for the space of locally constant compactly-supported functions $ : A —» C. If (TT, V) is a smooth irreducible (hence admissible) representation of G, we write (TT , V) for the contragredient or smooth dual of (?r, V), and ( , ) for the evaluation pairing V x V —> C. We denote by M(TT) the space of functions on G spanned by the 'matrix coefficients'

for v G V, v G V. Choose a Haar measure d*g on G, and let s be a complex variable. Define

Z(*,f,s)= [*(g)f(g)\g\'d*g, $6 5 ( 4

/ € M(*),

JG

where \g\ = || det g ||. There exists a0 = cro(7r) G R such that this integral converges for Re(s) > a0. From [7] (1.2) we have: (1.1) There exists a unique function L(n,s) satisfying (i) L(x,s) = P(q-°)-\ for some P(X) G C[X] with P(0) = 1; (ii) L(7T,5)- 1 Z($,/,6) G C[q~%qs] for all $ G 5(A), and all / G M(ir); (iii) there exist $ G 5(A), / G JM(TT) such that L(ir,s) = Z($,/,5). Moreover, if TT is supercuspidal and N > 2, then JL(TT,5) = 1. We now identify A with its Pontrjagin dual A via the pairing (x,y) »-> ^(xz/), x,r/ G A, and let dx be the self-dual Haar measure on A for this identification. Thus, if $ G S(A) and we put •*,

V e A,

then $ G 5(A) and $(x) = $ ( - x ) for a: G A. Also, for / G M(TT), define

feM(ic) by

(1.2) (See [7], (1.3.7)) There exists e(7r, $,) such that

2 Hereditary orders and strata for all $ G <S(J4),

63

/ G M(ir). Moreover, we have 6(TT, 5,^).e(if, 1 - 5, ij>) = ^ ( - 1 ) ,

where uT is the central quasicharacter of TT. In particular, £( 7 r, S ,^) = C( 7 r,V).9- m ', for some (7(TT, I\)) G C of absolute value 1 and some m = m(7r) G Z. Now let ^ f be some other nontrivial continuous additive character of F, and put V>i = V>f ° tr. There exists c € Fx such that ^lfa) = ^(ca0> # G A, and we have (1.3)

£(*,*,&) =«ir(c)|c|(-i>w.£(ir,a,^).

This completes the list of basic definitions we require. Notice that £(TT,S) is known when TT is supercuspidal, the case N — 1 being given by [11], and the others by (1.1). Jacquet shows in his survey article [7] how L(T,S) can be computed in the general case, using the supercuspidal one and the Zelevinsky classification of the representations of G. Using the same procedure, we need only compute e(7r,3,^) when ?r is supercuspidal. Indeed, we need only treat the case N > 2, appealing to [11] for the abelian one N = 1. In fact, our method is more general than this. 2 HEREDITARY ORDERS AND STRATA It is for the moment more convenient to identify our algebra A with End^ (V), where V is some JP-vector space of finite dimension TV > 1. An oF-lattice chain in V is a set C = {L{ : i G Z}, where each Li is an oF-lattice in V (i.e., a finitely generated o/r-submodule of V which spans V over F) such that (i) Li D L i+ i, Li ^ Z i+ i, i G Z, and (ii) there exists e G Z such that Li+e = ^PfX,-, i G Z. The integer e = e(C) is uniquely determined. We attach a ring ii = lt(£) to the lattice chain C by

a = H EndOF(Lt.) = f)End Ojp (^). *=0

This is an oF-lattice, hence an oF-order, in A. We can recover the lattice chain C from it(£), up to a shift of index, since C is precisely the set of all left ii-lattices in V. Thus we may view the 'period' e of the lattice chain as a

64

Bushnell - Gauss sums and local constants for GL(N)

function of it, e = e(it). Orders il in A obtained in this manner we refer to as hereditary orders. For a full account of these, see [10]. There is a convenient summary of the facts needed for the present situation in [2]. Now, for n G Z, we define Vn = VI = {x e A : xLi C Li+n,

i e Z}.

1

Then ty = ^P is just the Jacobson radical of il, and *$n is the nth power of jp. The meaning of this is obvious if n > 0; otherwise, we recall that ^p is an invertible (it, it)-bimodule. In particular, we have the relation pFU =
and

C/'(il) = 1 + V\

for i > 1.

Also, we put J?(it) = Aut(£) = {x e G : xC = £ } . This is an open subgroup of G, and it is compact mod centre, i.e., &(il)/Fx is a compact subgroup of G/F*, where we identify Fx with the centre of G. The groups (/'(it) are all normal subgroups of £(ii). Moreover, we have

U0(U)/U\iL) <*

'f[Aut.,/tr{Li/Li+1),

»=0

while

^(it)/f/t+1(a) s
for i > 1, via the map

I H J - 1 .

We now recall our additive character i\) = i\)F o tr^/^ of A. It is easy to work out the effect of changing the basic character ^ F , as we shall see below, and it is convenient for our present purposes to assume that the conductor of^F is pp. Thus ^>F is trivial on pF but not on op. If M denotes some oF-lattice in ;4, define M* = {x eA:

xl){xm) = 1,

m € M}.

With our choice of 0, we get This gives us a very useful description of the Pont rj agin duals of the finite groups U'/Uw.

2 Hereditary orders and strata (2.1) Proposition given by

65

The map b i-> ^ , where ij)h is the character of U{/Ui+1

establishes an isomorphism

for any i > 1. We write 0 H* £* for the inverse of the isomorphism of (2.1). A stratum in A is a triple (it, n, 0), where It is a hereditary order in A, n is a non-negative integer, and 0 is an irreducible representation of C/n(il)/£/n+1(it). Thus, in the case n = 0, 0 is of the form

where e = e(il) and 0,- is an irreducible representation of the finite general linear group Aut(X,-/L,-+i) over OF/PF- We say that the stratum (it, n,0) is fundamental if (2.2) (i) ra = 0, and each 0,- is a cuspidal representation, or (ii) n > 1 and ^ does not contain a nilpotent element of A. Now let 7T be some smooth irreducible representation of G = Aut F (F). We say that TT contains the stratum (it, n, 0) if the restriction of ?r to the subgroup Un(il) contains the representation 0 (inflated to a representation of Un in the obvious way). The definition of smoothness implies that ?r contains some stratum. (2.3) Theorem Let 7r be a smooth irreducible representation of G, and let (il, n, 0) be a stratum contained in TT. The following conditions are equivalent: (i) for any other stratum (it'jn 7 ,^) contained in TT, we have n/e(il) < n'/e(il') and, if n = n' = 0, then e(il) > e(lt'); (ii) (It, n, 0) is fundamental. This result was originally conjectured by A. Moy. It is proved in [1], It follows immediately from (2.3) that a smooth irreducible representation IT of G contains some fundamental stratum. This fundamental stratum is not unique in any absolute sense, but has strong and useful uniqueness properties. For the moment, let Hi be some subgroup of G, and pi an irreducible representation

66

Bushnell - Gauss sums and local constants for GL(N)

of Hi, i = 1,2. Recall that an element x £ G intertwines 0X with 02 if there is a nontrivial x~lHxx fl .HVhomomorphism from 0* to 02- K we do not wish to specify the intertwining element x , we simply say that 01 and 02 intertwine in G. Any two strata which are contained in the same irreducible ?r must intertwine, and we have a very strong necessary condition for two fundamental strata to intertwine. To state this, we first need another definition. Let (IX, n, 0) be some fundamental stratum with n > 1, and let 6 £ 6e, so that £0 = & 4- qj 1 "". Fix some prime element LJF of F, set g = gcd(n, e), e = e(il), define y £ il/9? by and let <^(£) £ (<>F/PF)M be the characteristic polynomial of y viewed as an endomorphism of, say, L0/UJFL0. This definition is independent of the choice of b £ 6$ and depends only trivially on the choice of uF. Further, choosing a different basic character ij)F would replace 6$ by c6$, for some c £ Fx. We would have to define y as u;™& e/tf, with the integer m chosen to ensure y £ H, ^ fp. All this has no material effect on the relevant properties of <}>e(i). Note that <j>$(t) is not a power of t , since the stratum is fundamental. (2.4) Proposition Let (il,n,0), (iX',n',0') be fundamental strata, and assume that 0, 0' intertwine in G. Then n e(il)

n e(il')'

If n = n' = 0, then e(il) = e(il') and the cuspidal tensor factors of 0' are the same as those of 0 up to isomorphism, possibly after a renumbering. Otherwise, we have

e(t) = M O When n > 1, we now have two cases to consider. A fundamental stratum (IX, n,0) is called split if the polynomial $(t) has at least two distinct irreducible factors (over the field oF/pF). (2.5) Theorem Let TT be an irreducible smooth representation of G, and suppose that TT contains a split fundamental stratum. Then TT is not supercuspidal. This version is proved in [8]. However, in recent joint work [3] of the author and P. C. Kutzko, it is shown that a representation TT containing a split fundamental stratum is in fact irreducibly induced from a maximal proper parabolic subgroup of G. It is therefore the other case, in which $(t) is a

3 Non-abelian congruence Gauss sums

67

power of some irreducible polynomial, which will concern us. We neglect the case in which n = 0. This is actually easier, but requires a different panoply of definitions. Now let 11 = ii(£) be some hereditary order in A, with C = {Li} as before. The lattice chain C is called uniform if the index (L,- : Li+i) is constant, independent of i. A hereditary order It = it(£) is called principal if the lattice chain C is uniform. There are various equivalent formulations of this. For example, it is principal if and only if *# = ilx, for some x E G. Moreover, it is principal if and only if £(it) is a maximal compact-mod-centre subgroup of G . We refer to [2] for the details of this. Now let (it, n, 0) be some fundamental stratum with n > 1. We say that it is simple if there exists an element a E S6 satisfying (i) E = F[a] is a field; (ii) Ex C i?(it) (i.e., Ex conjugates it into itself); (iii) a is minimal over F. In (iii), the term minimal means the following: (a) gcd(rc,e(£ | F)) = 1, and (b) for a prime element UJF of F, the residue class of ae^F\ujpa^ ptf) generates the residue class field of E over that of JP.

(modulo

We note that a simple stratum (it, n,0) is automatically not split. Moreover, we have 6e C i?(it). The next result is also taken from [8], but note that simple strata are there called alfalfa. (2.6) Theorem Let ?r be a smooth irreducible representation of G, and assume that 7T contains a non-split fundamental stratum (il,n,0) with n > 1. Then 7T contains a simple stratum. In particular, we note that if ?r is supercuspidal, then it must contain a simple stratum (or a stratum (it, 0,0)). 3 NON-ABELIAN CONGRUENCE GAUSS SUMS Now let il be a principal order in A = End F (V). Thus il = it(£) for some uniform lattice chain C = {L{} in the vector space V. So, there exists i G G such that xLi = L, +1 for all i E Z. For any such x, we have tp = itx = a:it, where, as before, ^P denotes the Jacobson radical of it. Indeed, we get ?pn =

68

Bushnell - Gauss sums and local constants for GL(N)

ilxn = xnii for all n G Z. Further, the compact-mod-centre subgroup £(H) of G is the semidirect product of itx with the infinite cyclic group generated by any such x. See [2] for details. Let p be a smooth irreducible representation of i?(il) on some complex vector space W. Then W is finite-dimensional, and the restriction of p to the centre Fx of £(il) is a multiple of some quasicharacter wp of Fx. Further, there is an integer / > 0 such that p is null on the open normal subgroup U*(tt) of Take / minimal for this property, and put (3.1) We also use the (slightly disreputable) convention from the abelian theory of Gauss sums that (3.2)

1 + f(p) = U*(U). F

Now let %l> = ip o tvA/F be our additive character of A, as before. We no longer make any particular assumption about the conductor of ij)F. Define

(3.3)

T(p, 0) = £ rtc-'sMc-1*) € Endc(W),

where, in the summation, x ranges over a set of representatives of il x modulo l + f(/o), and c is any element of £(il) such that c~xii = (f(/>))*. Schur's Lemma shows that the definition of T(p, ip) is independent of all these choices, and also that there exists r(p,ij>) G C such that (3.4)

T{p^)

= T{p,i>).lw.

This T(p,if>) is the non-abelian congruence Gauss sum of the representation p. As we shall see below, this Gauss sum has many properties in common with the abelian congruence Gauss sum defined analogously in the case TV = dimF(V) = 1. There is one serious difference, however, in that, for TV > 1, r(p, 0) can vanish. We now describe this phenomenon. Suppose for the moment that j(p) = ty? with / > 2, and consider the restriction of p to f//~1(it). This restriction decomposes as a direct sum of abelian characters 0 of U^~x/U^, and we may consider the sets 6e defined in §2. We say that p is nondegenerate if 6d C *(U) for some (equivalently any) 6 occurring in p \ i7^"1(il). Moreover, the set 8e is contained in i?(il) if and only if it meets it nontrivially. Immediately from the definitions we have

3 Non-abelian congruence Gauss sums

69

(3.5) Proposition Let p b e a smooth irreducible representation of £(il) with conductor ^ , / > 2. Let 0 be a character occurring in p | (/'""^(ii), and suppose that the stratum (it,/ --1,0) is simple. Then p is nondegenerate. We also say that p is nondegenerate if its conductor is f(/>) = it or $J. Now we have (3.6) Proposition The representation p of £(it) is nondegenerate if and only if T ( / ) , ^ ) is nonzero.

(3.7) Theorem Let ?r be a smooth irreducible representation of G and it a principal order in A. Suppose that ir satisfies the following conditions: (i) JL(TT,S) = 1,

and

(ii) the restriction TT | £(ii) contains a nondegenerate representation p. Then where p denotes the contragredient of p. The hypotheses (i) and (ii) hold when TT is supercuspidal. Remarks If TT is supercuspidal (and N > 2), we know from (2.6) that it contains a simple stratum (it, n,0) or a stratum (ii, 0,0). (It is easy to see, in the latter case, that we may choose it to be principal: indeed, we can take ii = MN(oF) here.) There is certainly an irreducible representation p of £(it) occurring in TT and containing this 0. Any such p is nondegenerate in the above sense. This proves the last assertion of (3.7), and gives the desired formula for the local constant in the supercuspidal case. Note that it is an exact generalisation of Tate's formula in the case N = 1. A different proof of the last assertion is given in [1], involving a more detailed analysis of the functional equation. The one we have given here, via Kutzko's theorem (2.6), seems more direct and its ideas have proved more useful. The hypotheses of (3.7) hold for more general representations TT. It follows from the forthcoming [3] that a representation TT for which they fail has either to be irreducibly induced from a maximal parabolic subgroup of G, or to have a fixed vector for an Iwahori subgroup of G.

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The conditional version of (3.7) holds also when G = GLN(F) is replaced by GLM(D), where D is a finite-dimensional central F-division algebra of dimension cP, dM = N, except that a factor (—l)^**""1) has to be inserted in the formula for e. However, in this degree of generality, no results of the type (2.6), or even (2.5), are known. Finally, it has to be remarked that the proof of (3.7) in [2] apparently demands that the field F be of characteristic zero. However, once one defines, as here, a Gauss sum attached to an arbitrary additive character ij>F, the proof goes through in all characteristics. A different description of the exponential factor of £(TT,S,^) has been given by Jacquet/Piatetski-Shapiro/Shalika in Math. Ann. 256 (1981) 199-214.

4 ARITHMETICAL PROPERTIES OF GAUSS SUMS Again let p be an irreducible smooth representation of the normaliser &(il) of some principal order il in the algebra A = MN(F), and r(p,ij;) its Gauss sum, as above. We note first that this behaves very simply under change of the additive character if). If if>' is any other additive character of A of the requisite type, then there exists some b G Fx such that if>'(x) = if>(bx), x G A. We find (4.1)

r{p^) = r{p^).ujp{b)-\

where u?p is the central quasicharacter of p. Now let x be an unramified quasicharacter of G. Thus X = Xo° det for some unramified quasicharacter Xo of F x , and we may view x a s a quasicharacter of £(il) by restriction. It is then immediate that

for any c G £(ll) such that clt = (f(p))* (where 'star' is defined relative to if? as in §2). Thus, in many questions concerning Gauss sums, it is enough to treat the case in which the central quasicharacter u>p is of finite order. (4.2) Proposition Let p be an irreducible representation of i?(il), as above, and let p be the contragredient of p. Let p be the residual characteristic of F, and let f(p) =
where u(p) is given by

4 Arithmetical properties of Gauss sums

71

(i) 0 if p is degenerate; (ii) (ii : f(p)) if p is nondegenerate and / > 2; (iii) pk for some nonnegative integer k such that pk divides (IX : f(/?)) otherwise. (4.3) Corollary Suppose that the central quasicharacter u>p of p has finite order. Then the absolute value of r(/>,?/>) G C is For a detailed interpretation of case (4.2)(iii), see [2] and [5]. If we stick to the case of representations p attached to supercuspidal representations of G, we can neglect some of these possibilities. If f(/>) = *# or il, we say p is nondeficient if v(p) = (ii: f(/>)). As a supplement to (3.7) we have (4.4) Lemma Let p be an irreducible representation of the normaliser of a principal order il such that f(p) = il or *#. Suppose also that p occurs in some irreducible representation TT of G with £(TT,,S) = 1. Then j(p) = *# and p is nondeficient. This is proved in [2], and provides some justification for regarding nondeficient nondegenerate representations as somewhat more interesting than the others. In particular, the property that the absolute value of the Gauss sum of such a representation is the square root of the norm of the conductor is exactly what one would expect from the classical case. Now suppose that F has characteristic zero. Thus F is a finite extension of Qp. Let QQ denote the Galois group Gal(Q/Q) of an algebraic closure Q/Q. We write \P f° r the p-adic character Q,Q —> Z£ given by

for a G OQ and any p-power root of unity ( in Q. If our representation p has UJP of finite order, then it is easy to see that the image />(i?(il)) is finite. It follows that p may be realised over the ring of integers in some algebraic number field. We draw two conclusions. First, we can 'twist' p by Galois automorphisms a G SIQ and, second, r(/?, ij)) is an algebraic integer. We then get a formula which, although very easy to prove, is nontheless remarkable.

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(4.5) Proposition Let p be an irreducible representation of £(il) as above, and suppose that UJP has finite order. Then r(p*~\tl>Y =

T(^)^P(XP(<7)),

for all a E fiQ.

There is a strikingly similar formula for the so-called Galois Gauss sums. Let fiF be the Galois group of some algebraic closure of JP (which we continue to assume to be a finite extension of Qp). It is also convenient to take our basic character xj)F of F to be the Iwasawa-Tate character. This is the composite of the field trace F —> % with the obvious embedding of Qp/Zp in the unit circle. In the formula for £(TT,S,V0 given by (3.7), the factor ((f(p)* : It) is then just where %)p is the absolute different of F. Now let /i be a continuous finite-dimensional representation of fiF, and r(//,^ F ) its Galois Gauss sum (with ipF as above). (See, for example, [9] for a treatment of Galois Gauss sums.) This Gauss sum is related to the Langlands-Deligne local constant (see [4] or [12]) e(//,s,^ F ) by the formula £ ( M , S ,V

F

) = (oF : f ( p ) ^ m ( " ) ) ( 1 / 2 - ) . r ( / i , ^ ) . ( o F : f^))" 1 ' 2

where f(//) is the Artin conductor of //. Apart from the similarity between this formula and (3.7), we get a Galois action formula for the Galois Gauss sum: for all a £ fiQ. Here det^ denotes the character of F* associated to the one-dimensional representation det(//) of Clp via class field theory. Note here that Fx contains Zp and hence the values of Xp • Comparison of this formula with (4.5) is particularly interesting in the context of the local Langlands conjectures. REFERENCES 1 C. J. Bushncll, Hereditary orders, Gauss sums and supercuspidal representations of GLN, J. reine angew. Math. 375/376 (1987) 184-210. 2 C, J. Bushntll and A. Frohlich, Non-abelian congruence Gauss sums and p-adic simple algebras, Proc. London Math. Soc.(3) 50 (1985) 207-64.

References

73

3 C. «/. Bushnell and P. C. Kutzko, The admissible dual of GLN via compact open subgroups, in preparation. 4 P. Deligne, Les constantes des equations fonctionelles des fonctions L, Lectures Notes in Math. 349, Springer 1973, 501-95. 5 A. Frohlich, Tame representations of local Weil groups and of chain groups of local principal orders, Heidelberger Akad. Wiss., Springer 1986, 1-100. 6 R. Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math. 260, Springer 1972. 7 H. Jacquet, Principal L-functions of the linear group, Proc. Symposia Pure Math. 33, Amer. Math. Soc. 1977, 63-86. 8 P. C. Kutzko, Towards a classification of the supercuspidal representations of GLN, J. London Math. Soc.(2) 37 (1988) 265-74. 9 J. Martinet, Character theory and Artin L-functions, Algebraic Number Fields (A. Frohlich, erf.), Academic Press 1977, 1-87. 10 /. Reiner, Maximal Orders, Academic Press 1972. 11 J. Tate, Fourier analysis on number fields and Hecke's zeta functions, thesis, Princeton University 1950. Also Algebraic number theory (J. W. S. Cassels & A. Frohlich ed.) Academic Press 1967. 12 J. Tate, Local constants, Algebraic Number Fields (-4. Frohlich ed.), Academic Press 1977, 89-132.

L-functions and Galois modules PH. CASSOU-NOGUES, T. CHINBURG, A. FROHLICH, AND M. J. TAYLOR Notes by D. BURNS AND N. P. BYOTT

0 INTRODUCTION 1 THE TAME ADDITIVE THEORY - based on a lecture given by A. FROHLICH 2 ARTIN ROOT NUMBERS AND HERMITIAN GALOIS MODULES - based on a lecture given by Ph. CASSOU-NOGUES 3 ON SOME PARALLEL RESULTS IN ADDITIVE AND MULTIPLICATIVE GALOIS MODULE THEORY - based on a lecture given by A. FROHLICH 4 ADDITIVE-MULTIPLICATIVE GALOIS STRUCTURES - based on a lecture given by T. CHINBURG 5 EXPLICIT GALOIS MODULES - based on a lecture given by M.J. TAYLOR 0 INTRODUCTION Let N/K be a finite Galois extension of number fields and let T = Gal(N/K) denote its Galois group. In these lectures we study the structures of certain 'Galois modules' arising in this context such as the ring of algebraic integers ON of iV, the multiplicative group UN of units of ON modulo torsion elements, and the ideal class group CIN of N. A striking feature of the theory discussed is the close interplay between the module structure and the arithmetic of the number fields. In particular (Artin) L-functions play a fundamental role in various places. What results however is much more than theorems and conjectures relating two very abstract and sophisticated points of view - the module theoretic aspect is frequently very down to earth, and indeed some indication of the power of the theory lies in its capacity to provide concrete and explicit information.

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There are several approaches to these Galois module theoretic problems and we shall discuss different approaches in separate paragraphs. The case which is best understood is that of ON when N/K is at most tamely ramified. Here, ON is a (locally-free) Z[F]-module and in §1 we discuss the fundamental result which shows that its structure as such is determined, up to stable isomorphism, by the root numbers occuring in the functional equation of the Artin L-functions of N/K attached to the irreducible symplectic characters of F. The material of this section is also useful in providing motivation for many of the approaches adopted in more general situations. In §2 we again restrict to Galois extensions which are at most tamely ramified but here consider the converse problem of recovering the symplectic root number values from the Galois module structures attached to N. The Z[F]structure of ON alone is not sufficient for this. However the root numbers are completely determined by the isometry class of the Hermitian Z[F]-module consisting of ON together with the trace form, and this is the subject of §2. For wildly ramified Galois extensions there is even today no general theory analogous to that discussed in §1 and §2 for extensions which are at most tamely ramified. In §3 we recall the notion of factor-equivalence between modules which provides additional insight into the module structure of ON for wildly ramified extensions. Furthermore factor-equivalence allows one to exhibit some remarkable, and at the present time unexplained, similarities between the 'additive' structure theory of ON and the 'multiplicative' structure theory of UN and this is an underlying theme throughout §3. In particular the results discussed here raise integral variants of theorems, problems and conjectures relating to Tate's formulation of the Stark conjectures. A basic technique in studying arithmetic Galois modules is to compare them with given 'standard' modules which have better understood structures. In §1 and §3 this is done by constructing suitable module invariants. In §4 we continue with this general approach but now use exact sequences and comparison of the resulting cohomology theories. In particular we discuss the construction of certain cohomological invariants of a Galois extension which simultaneously reflect both the 'additive' and 'multiplicative' Galois module structures attached to N. The invariants of §4 are constructed in the context of general global fields N/K and without any hypotheses on ramification (of N/K). They are elements of the locally-free class group C7(Z[F]) of Z[F]. Conjecturally at least these invariants are closely related to other aspects of the arithmetic of N/K and, in particular, to the root number

1 The tame additive theory

77

classes introduced in §1. The Stark conjectures (again in Tate's formulation) also have interesting implications for the material of this section. Finally, in §5, we consider the problem of finding explicit Galois generators for Opf (and other related modules) for certain special classes of extensions. The classical example of this type of result is Leopoldt's Theorem which gives such a generator for any abelian extension of the field of rationals Q. We discuss how, under suitable hypotheses, an elliptic curve with complex multiplication can be used to provide analogous generators for the rings of integers of extensions arising from its torsion points. In particular we give a result which may be considered as an integral contribution to the famous 'Jugendtraum' of Kronecker. We mention here two of the conventions in use throughout this article. Let K be a number field and let OK denote its ring of integers. Let G denote an arbitrary finite group, with A any C^-order in the /^-algebra if [G]. With these notations all .4-modules to be considered in this article are finitely generated right ^4-modules. By a 'prime divisor', or equivalently a 'place', of if, we shall mean an equivalence class of non-trivial valuations of K. Such a prime divisor (place) is 'finite' if it comes from some non-zero prime ideal of OK- Otherwise it is said to be 'infinite', or equivalently, 'archimedean'.

1 THE TAME ADDITIVE THEORY In this paragraph our concern is with the Galois structure of ON, primarily in the context of tamely ramified Galois extensions. Of course ON naturally hcts the structure of 0/f[F]-module but for our purposes it is more fruitful to restrict scalars and consider ON qua Z[F]-module. At the field level then, the Normal Basis Theorem implies an isomorphism of Q[r]-modules N ^ K[T] (1.1) and hence, on the integral level, it is natural to compare the Z[F]-module structures of ON and (9# [F]. More specifically, writing ~z\r] for the relation of Z[F]-genus equivalence (that is, isomorphic F-module structure after tensoring with lp for each rational prime p) one might first ask for the conditions under which ON ~i[r] OK[T]. In the sequel we shall say that an extension is 'tame' if it is at most tamely ramified.

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Theorem 1 (Noether, 1932) Let N/K be a finite Galois extension of number fields of Galois group F. Then ON ~Z[r] 0*r[F] if, and only if, N/K is tame. But 0 # [ r ] is a free Z[F]-module and hence, for a tame extension N/K it is a natural problem to determine the global structure of the locally-free Z[F]-module ON- In the 1970s a theory was developed in response to this problem, the 'tame additive theory', and is now more or less complete. In this paragraph we shall nevertheless give a brief survey of certain of its aspects which will provide us with a guide for the underlying strategy and for possible results to aim for in the general case.

1.1 The locally-free class group We must first introduce a suitable classifying group C7(Z[F]), the 'locally-free class group' of Z[F]. If X and Y are any two locally-free Z[F]-modules such that I ®i Q = F ®i Q then C7(Z[F]) will classify them to within stable Z[F]-isomorphism. That is, X and Y will be represented by the same element of C/(Z[F]) if, and only if, there is an isomorphism of Z[F]-modules

Remark 1.2 Under certain conditions on (the absolutely irreducible complex valued characters of) F - satisfied for example if F is either abelian, dihedral, or of odd order - stable Z[F]-isomorphism implies Z[F]-isomorphism but this is not true in general. Let jftT0(Z[r]) denote the Grothendieck group of the category of locally-free Z[F]-modules modulo relations given by direct sums. Each locally-free Z[F]module X has a rank n(X) E N defined by X <8>z Q £*

)

and extending the map X i—> n(X) to a homomorphism rk : K0(Z[T}) —+ Z one defines C/(Z[F]) by the exactness of the sequence 0 — • C7(Z[F]) —-> KO(I[T]) - ^ Z —+ 0. Equivalently, defining an injective group homomorphism 0 —> Z - £ • KO(1[T])

(1.3)

1 The tame additive theory

79

by n A n(Z[F]),

n> 0

where here (X) denotes the class of a locally-free Z[F]-module X in /
(1.4)

Both descriptions (1.3) and (1.4) are however inappropriate for arithmetical computations and so we must work in terms of a description, the 'Horndescription' of C7(Z[F]), in terms of functions on characters of F. For this we let E denote any Galois extension of Q (contained in the algebraic closure Qc of Q in C) over which all absolutely irreducible complex representations of F can be realised. For convenience we also assume that N £ E. We write J[E) for the multiplicative group of ideles of i?, and Rr for the additive group of (complex) virtual characters of F. For any number field K we shall write CtK for the absolute Galois group Ga/(QC/K). For K == Q we shall frequently abbreviate this by Ct = fiQ. Then there is an exact sequence Homa(Rr, J{E)) -£* C7(Z[F]) —•> 0

(1.5)

where for an explicit description of the projection map ?rr = 7rr>£; the reader is referred, for example, to Frohlich (1983). For the reader's convenience however we shall now briefly recall, for any given locally-free rank one Z[F]-module X, the recipe for constructing a homomorphism / £ Homn(Rr,J(E)) such that ?r r (/) is equal to the element (X)z[r] of C^(Z[F]) corresponding to X. Consider a representation T : F —> GLn(E). Let i? denote either Q or Qp for some rational prime p. The representation T extends by J9-linearity to give a B-algebra homomorphism T:B[T]—+Mn{E®QB) which in turn induces a homomorphism DetT : (B[T]Y -2U GLn(E ®Q B) -^ {E ®Q B)* . In fact DetT = Detx depends only upon the character x °f ^ n e representation T. Choose now a free generator x of X ®2 Q over Q[F] and, for each rational prime divisor p choose a free generator xp of X ®2 Zp over Zp [F]. Both x and xp are free generators of X ®2 Qp over Qp [F] and hence xp = x\p , Ap G (<

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Cassou-Nogues et al. - L-functions and Galois modules

defined componentwise at each character The map / G Homn(Rr,J(E)) ofTby (f(X))p = Detx(Xp) for each rational prime divisor p, then satisfies

The advantage of this 'Horn-description' is that it allows us, as we shall presently see, to relate the algebraic and arithmetical aspects of the theory and so to state and prove the basic results. Apart from this it is also useful for computation and is a powerful tool for the derivation of functorial properties under change of group and so on. Before proceeding we note that the double role of number fields apparent here - both as the objects of study and as the ranges in which the classifying functions take their values - is a typical feature of the theory. The Horndescription also gives us a first indication of the absolutely fundamental role played throughout the theory by functions on Galois characters. 1.2 The arithmetic theory Now given a tame Galois extension N/K a central problem is therefore to find a homomorphism / = fN/K €Homn(Rr^J(E)) with

Mf) = (Os)mQuite remarkably, in order to describe such a homomorphism we must turn to the Artin L-function attached to each complex character x of T. For the moment we may also relax the restriction that N/K is tame. For any character x of F we write L(s>x) f°r th e extended Artin L-function attached to N/K and to x* For each such x we denote by x its contragredient. Then Z(s, x) has a meromorphic continuation to the entire complex plane and satisfies a functional equation L(s,x) = w(x)A(X)l-'L(l-s,x)

(1.6)

where A(x) = A(N/K,x) > 0 is a positive real constant and w(x) = w(N/Kyx) ls a complex constant of absolute value one. In the literature io(x) is known as the '(global) Artin root number' of x« In particular then if x is real valued then x = X a n d w(x) = ±1. To decide this parity question it is convenient to consider two sub-classes of real-valued characters. We call a character x 'orthogonal' (respectively 'symplectic') if it is attached to a

1 The tame additive theory

81

representation that factors through the group of orthogonal (respectively symplectic) matrices. Orthogonal and symplectic characters are real-valued and, conversely, any real-valued character can be written as a Z-linear combination of orthogonal and symplectic characters. If the character x is orthogonal then in fact to(x) = +1- On the other hand if x is symplectic then w(x) may be either +1 or —1 and, if N/K is tame, the values of w(x) f° r all symplectic characters x of F actually determine the element (ON)Z^\ To make this striking remark more precise we define an element

WNIKeHomu(RT,J{E)) in terms of the values w(x). (This definition was originally due to Ph.CassouNogues). For this we need an 'adjusted' root number defined at each absolutely irreducible character x of F by >(x), ,

if X is symplectic; otherwise.

and extended to Rr by linearity. In fact if N/K is tame then for each r/ £ 0 and character x of F one has w'(xv) = w'(x)

(1.7)

so that the homomorphism X i—> w'(x) is ft-equivariant. Note however that equation (1.7) is not in general true for wild (that is, non-tame) extensions. Next, for each virtual character x of F we define an idele wN/K(x) e J(Q) c j(E) via its local components:

(WN/K(x))p

= w'(X), all finite p.

We write WN/K for the corresponding element of Hom^(Rr^J(E)) (for the fi-equivariance of WN/K recall (1.7)). Then the main theorem in this section gives, for any tame Galois extension N/K, a complete classification of the global structure of the locally-free Z[F]-module ON in terms of the homomorphism WN/K. The result of this theorem was originally conjectured by Frohlich.

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Theorem 2 (Taylor (1981)) If N/K is a tame Galois extension, then

Of course if F has no irreducible symplectic characters then WN/K is trivial. Furthermore under these conditions stable Z[r]-isomorphism is equivalent to Z[F]-isomorphism (see remark 1.2) and therefore Corollary 3 If N/K is a tame Galois extension which has Galois group possessing no irreducible symplectic character, then oN s '*Ql In particular then if F is abelian, or dihedral, or of odd order then ON is a free Z[F]-module. This justifies to a considerable degree our claim that the theory produces explicit concrete information. Thus for tame extensions the Artin root numbers determine the Z[F]-module structure of ON. However as the following example shows the converse is not true. Example 1.8 For any integer m we write if4m for the generalised quaternion group of order 4m. Then Cl(l[H8]) has order two, and (as first demonstrated by Martinet) there exist tamely ramified extensions JV/Q with Ga/(iV/Q) = Hs and (ON)Z[T] = + 1 , and also tamely ramified extensions 7V/Q with Gal(N/q) = H8 and (ON)m = - 1 . On the other hand however, Cl(l[H16\) also has order two, and whilst there exist tamely ramified extensions N/Q such that Gal(N/fy) = ifi6 and the symplectic root numbers are of each possible value, one knows that for any such extension (^jv)zrri = + 1 . Hence, in order to determine each of the root numbers w(x) by means of the Galois module structures attached to JV, one must consider more than just the Galois module structure of ON. In fact the root numbers are completely determined by the Z[F]-structure of the Ox[r]-Hermitian module (ON, TN/K) where TN/K : N x N —> K[T] is derived from the trace pairing TrN/K :N x N—> K, which is given by TrN/K(x,y)

= traceN/K(xy),

all x, y G N.

1 The tame additive theory

83

We shall discuss this more fully in §2. 1.3 The basic techniques For the remainder of §1 we shall discuss some of the ideas behind the proof of Theorem 2. This will in fact provide useful motivation for many of the approaches to be adopted in subsequent sections. At the heart of the proof of Theorem 2 are invariants constructed to interpret the connections between the module theory and the arithmetic of tame Galois extensions N/K. In fact these invariants, or{resolvents', generalise the classical notion of Lagrange resolvents to the case of non-abelian characters of F. For a complex character x °f I\ corresponding to the representation T, and any element a of N such that a.K[T] = N one defines the resolvent of a and x to be (a | X ) = where here E is as before any 'sufficiently large' Galois extension of Q. (Of course one can similarly define a resolvent attached to any complex valued character and normal basis generator of a finite Galois extension of local fields). This definition of resolvent is extended to virtual characters of F by linearity. As we are investigating the Z[F]-structure of ON (i.e. not the C?K[F]-structure) we are forced to work with the 'Norm-resolvent' defined for each a and % by 1

lx)= II Mx*" )'

where the product is taken over a set of coset representatives {a} for f2# in fi. This definition does depend upon the choice of representatives {a} but only to within a root of unity. Working now at the integral level there exists an idele a = (av)v £ J{N) such that for each prime divisor V of N = av.(OK)VnK[T].

(1.9)

Thus for each virtual character x °f T one can define an idelic resolvent (a | X ) € J(E) , by N I X)

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Cassou-Nogues et al. - L-functions and Galois modules

for each prime divisor V of E. Similarly one defines an idelic norm resolvent

•A/W/QHX)=

I I Mx*" 1 )'

which again only depends upon the choice of coset representatives {a} (of Q,K in ft) to within a root of unity. Theorem 4 (Frohlich (1976)) Let N/K be a tame Galois extension and let the elements a £ N and a £ J(N) be chosen as above. Then the map X •—> -V*/Q(a I X)'NK/Q(O> I X)"1

is an element of Homn(Rr, J(E)) which has image under the projection map 7rr of (1.5) equal to (ON)l[r]. This theorem is the main reason for introducing resolvents into the subject and also justifies the use of the 'Horn-language' for classgroups. A second basic ingredient of the theory is the notion of the Galois-Gauss sum. To introduce this we may again relax the restriction that N/K be tame and work in complete generality. We shall deal initially with the local case and define a complex valued function rp on the set £p of pairs (E/F, x) where F runs through the set of finite extensions of Qp, E through the finite Galois extensions of F , and x through the group RGai(E/F) of virtual characters of Gal(E/F). Suppose for the moment that x is a n abelian character of Gal(EfF). Then, by local class field theory, there is a multiplicative character 0X of F* corresponding to x a n d w e write J"(x) for the conductor of 0X. Let i}>F : F(additive)

— • C*

denote the canonical (Tate-Iwasawa) additive character of F. The GaloisGauss sum of the pair (E/F, x) is then Tp(EF,x) = \

(1.10)

where DF is the different of the extension F/Qp and, if J~{x) ^ OF, then c G F* is chosen such that

and u runs through a complete set of representatives of OF modulo 1 (This definition is independent of the choices of u and c within the stated

1 The tame additive theory

85

conditions). By a reformulation of the theorem of Langlands proving the existence of local constants, there exists a function rp defined on £p which agrees with (1.10) for the abelian characters and satisfies in addition that rp(E/F,Xl

+ X2) = TP(E/F,XI).TP(E/F,X2),

(1.H)

and that, if L 2 M are intermediate fields of E/F with A = Gal(E/L) and S = Gal(E/M), and is any character of A of degree 0, then rp(E/L, ) = rp(E/M, ind°(*))

(1.12)

where here indf is the induction map. Moreover, as a consequence of Brauer's induction theorem, formulae (1.10), (1.11) and (1.12) will together determine the function rp uniquely and this then is the local Galois-Gauss sum. One now defines a global Galois-Gauss sum by means of local components. Let then N/K denote a Galois extension of number fields (not necessarily tame) with F = Gal(N/K). Let p be a prime ideal of K (of residue characteristic p say) with V a prime ideal of N lying above p. Taking completions one has a Galois extension of fields Nv/Kp whose group Tv embeds into V as the stabiliser of V, with this embedding uniquely determined by p to within conjugacy. The local component \P °f a character x °f T is then simply the restriction of x to Tv. For each character x °f T o n e defines the global Galois-Gauss sum to be

r( X ) = r(N/K,X) = JITP(NV/KP,XP)

(1.13)

where the product (taken over all prime ideals p of K) is convergent since T(N

(1-14)

where here itfoo(x) ls the product of wp(x) over all archimedean places p of K (in fact wp(x) is a 4th root of unity) and NJ7(x)^ € (Qc)* 1S the positive square root of NJ-(x)- From equation (1.14) it is apparent that the GaloisGauss sums arise essentially from the functional equation of the extended

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Cassou-Nogues et al. - L-functions and Galois modules

Artin L-function. We stress that (1.14) remains valid even for wildly ramified extensions N/K. The fundamental connection between the resolvents and the Galois-Gauss sums is provided by Theorem 5 (Frohlich (1976)) Assume the conditions and notation of Theorem 4. Then (1) The map x •—> -Ak/g(a I x ) M x ) belongs to Homa(Rr,J(E)). (2) If x is a symplectic character of F and V is an archimedean place of E then

( J W « I x)/r(X))v is real and has the same sign as {WM/K{X))V (3) MK/^{a | x)/ T (x) i s a u n i t idele for each character x of T. This theorem lies at the very core of the tame theory, implying an entirely unsuspected interpretation of the Galois-Gauss sum, and hence of the functional equation of the extended Artin L-function, in terms of the module structure of the rings of integers. Indeed if NjK is a tame extension then by Theorem 4 the class (Ojv)z[rj is equal to (

But the second map in (1.15) is actually one from Rr to the multiplicative group E* and any such map can be shown to lie in ker(7cr). Hence Corollary 6 Assume the conditions and notations of Theorem 4. Then (n

\

(ON)m

nr

= 7r

The proof of Theorem 5 depends critically upon the explicit decomposition (1.13) of the Galois-Gauss sum r(x) into local factors. This analysis is indeed a major part of the tame theory but we do not discuss it here since at present no trace of a possible generalisation has become apparent. We shall also not discuss the proof of Theorem 2 in any further detail save to say that, given the above ingredients, it is proved by combining (refined) congruences for the Galois-Gauss sums with Taylor's logarithm map for group rings. For more details the reader is referred to either Frohlich (1983) or to the original paper of Taylor (1981). To end this section we note a further consequence of the above techniques. Let A4 denote any maximal Z-order in Q[F] with A4 2 Z[F]. In the case that N is

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a Galois extension of Q (with F = Gal(N/Q)) Martinet had conjectured that the .M-module ONM generated by ON was always a stably-free .M-module. In fact for N/K tame, and even with K ^ Q, this is an easy consequence of the Hom-description of Cl{M) (corresponding to (1.5)) together with Corollary 6 and Theorem 5 (parts (2) and (3)). Corollary 7 Let N/K be a tame Galois extension of group F. If Ai is any maximal Z-order in Q[F] with Ai 2 Z[F], then O^M. is a stably-free M. -module. Even in case K — Q, the result of Corollary 7 is in general not true without the tameness hypothesis. However, we shall later derive an analogue of Corollary 7 for certain wildly ramified Galois extensions (§3 (3.23)). There is an alternative way of stating the result of Corollary 7 which is useful for later reference. For any maximal Z-order M as above, tensor product with M over Z[F] induces a homomorphism TTM : Cl(l[T]) —> Cl(M) and we let D(Z[F]) denote the kernel of the map TTM . In fact, X)(Z[F]) is independent of the choice of maximal order M as above. Now if X ~i[r] Z[F] then XM naturally identifies with X ®z[r\ M , and hence an equivalent formulation of Corollary 7 is Corollary 7 Let N/K be a tame Galois extension of group F. Then (ON)m

eD(i[T}).

2 ARTIN ROOT NUMBERS AND HERMITIAN GALOIS MODULES Let N/K be a finite tame Galois extension of number fields of Galois group F. In this paragraph we examine more closely the connection between the structure of ON as a module over the integral group ring Z[F] and the values of the Artin root numbers w{\) attached to the symplectic characters x of F. From Theorem 2 of §1 one knows that the Z[restructure of ON is determined (up to stable isomorphism) by the root numbers w(x) of the irreducible symplectic characters of F. On the other hand the Example 1.8 indicates that the Z[restructure of ON alone is not sufficient to determine the symplectic root numbers and hence we shall consider the finer structure of the Hermitian module (ON,TN/K) as mentioned in §1.

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The idea of relating the Hermitian structure to the symplectic root numbers originated with Frohlich, who made the following conjecture: Conjecture 2.1 For a tame Galois extension N/K of number fields (respectively local fields) the global (respectively local) Hermitian module structure of (ONi TNjK) determines the global (respectively local) symplectic Artin root numbers. In this paragraph we shall describe the precise form in which Conjecture (2.1) was proved. However for this we must first be more precise about the notion of Z[F]-Hermitian modules and also introduce some general notation and algebraic preliminaries.

2.1 Hermitian modules From now on K is a finite extension of Q or of the p-adic field Qp for some prime number p. We let G denote an arbitrary finite group. If M is any right OK[G]-module then we shall write MK for the K[G]-module M ®oK K, and for an element a = Y^geG ag9 € K[G] w e shall write a = J2geG ag9~l • Definition 2.2 A Hermitian 0#[£r]-module (M,/&) consists of a locally-free right OK[G]-module M and a map h : MK x MK —> K[G] such that (i) (ii) (iii) (iv)

h is iiMinear in each variable; h is non-degenerate; h(x,yg) = h(x,y)g for x,y € MK and g G G; h(y,x) = h(x,y) for x,y G MK .

Remark 2.3(i) Giving such a map is equivalent to giving a non-degenerate G-invariant symmetric if-bilinear form h' : MK x MK —> K , the relation between h and h! being that h(xiV)

= IC^'C^ra" 1 )^ for each x,y G MK .

Remark 2.3(ii) The map h is defined at field level and need not map M x M into OK[G). The non-degeneracy condition means that the discriminant of h is a unit of K[G] - it need not be a unit in, or even an element of, (!?#[(?]. Example 2.4(i) The standard Hermitian OK[G]-module is where fiK : K[G] x K[G] —> K[G] is given by fjLK(x,y) = xy.

(OK[G],IIK)

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Example 2.J[(ii) If N/K is a tame Galois extension of number fields with Galois group F then {ON^TNJK) is a Hermitian (9K[r]-module, where TS/K •' N x N —v K[T] is derived from the trace pairing TrN/K : N x N —> K; explicitly, TNfK{x,y) = YjiTrNIK{^,yTl))l

for each x,y e N.

2.2 The Hermitian class group We shall now outline the theory of the Hermitian class group as developed in Frohlich (1984). To classify Hermitian OK[G]-mod\i\es one can certainly form the Grothendieck group K0H(OK[G]) of isometry classes of such modules with relations given by orthogonal sums, but this group is too big for our purposes. To obtain an invariant for the Hermitian Galois modules which is more useful than the class in K0H(OK[G]) we will define a suitable notion of the discriminant of such a module. We introduce the Hermitian class group HCL(OK[G\) as the group in which these discriminants lie. This group has a 'Horn-description' analogous to that described in §1.1 for the locally-free class group of Z[G]-modules. For this we must recall some standard notation. We let RSG denote the subgroup of the group RG of virtual characters of G that is generated by the symplectic characters. For any virtual character x € RG we set Tr{x) = X + X ' then Tr(x) G R°G and w(Tr(x)) = + 1 . As in §1 we shall also fix a 'sufficiently large' number field E and let J(E) denote its idele group. For the purposes of this exposition, we now restrict attention to the global case and, for simplicity, assume that K = Q. The reader is referred to Frohlich (1984), in particular to chapter 2, for a more general and a more detailed treatment of the construction of the Hermitian class group. Again we abbreviate the absolute Galois group Gfa/(QC/Q) by fi. We will define HCL(l[G]) in terms of certain subgroups of We set

U(L[G\) = I[( p

the 'unit-ideles' of Z[G], where here the product is taken over all places of Q (with ZQO = R). For u = (up)p £ U(l[G]) we may define as in §1.1 an element Det(u) e Homa(RG,J(E)). We thus obtain a subgroup Det(U(l[G])) of Homn(RG,J(E)). Now let A IT ™ (n TP*\ , A : Homn{RG,E ) —>

Homn(RG,J(E)) Det(U(l[G]))

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be the homomorphism defined by A(/) = (/- 1 modulo Det(U(l[G})), f ) where /* denotes the restriction of / to RSG. We then make the Definition 2.5 The Hermitian class group HCL(l[G]) tient group coker(A) .

of l[G] is the quo-

The group HCL(OK[G]) is defined similarly and, in general, the Hermitian class group has good functorial behaviour with respect to changes of group or of base field. Our next task is to relate HCL(l[G]) to K0H(l[G]) by means of a discriminant map d : KOH(1[G)) —> HCL(1[G}) . Given any class in K0H(l[G]) we choose a representative (M, h) of that class. Assuming for simplicity that M has Z[G]-rank 1, we take v a basis of MQ over Q[G], and set c = h(v,v). It follows easily from the definition of a Hermitian module that c = c G Q[G]*. Now let x be any symplectic character of G and let T : G —> GL(V) be a representation of G, affording x o n a n 2?-vector space V. As T is symplectic there is a non-degenerate G-invariant skew-symmetric ^-bilinear form /3 on V. Taking adjoints with respect to /? gives an involution j on GL(V): j is determined by the rule p{xA,y)

= P{x,yA>) for each x,y G V, A e GL{V).

Moreover, as /3 is G-invariant, we have that T(z)J = T(z) for all z G Q[G]% so that in particular T(c)> = T(c) G GL(V). For any A G GL(F) with A3 = .A we can define a new non-degenerate skew-symmetric 22-bilinear form a = a(A) on V by setting a(x,y)

= P(xAyy)

for a?,y G V.

Since all such forms are equivalent there is an automorphism P = P(A) of V, unique up to composition with a symplectic automorphism, transforming P into a. Thus

a(x,y) = P(xP,yP)

for all x,y G V,

i.e., /9(a?i4,y) = P{xPP',y) and we deduce that A = PP3.

for all s,y € V,

2 Artin root numbers and Hermitian Galois modules

91

Applying this to A = T(c) we have T(c) = PP3 for some P G GL(V), unique up to a symplectic automorphism. We define the Pfaffian Pfx(c) of x at c to be det(P). Since any symplectic automorphism has determinant +1 this is indeed independent of the choice of P and so gives a canonical square root of p det(T(c)). We define Pf(c) G Homa(RG,E*) as the map x •—• fx(c)Finally, we choose an idele a = (ctp)p G i7(Q[G]) such that, for each place p ofQ, Mp = apV.lp[G] .

We now define the discriminant map d by setting d([(M,h)}) = the class of (Det(a),Pf(c))

G HCL(Z[G}).

The next stage is to use d to construct a subgroup K'0H(l[G}) of K0H(l[G]). For this we first define H(G) = Given (/,#) G H(G) let / be the element of Homn(RG,J(E)) f(v\ Jyxjp

_ / /(X)J \1,

defined by

if

X is symplectic and p is finite, otherwise,

and let g be the element of Horrin(RG, E*) obtained by composing g with the natural embedding Q* —> E*. Now define hG : H(G) —-* HCL(Z[G]) (f>9) '—* the class of (/, #). The map hG can be shown to be inject ive and we use it to regard H(G) as a subgroup of HCL(l[G]). The required subgroup of K0H(l[G]) is then obtained by setting K'0H(l[G}) = d~\H(G)). Finally, composing d with the projections of H{G) onto its two factors, we obtain homomorphisms 0 : K'OH(1[G\) —• and 7/ : K'OH(1[G})

Hom n(R'G/Tr(Rc),±l)

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2.3 The arithmetic case We now return to the Hermitian OK[F]-module (ON ,TN/K), where N/K is a finite tame Galois extension of number fields of Galois group F. We want to compare this module with the standard Hermitian module (C^fF] ,I*K) ? regarding both as Hermitian modules over Z[F]. Thus we define XNIK

= Res([(ON,TN/K)]

where here Res : K0H(OK[T]) by restriction of scalars.

-

[(OK[T],ixK)}) €

K0H(l\T])

—> KoH(l[T)) is the homomorphism given

The map X >-+ «;(x),

xeR'r

induces an element of Homcl(R^/Tr(Rr), ± 1 ) . For any symplectic character X of F we let Woo(x) and NP(x)* € Q* b e a s i n equation (1.14) of §1. We then have Theorem 1 (Cassou-Nogues and Taylor (1983a)) (i) XNIK

€ K'OH(1[T]);

(ii) 0(XN/K)

is the map induced by x '— y

w

(x)\

(iii) rj(XN/K) is the map given by x'—> Nf(x)*u>oo(x) • Remark 2.6 Recall that, for each symplectic character x of T? the complex algebraic numbers ttfoo(x) a n d Nf{x)^ were defined immediately prior to equation (1.14) of §1. In particular then the class XN/K determines the symplectic root numbers so that we have a precise statement (and a proof) of the global version of Conjecture (2.1). A similar result also holds for the local version of the conjecture (Cassou-Nogues and Taylor (1983b)). The two main ingredients of the proof of Theorem 1 are the algebraic fact that the map hr : H(T) —> HCL(1[T]) is injective, and the comparison of the arithmetic behaviour of Normresolvents and Galois-Gauss sums. As we have seen the second of these is also at the heart of the proof of Theorem 2 of §1, and indeed Theorem 2 of §1 can be recovered from the above Theorem 1 by applying the forgetful functor KOH(1[T]) —> K0(Z\T]).

2 Artin root numbers and Hermitian Galois modules

93

2.4 The isometry class of (ON,TN/K) If, for example, F has no irreducible symplectic characters then, by Theorem 2 of §1, ON is known to be isomorphic as a Z[F]-module to CTR^F]. On the other hand for example the factor NJr(x)^ in Theorem l(iii) shows that we do not, however, have an isometry of Hermitian modules between (ON ,TNjK) and (C?K-[r], fxK). We now consider the problem of finding a Hermitian module, related to the group ring, which is isometric to (ON, TN/K). We begin by comparing (ON,TN/K) and (OK[T]yfiK) at the field level. Thus we ask whether there is an isometry of Hermitian if [F]-modules between (N,TN/K) and (K[T],fiK). This is equivalent to asking whether there exists an element c of N such that {cy : 7 E F} is a basis for N/K satisfying the condition

{ Oi 1

*-f

1

otherwise.

Such an element, if it exists, is called a self-dual normal basis of N/K. has odd order then the answer to our question is given by

If F

Theorem 2 (Bayer-Fluckiger and Lenstra (1989)) If F has odd order then N/K has a self-dual normal basis. Remark 2.1 If F has a quotient of order any power of 2 then there is no self-dual normal basis. In the cases T = A4 and T = A5 Serre has shown that the existence of a self-dual normal basis is connected with the Witt invariant of the trace form. From now on we assume that F has odd order. Thus the Hermitian 0jr[F]modules (ON, TN/K) and (C?K[F], JJ,K) become isometric on extension of scalars to K[T]. They are not however in general isometric at integral level, even locally, since (OK\F],tiK) is unimodular whereas (ONITN/K) is not (its discriminant being the relative discriminant of the extension N/K). To take account of the ramification in N/K we therefore replace <9# [F] by a so-called Swan module. For simplicity, we assume that the prime divisors of K lying above 2 do not ramify in the extension N/K. Let F b e a quadratic extension of K such that every prime ideal of OK which ramifies in N/K also ramifies in F/K. For each prime ideal V of OF we let p be the prime ideal of OK below V and let V be some prime of ON lying over p. We write Iv, for the inertia group of V in F.

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We now define the Swan module (SF^F) to be the Hermitian (^[Fj-submodule of [OF[F], fiF) which has local completions SFtP (at each prime ideal °f ®K) given by if p is unramified in N/K; "~ Y^seiv, ^ "* JFl, otherwise. I {P> The module SF therefore depends on the choices of the prime ideals V of ON lying above the ramified prime ideals p of OK, but the isometry class of (SFJ^F) is independent of these choices. By extending scalars from QK to QF we obtain from (ON, TN/K) a Hermitian (!?p[r]-module {ON,TN/K)F and so we can compare the classes [(ON,TNjK)F] and [(SF,nF)] in K0H(OF\T]). Writing Res : for the homomorphism given by restriction of scalars we set YN/K = Res([(ON,TN/K)F]

-

We then have Theorems

(Taylor (1989)) YN/K = 0.

Remark 2.8(i) If F has even order then one can still obtain a result about the isometry class of (ON,TN/K) (Cassou-Nogues and Taylor (1989)). Remark 2.8(ii) Instead of allowing for the ramification of N/K by replacing with a Swan module, one could replace ON by a suitable fractional Ojv-ideal. In this context a special case of a theorem of Erez and Morales (1989) gives the following result. OR-[F]

Let N/Q be a tame abelian extension of odd degree, and let 2 ^ } Q denote its inverse different. Then there is a fractional ideal A of ON such that A2 = the Hermitian Z[F]-module (A,TN/Q) is isometric to (Z[F],//Q).

3 Additive and multiplicative Galois module theory

95

3 ON SOME PARALLEL RESULTS IN ADDITIVE AND MULTIPLICATIVE GALOIS MODULE THEORY In this paragraph we shall continue the study of the Galois structure of rings of algebraic integers but in the context of wildly ramified extensions. We shall also discuss some remarkable, and at the present time unexplained, similarities between this 'additive' theory and the 'multiplicative' theory of the Galois structure of groups of units. Indeed the close analogies between these additive and multiplicative theories is an underlying motivation and theme for much of the material in this paragraph. We shall work in the setting of arbitrary finite groups. In the particular case that V is abelian proofs for most of the results discussed here can be found in Frohlich (1989). We first introduce the elementary but nevertheless powerful notion of factorisability. There are a number of variants of this notion, some of which are needed elsewhere. We shall use only that which is best for the present lecture - this is, for example, stronger than that discussed in Frohlich (1988). 3.1 Factorisability and factor-equivalence Let G denote an arbitrary finite group (which need not arise as a Galois group). Each subgroup H of G gives rise to a G-set H \G which in turn defines a permutation representation of G of character pn - We set S(G) = {G-isomorphism classes of H \ G : H < G}. We write G* for the set of absolutely irreducible complex valued characters of G, so that Gf gives a Z-basis of RG. For a number field F let T(F) denote the group of fractional (9^-ideals. For number fields F' and F with F' C F we shall identify T(F') with a subgroup of T(F) in the natural way. We consider functions on S(G) taking values in I(Q). Such a function / will be said to be 'factorisable' if for some 'sufficiently large' number field E which is Galois over Q (and hence for every sufficiently large such E) there exists a homomorphism g : RG — 9(Xi + Xa) = such that, for each subgroup H < G, f(H\G)=g(PH) and also, for each x £ G ( and rj 6 Gal(E/Ci), 9(xt>)=9(x)" •

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Cassou-Nogues et al. - L-functions and Galois modules

Note that, writing < , > for the Z-bilinear pairing on RG defined on elements of Gt by <X^>=|0|

otherwise,

then condition (3.1)(b) is equivalent to the condition that, for each subgroup H < C1

f(H\G)= U To introduce a general procedure for producing suitable maps / on S(G) we let L and M denote finitely generated l[G]-lattices (that is, Z[G]-modules that are Z-torsion free) such that there exists an isomorphism of Q[G?]-modules i : L ®z Q £ M ®z Q.

(3.2)

For each subgroup H < G let LH (respectively MH) denote the Z[G]-sublattice| of L (respectively M) consisting of those elements invariant under the action of each element of H. Define a map / = /L,M,» on S(G) by f(H \ G) = [(iL)H : MH]t , all H < G where here [ : ] z denotes the Z-module index as defined for Z-lattices that span the same Q-space. Definition 3.3 Two Z[G]-lattices L and M will be said to be G~factorequivalent, written L AG M, if for some choice of isomorphism i as in (3.2) the function fL}M,t is factorisable. In fact if j is any other isomorphism as in (3.2) then /L,M,I is factorisable if, and only if, JL^MJ is factorisable. One can also show easily that AG is an equivalence relation on the set of Z[C?]-lattices (that span the same Q[G]space). Furthermore AG is a weakening of the equivalence relation of Z[G]genus equivalence - i.e., if L ~Z[G] M then L AG M. Example 3.4 Let if be a number field. If G is abelian and M(OK, G) denotes the maximal C^-order in K[G) then M(OKjG) AG OK[G] if, and only if, G is cyclic. Thus the notion of factor equivalence is far from trivial. Despite this, and without any hypothesis on ramification, one has the following theorem. To state this and subsequent theorems we shall say that a group is 'admissible' if it satisfies a certain condition on its complex representations about which

3 Additive and multiplicative Galois module theory

97

we shall say more presently. At the moment it is not known whether all finite groups are admissible but, for example, this is certainly the case for all abelian, dihedral, and (generalised) quaternion groups and also for all p-groups. Theorem 1 (Frohlich) Let N/K be a finite Galois extension of number fields of Galois group F. If F is admissible then ON Ar C?/r[F]. Corollary 2 Assume the notation and conditions of Theorem 1, and in addition that F is abelian but not cyclic. If M{OK,Y) denotes the maximal C^-order in K[T], then (ON)M(OK,T) % ON. Proof of Corollary 2 From Theorem 1 and Example 3.4 one deduces immediately that M(OK,T) Ar ON. Hence, a fortiori, M(OK,T) 9^>K[r] @N > &nd the claim follows by a standard property of M(OK, F)-lattices. • If N/K is wildly ramified then it is natural to compare ON not with OK[F] but with the full set A(N/K) of elements of K[T] that induce endomorphisms of ON (see Theorem 1 of §1). This set A(N/K) is an (9^-order in K[T], the 'associated order' of ON in K[T]y and is strictly bigger than 0/r[F]. Theorem 1 also gives some information on the old vexed question of when ON is locally-free as an A(N/ K)-module. Indeed if ON is a locally-free A(N/K)module then necessarily A(N/K) Ar O^[F]. But given an explicit description of A(N/K) this is a purely computational question. Even better, if F is abelian and no prime ideal (of OK) which ramifies wildly in the extension N/K divides the different of the extension iiT/Q then under certain conditions the relation A(N/K) Ar ©^[F] is actually sufficient to ensure that ON is locally-free as an A(N/K)-module. For more details of this last result see Burns (1989). Before discussing the proof of Theorem 1 we shall briefly consider the multiplicative theory. Here our aim is to understand the Galois structure of the multiplicative group UN of units of ON modulo torsion elements, and again it is fruitful to compare this arithmetical lattice with a lattice which has a better understood structure. For any number field F we let Soo(F) denote the set of archimedean places of F. We let YF = Divz(Soo(F)) denote the additive group of Z-divisors supported on SOO(F) and write XF for the subgroup of YF consisting of divisors of degree 0. That is, XF is defined by means of the exactness of the sequence 0 —> XF —> YF - U Z —> 0,

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in which e : YF —• Z is the homomorphism defined at each place v 6 by By a theorem of Herbrand one knows that XNiQ and £/w<8>iQ are isomorphic as Q[F]-modules. Thus choosing a Q[r]-isomorphism « : XN ®z Q S W* ®j Q,

(3.5)

one can define a function /K = fuNtxN,K on 5(F) by

fK(A \ F) = [*(**)* :

(UN)\

where, for any number field F, /&/> denotes the class number ord(ClF) of F. In analogy to Theorem 1 one has Theorem 3 (Frohlich) Assume the notation of Theorem 1. If N is totally real and F is an admissible group of odd order then, for any choice of isomorphism K as in (3.5), the function fK is factorisable. Remark 3.6 In the statement of Theorem 3 the hypotheses that TV is totally real and that F is of odd order are made purely for the sake of simplicity there is a more general theorem of which Theorem 3 is a specialisation. Apart from the assertion of factorisability in Theorems 1 and 3 we shall also presently see that a great deal of interest lies in the precise arithmetic nature and in the interpretation of the respective factors #(x) (see (3.1)) giving the factorisations of the functions fuN,xN,K and foN,oK[r],t for any suitable isomorphisms K and i. But to discuss this further we must introduce invariants that connect the arithmetic and the module theory. Recall that in the tame additive theory, as discussed in §1, the necessary invariants were the resolvents. The invariants to be introduced here both for the wild additive theory (again labelled 'resolvents') and also for the multiplicative theory (labelled 'regulators') can be considered as generalisations of the fractional ideals generated by the 'tame' resolvents. Indeed the resolvents and regulators to be introduced here are each defined by the same formal procedure. We shall deal firstly with the additive theory.

3.2 The generalised resolvents

Writing SN for the set of field embeddings N <—> • C, one has a non-degenerate Q-bilinear F-pairing b : N x DivQ(SN) —> C

3 Additive and multiplicative Galois module theory defined by

s

99

x G N,s G SN.

Let E again denote a number field, Galois over Q, over which all elements of n can be realised and in addition such that N C E. Thus to each actual character of F there corresponds an jE[F]-module V^. Associated to b and to V : HomE[r](V^ E®QN)

V+ ®E[r] (E ®Q DivQ(SN)) —> C

X

defined by (h,v(g)Qs) i—• h(v)

s

for each h G HomE{r\(V+,E ®Q N), v G V+, and s G SN. For any OE[T]lattice H spanning V^ one defines the resolvent R(H, ON) as the discriminant with respect to the pairing bv of the sub-lattices HomOE[r](H,0E®zON)

C

and DivQ(SN)). If these lattices are free over OE then this discriminant is as usual the basis discriminant modulo O*E. In general the discriminant is defined via localisations. Let % £ I"*. We assume that there exists an 0^[r]-lattice TX spanning Vx and with the following property: for every u G G?a/(E/Q(x)) the a;-semilinear translate (Tx)" of Tx is isomorphic to Tx as an O£[r]-module. This is certainly the case if x c a n be realised over Q(x). It is also true for all representations of both (generalised) quaternion groups and p-groups. In general we say that a group T is 'admissible' if this condition can be satisfied for each element of Ff (recall the remarks prior to Theorem 1). Assuming that F is admissible choose for every x € T1 an O^[r]-lattice Tx as above and furthermore such that for each LU G Ga/(Q(x)/Q) and each x £ Tf Tx» = (T x)u

(3.7)

where now the lattice on the right is unique to within isomorphism. For each actual character <j> of F we now set xert

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Cassou-Nogues et ah - L-functions and Galois modules

Definition 3.8 Let F be admissible. Then for any (complex) character <j> of F the generalised resolvent of ON with respect to is

Each resolvent R((/>^ ON) is therefore a fractional (9^-ideal. From the explicit definition of i?(^>, ON) in terms of the pairing bv

R(,ON) = I I R(X,ON)<X^

(3.9)

xert

for each character <j> of F. Via the product expression (3.9), the definition of a generalised resolvent can easily be extended to all virtual characters G RrIf F is abelian then the set {Tx} € r t can be chosen canonically. In general however this definition of each R((j), ON) is dependent upon the choice of lattices {T x } x6r t but the theorems to be stated remain valid with any set of lattices chosen as above. Furthermore if N/K is tamely ramified and ot G J{N) is chosen to satisfy equation (1.9) then, for each virtual character ð G Rr one has K{(P)UN)

=

\JvK/ty\a I
(O.IUJ

where cfc is the absolute discriminant of the field K and equation (3.10) is to be interpreted as an equation between fractional C^-ideals.

3.3 The generalised regulators Turning now to the multiplicative theory we shall introduce pairings and discriminants in an exactly analogous fashion. Defining a Z-lattice QN by the exactness of the sequence 0

—> ( E

s

) —*

\seSooiN) /

Y*

—»

SN

—»o

one has a natural identification of QN with the Z-linear dual Hom,z(XN,Z). Thus GN®IQ and #omQ(ZVjv®zQ,Q) a r e isomorphic Q[F]-modules (see (3.5)) and there exists a non-degenerate Q-bilinear F-pairing V : UN ®2 Q x QN <2>Z Q —> C

given by (u,v)

i—y log \\u\\v

u G U N,

v G S'0O(Ar)

3 Additive and multiplicative Galois module theory

101

where here || ||v denotes the canonically normalised absolute value at the place v. For any (complex) character of F, corresponding to the E\T]module V^, and any (^[Fj-lattice H spanning V+ one has a regulator R(H,UN) defined just as in the additive case but here with respect to the extended pairing b'v and suitable sublattices of HomE[T](V
(Note that in this case, if K = Q and = e is the identity character of F then HomE[r](V€,E®zUN) = Ve®m(E®EgN) = (0) and we set R(H,UN) = OE). Similarly, if. F is admissible then, after making a choice of OE [F]-lattices {T^}xert as above, one defines lattices T^ for each (complex) actual character ð of F, and then defines a regulator. Definition 3.11 Let F be admissible. Then for any (complex) character of F the generalised regulator of UN with respect to is

Just as with the additive theory one has the product expression

valid for each character <j> of F, and this allows a natural extension of the notion of generalised regulator to all virtual characters (/> G i?rRemark 3.12 Each regulator i?(<^>,i/jv) is a rank one (^-lattice in C. Multiplication here therefore is that induced on the set of such lattices by the usual product in C. Whilst the definition of regulator given here is in general dependent upon the choice of lattices {T^}xGrt the theorems to be stated remain valid for any choice as above. Of course if F is abelian then the definition can again be made canonical. We add a number of remarks concerning the comparison of the above regulator with that introduced by Tate in his treatment of the Stark conjectures

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(see Tate (1984)). Our remarks are not very precise as this would require further lengthy definitions. Firstly our regulator is not as in Tate's approach a complex number modulo E* but is a (transcendental) rank one (9^-lattice. Such rank one lattices inside a given one dimensional jS-subspace of C have divisibility and integrality properties and, as we shall presently see, this enables one to formulate integral variants of theorems, problems and conjectures relating to the Stark conjectures. Aside from this any rank one (^-lattice defines a class in C/E, its Steinitz class, and we shall later discuss an interpretation of the class of a regulator in terms of Galois modules. Moreover there is also an action of Gal(E/Q)] for rj E Gal(E/ty) there is an 77-semilinear isomorphism R{H,UN) - ^ R(H\UN) ( or N)

^ R(\UN).

(

For each (complex) character <^> of V we now let L(s,<j>) denote the Artin Lfunction obtained by omitting from L(s, ) the Euler factors corresponding to all archimedean places of N. Then the E-vector space in which R(,UN) lies is generated by the regulator of Tate at <£, and conjecturally (Stark's Conjecture) by the leading non-zero coefficient at s = 0 of L(s, (/>). Moreover the 0^-lattice spanned by (any choice of complex representative of) Tate's regulator can be recovered as a regulator R(
£

h

3 Additive and multiplicative Galois module theory

103

Recall that, for each subgroup H < G, pn denotes the character of the permutation representation of G obtained from the G-set H \G. For each subgroup H < G, we now set Jff(L) = [HomOE[G](eH0E[G],L®z

OE) : HomOE[G]{TPH,L ®Z OE)) OE

where here TPH is the ©^[GJ-lattice (spanning eHE[G\) as defined immediately prior to Definition 3.8. For each subgroup H < G we shall also set JH{L,M) = JH{L). ( where, for each H < G, Jd,H(L,M) = [Homo^iT^L^OE)

:

HomOB{G]{T{Pll),M®Z

and Jn,H(L,M) = G},L®10E)

: HomOElGh(eHOB[G],M

Lemma 3.14 Using the above notation, the function on S(G) defined for each subgroup H by

H\G^JiiH(L,M)

is factorisable. Proof For each subgroup H < G one has

U9(x)<x'PH>

Jd)H(L,M)=

where g is the homomorphism from RG to I(E) defined at each % £ Gf by g(x) = [HomOs[Gi(Tx,L®10E)

: HomOE[G](Tx,M ®t OE)]OE

. D

Furthermore, for each subgroup H < G,

and hence, defining a function J(L, M) on S(G) by

JH(L,M)

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one has the equivalence LAGM

<=> J(L, M) is factorisable.

(3.15)

If now L and M span isomorphic (but unequal) Q[G]-spaces then, for any Z[G]-embeddings i and j of L into M ®z Q one has

and the symbol J(L, M) is to be interpreted as this function. Equivalence (3.15) is then still valid. We now discuss the proof of Theorem 1, the notation of which we shall continue to use. Using equivalence (3.15) it is clearly sufficient to give a factorisation of the function J(C?K[F], ON). For this we shall introduce the 'adjusted' global Galois-Gauss sum, defined for any virtual character <j> £ RT by

where here T(N/K, <j>) is the Galois-Gauss sum as defined in §1 (1.13) and dK is an appropriately normalised square root of the absolute discriminant dK of the field K. In other words f (K, <j>) is the Galois-Gauss sum of the character of $IQ induced from the character of flK. (Recall also equality (3.10)). We define a homomorphism g%jK on Rr taking values in T{E) by specifying its image at each % € F*:

We claim that g^K satisfies the conditions (3.1) with respect to the function J(OK[T]yON). Explicitly therefore, at each subgroup H < G, we claim that x€rt

while for each \ € Ff and rj G Ga/(E/Q), one has

where here the Galois action on each i?(x, ON) is as in (3.13). Of these conditions we shall discuss only (3.16)(a) - the other being a fairly straightforward consequence of known results. Before this however we note that by means of (3.16)(a) together with the known localisation behaviour of the function J(OK[T],ON) one can in fact say much about the C^-ideals g%/K(x):

3 Additive and multiplicative Galois module theory

105

Theorem 4 (Frohlich) Let N/K denote a finite Galois extension of number fields of group F. Assume that F is admissible and let x be a complex character of F. Then any prime ideal of OE which occurs in the decomposition of the (fractional) C^-ideal g^/Kix) n a s the same residue characteristic as some prime ideal of K which wildly ramifies in the extension N/K. In particular therefore the support of g%/K(x) o n ly involves prime divisors of ord(T). We shall later see a multiplicative analogue of Theorem 4. Now to prove formula (3.16)(a) one uses two different product expressions. The first is of course (3.9) for the character <j> = pH. But on the other hand one has by definition f(K,pH)

xert

(3.17)

and, by direct computation,

Thus (3.16)(a) follows from (3.9) (with cj) = pH) and (3.17) together with the obvious (formal) behaviour of discriminants under change of lattices. Remark 3.18 With a strengthening of the notion of factorisabilty one can prove that the Galois Gauss sum ideals are actually uniquely determined by the factorisation (in the stronger sense) of the function J(OK\JL])ON)- This result therefore gives a generalisation of Theorem 5 of §1 to the case of wildly ramified extensions, but lies outside the scope of this article. To develop the analogy between the additive and multiplicative theories we now discuss Theorem 3 and a multiplicative analogue of Theorem 4. Thus until explicitly stated otherwise we shall assume the notations and conditions of Theorem 3. In particular therefore N is a totally real number field and F = Gal(N/K) is an admissible group of odd order. In this context one can prove the existence of a homomorphism g on Rr which takes values in T(E) and satisfies conditions (3.1) but here with respect to the function defined on S(T) by A \ F i—> JA(XJVA).JA(JUMA)

.hjyA

But, for each subgroup A < F, one has (XN) so that JA(XNA)JA(UNA)~

= JA(XN)JA(MN)~

= XNA and (liN) = UNA , =

JA(XNUN)

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and hence (using Lemma 3.14) the existence of such a homomorphism g is indeed sufficient to prove Theorem 3. However for a more natural interpretation of such a factorisation we consider the structure of C1N as a Z[F]-module. We write Cl*N for the maximal subgroup of ClN of order coprime to F and set h*N = ord(Cl*N). We define a homomorphism h* on Rr taking values in I(Q(X)) C I(E) by

where here Z[x] denotes the ring of integers of Q(x)5 <>rdi[x] denotes the order ideal of a torsion Z[x]-module, and a superscript (x) denotes the x~is°typic component of a Z[x]-module. Remark 3.19 Since C1*N has order coprime to ord(T) it is not difficult to show that h* satisfies conditions (3.1) with respect to the function defined on 5(F) by A \ F i—> h*(NA).

Theorem 5 (Frohlich) Let N/K denote a finite Galois extension of number fields of group F. If N is totally real and F is an admissible group of odd order then there exists a homomorphism g on Rr which satisfies conditions (3.1) with respect to the function fK and is such that, for each character x of F, the support of the fractional ideal (g(x)\ only involves prime divisors of ord(T). Remark 3.20 The reader should recall the result of Theorem 4. As a specific example we take N to be an absolutely abelian field of odd degree with K = Q. For each (complex) character of F = Gal(N/Q) we write L*(0,<j>) for the leading non-zero coefficient in the Taylor expansion at s = 0 of the Artin L-function L(s,) as defined as the end of §3.3. In this case the homomorphism g^/Q defined on Rr by

3 Additive and multiplicative Galois module theory

107

satisfies conditions (3.1) with respect to the function fK (notation as in Theorem 3). However at the present time one knows only that any prime ideal occuring in the decomposition of the fractional ideals 9N/Q(X)

must divide 2.ord(T) . (For more details of this special case see Frohlich (1989)). This latter result is essentially a reinterpretation of the Gras conjecture, which is now known to be true (at least for any prime p ^ 2). But this new formulation has the advantage that it admits a natural conjectural generalisation in any case in which (some weak form of) Stark's conjectures are known to be true. That is, a natural conjectural generalisation exists for any class of extensions for which the function corresponding to (3.21) is known to satisfy condition (3.1)(c) - for example abelian extensions of quadratic imaginary fields or arbitrary Galois extensions of exponent 2. In this way one is therefore led to pose integral variants of conjectures and problems relating to Stark's conjectures. We now turn to consider module theoretic interpretations for the factorising functions g^/K a n ( i 9N/K- However for simplicity of exposition we shall restrict to the case that F has odd order. As usual we shall begin with the additive theory. Thus N/K is a finite Galois extension of number fields of group F which is now assumed to have odd order. As before we write Ai for an arbitrary maximal Z-order in Q[F] satisfying M 2 Z[F]. The locally-free class group Cl(M) of M is a quotient of C/(Z[F]) and (following the ideal theoretic version of the Horn-description (1.5)) each class is represented by homomorphisms g on Rr satisfying for each character x € Ff (3.22)(a) whilst for each TJ 6 Ga/(Q(x)/Q) (3.22)(6)

In particular, an element of Cl{M) is trivial if, and only if, a representing homomorphism takes only principal ideal values (in the fields Q(x))- For any finitely generated Z[F]-lattice X we shall write XM for the maximal Z[F]-sublattice of X which is also an .M-module and let (XM)M denote the corresponding element of Cl(M).

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Theorem 6 (Frohlich) Let N/K denote a finite Galois extension of number fields of group F. Assume that F is admissible and has odd order. Then the class (ON M)M is represented by the homomorphism Thus one has an explicit module theoretic interpretation of a factorisation of the function «/((!?# [F], ON). Moreover this has concrete arithmetical implications. For example, if F has prime power order (and is therefore admissible) then, as an immediate consequence of Theorem 4 together with the fact that all p-primary ideals in a cyclotomic field of p-power conductor are principal, one deduces that (ONM)M = 1, (3.23) i.e., that ONM is a stably-free .M-module. Note that this result is related to that of Corollary 7 of §1. Indeed, if, for example, N/K is a tame abelian extension, then ON M is isomorphic to ONM as an .M-module. In general however, and even for abelian extensions, neither (ONM)M or (ONA4)M is always trivial (c.f. Burns (1990)). Yet again one can find an analogous multiplicative theorem. For this we impose the restrictions that N is a totally real absolutely abelian number field and that F = Gal(N/Q,) has odd order. We write T r for the principal ideal of Z[F] generated by the trace element

We define the Z-lattice Ar by the exactness of the sequence 0 —> T

r

—> Z[F] —> A

r

—> 0.

This lattice Ar is a Z-order in the Q-algebra Q[F]/(Tr) , and UN is a rank one ^4r-lattice. Let M1 denote the maximal Z-order in Q[F]/(T r ). By means of the Horn-language, elements of the locally-free classgroup Cl(M') of M' are again represented by homomorphisms on Rr satisfying the conditions (3.22). Theorem 7 (Frohlich) Let TV denote an absolutely abelian number field of odd degree and set F = Ga/(Ar/Q). Then the class (UNM')MI is represented by the homomorphism Thus, for example, if Gal(N/Q) is an abelian group of odd prime power degree then the support of the class (UNM )M' m u s ^ li e above 2h*N - this can be a very strong restriction on possible module-structures. In general the question of the precise support of {UNM')M, lying above 2 remains unanswered.

3 Additive and multiplicative Galois module theory

109

3.5 Open problems We end this section by making explicit mention of three open problems. Problem 1 Let N/K be a finite Galois extension of number fields with F = Gal(N/K). For the additive theory, Theorem 2 of §1 implies that if F is abelian then ON is a locally-free Z[F]-module if, and only if, it is a free Z[F]-module. We now raise a possible multiplicative analogue of this result. Assume then that AT is a real absolutely abelian field and that K — Q. The group Cl*N (as defined immediately prior to Remark 3.19) is an ,4r-module, the quotient of two locally-free ^4r-modules. Thus Cl*N defines in a natural way a class (Cl*N)A G Cl(Ar)> HUN is locally-free as an ,4r-module, is the equality necessarily true? In all known examples when UN is locally free over Ar (for example extensions iV/Q of prime degree) this is certainly true (except possibly for the 2-primary parts), and the verification of this points to a systematic general method. Problem 2 The additive theory discussed in this paragraph is more straightforward than the corresponding multiplicative theory because the localisation behaviour of the function which factorises J(OK[T], ON) (i.e., g%fK) is completely understood. Thus we ask, for the multiplicative theory (for example, N real absolutely abelian and K = Q) can one obtain a factorisation of the function A\r.—•jA(xN,uN)hNA

,

A < r

each localisation of which has a natural module theoretic interpretation in terms perhaps of either local units or of local principal units? More specifically, can one decompose the function g^fK into a product of local factors in any 'natural' way? Problem 3 Again assume that TV is a real absolutely abelian field with K = Q and suppose in addition that iV/Q is of odd degree. In view of the discussion following Theorem 5 and, for example, the result of Theorem 7 it is of considerable interest to clarify the role of 2 in the decomposition of the fractional e>Q(x)-ideal $&/Q(x)/**(x) for each character x € F f . Remark 3.24 (added 10 December 1989) Chinburg has recently indicated how one can deduce the Gras conjecture at p = 2 from work of Greenberg, Gillard and Wiles (Chinburg (1989d)). By combining this argument of Chinburg together with the approach adopted by Frohlich (see §6 and §7 of Frohlich

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(1989)), one can now satisfactorily answer all of the questions concerning behaviour at the prime p = 2 that were explicitly mentioned in this paragraph. For example, in the setting of Open Problem 3, one can now prove that no prime ideal which lies above 2 can occur in the decomposition of any fractional ideal ^ / Q ( X ) / ^ * ( X ) • The point in all of this is that there are various differing definitions of cyclotomic units which, a priori, may affect the truth or otherwise of the 'Gras conjecture'-type result at p = 2 that is needed to pursue the Frohlich approach. In fact one can check that the argument suggested by Chinburg does indeed work for the definition of units underlying Frohlich's approach. For more details of Chinburg's argument see Remark 4.23. Remark 3.25 (added 14 May 1990) J. Ritter and A. Weiss have now proved that all finite groups are admissible (in the sense of this paragraph).

4 ADDITIVE-MULTIPLICATIVE GALOIS STRUCTURES Let N/K denote a finite Galois extension of number fields of Galois group F. (We shall later also consider global function fields). As in §3 we write Soo(N) for the set of infinite places of N. Let S denote any finite F-stable set of places of N large enough to satisfy both of the following conditions: (4.1) S contains S^N) and the places of N which ramify over K, as well as at least one finite place of N', and (4.2) The S'-class number of each subfield of iV containing K is equal to 1. In this paragraph we study the F-module structure of objects which incorporate the F-module structures of both ON and of the multiplicative group Us — UN)S of 5-units of N. To be more precise we need more notation. We denote by Nv the completion of N at the place v (of N) and by N* the multiplicative group of Nv. For any non-archimedean place v we let Uv denote the group of units of the valuation ring of Nv. The group Js = JN,S of 5-ideles of N is therefore

The objects we consider in this section are exact sequences 0 — • Us —+ JSj —•+ CSJ —* 0

(4.3)

of certain finitely generated Z[F] modules constructed from ON, Us and Js , and such that the sequence (4.3) has the same cohomology as the basic exact

4 Additive-multiplicative Galois structures

111

sequence 0 —> tf5 —>J S —>C —> 0 ,

(4.4)

in which, as a consequence of condition (4.2), C = CN is the idele classgroup of N. In §4.1 we discuss how by means of an 'approximating sequence' (4.3) one can define elements Q(N/K, 1), Sl(N/K, 2), and Sl(N/K, 3) of the locallyfree class group C7(Z[F]) which are associated to the F-module structures of Cs,/5 Jsj and Us respectively. Remarkably these elements do not depend on the precise choice of the sequence (4.3) or on any of the other arbitrary choices made in the course of their definitions and are therefore invariants of the extension N/K. Having defined the invariants, in §4.2 we discuss conjectures and theorems relating them to other aspects of the arithmetic of N/K. In particular we discuss the conjectured relationship between these invariants and the root number class defined in §1. In fact much of the theory to be discussed here is also valid mutatis mutandis for the case in which N/K is a finite Galois extension of global function fields and in the text we shall make remarks to this effect whenever appropriate. Complete proofs of the new assertions made here about function fields will appear in Chinburg (1989c).

4.1 Definition of the Sl(N/K,i) Let N/K be a finite Galois extension of number fields of Galois group F. We shall write Sf for the set S \ S^N) of finite places contained in S. By condition (4.1) therefore Sj ^ 0. In order to construct the sequences (4.3) we need two technical lemmata. The first of these is not difficult to prove and the second is taken from Chinburg (1985). Lemma 4-5 After enlarging S (if necessary), one can find an element a G Us fl K and a free rank one 0#[r]-submodule F of ON with the following property: (*) a is a non-unit at each place in S; and a3Ojv C F C OC2ON . Furthermore, for all F and a for which (*) is true the closure 1 + F of 1 + F in FLeS/ N* has a filtration by the modules

T{m) = (1 + amF)/(l for m > 0, where each T(m) is F-isomorphic to F/aF. In particular, 1 + F is a cohomologically trivial F-module of finite index in lives* Uv. For each place v of N we write Tv for the decomposition group of v in F. In particular therefore if v is an infinite place then Tv has order 1 or 2.

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Lemma 4-6 Assume the above notations. Then for each infinite place v 6 5oo(iV) there exists a finitely generated Z[rv]-submodule Wv of N* such that Us Q Wv, the quotient group Wv/Us is Z-torsion free, and the injection Wv c—• N* induces an isomorphism in Tv-cohomology. Furthermore, for each element 7 G F and place v G Soo(N) one may also require i(Wv) = W^y Remark 4-7 Of course if Yv is trivial then one can take Wv = Us to satisfy the conditions of Lemma 4.6! Using Lemmata 4.5 and 4.6 we can now define groups J's and JSj that are 'approximations' to Js by T~T

7 J

s

—

TT

w

Llvgsuv

AT*

livesfiyv

x

1

T~[

w x

i

AT*

HveSoo(N)

1

I

X

j

T~T iV

v

llvG5oo(iV) ™v

(A Q\

y

Note that with this definition JSj is finitely generated and has the same cohomology as J 5 . Indeed for the second and third factors in the product decomposition of Js and JSj in (4.8) the equality of cohomologies is clear from the assumptions on F and on each Wv. But if v $• S then v is unramified over K and hence Uv is a cohomologically trivial Fv-module so that the cohomologies do indeed agree. By the assumptions on each Wv, the diagonal embedding Us <—> Js induces embeddings Us ^ Jsj and Us <-* J's - With respect to these embeddings, we define quotients C's and Csj of J's and JSj respectively by means of the exactness of all rows in the following commutative diagram: 0 —> Us

—>

Js

0 —.

—.

J>

0

«,

i

I

—> Us

—> J 5)/

i

I

—>

C

—,

CJ

—»• C 5|/

—> 0

(4.9)(a)

—.

(4.9)(6)

i

1

—> 0

0

(4.9)(c)

In diagram (4.9) all vertical homomorphisms, either up or down, are the natural projection maps and hence induce isomorphisms in cohomology with

4 Additive-multiplicative Galois structures

113

respect to all subgroups of F. Moreover each module in row (4.9)(c) is by definition finitely generated. In this way therefore one constructs 'approximating sequences' of the form (4.3). In the above construction one could consider replacing 1 + F by an arbitrary cohomologically trivial F-submodule T of finite index inside l\veSf Uv. The reason for considering T of the form 1 + F is that the invariants fl(N/K, 2) and Q(N/K, 1) which we define below using the sequence (4.9)(c) will then not depend on the choice of F. The role played by O^ in the above construction is thus to provide a family of T of the above kind which all lead to the same invariants Q,(N/K,2) and i^Af/if, 1), namely the family of all 1 + F with F as in Lemma 4.5. We let Ys denote the additive group Divz(S) of Z-divisors supported on the set S with Xs the subgroup of Ys consisting of divisors of degree 0. That is, Xs is defined by means of the exactness of the sequence 0-^Xs-^Ys-^I—•*

0

(4.10)

in which e : Ys —> Z is the homomorphism defined at each place v G S by

In particular therefore, recalling the notation of §3, XSoo^N^ = XN and ^Soo(iv) = YN •

In fact the canonical exact sequence (4.10) is closely related to the 'approximating sequence' (4.9)(c) as constructed above. Using the techniques of Tate (1966) this connection is expressed by means of the following exact diagram of Z[r]-modules 0

0

—-»

0

—»

0

—»

0

0

0

i i i i A —> B —•» X 1 1 1 1 J —• A —• B —• Y 1 1 1 1 ' C —» A —-y B —» Z 1 1 1 1 Us —»

SJ

SJ

0

3

3

s

—*

0

(

2

2

s

—>

0

(

x

x

—+

0

(

0

0

0

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in which the left hand column is the sequence (4.9)(c), the right hand column is the sequence (4.10), and in addition the following three conditions (4.12), (4.13) and (4.15) are satisfied. Condition J^.12 The Z[F]-modules A{ and B{ for i = 1,2, and 3 are finitely generated and of finite projective dimension. In the sequel we shall write Hl(T, M) for the ith Tate cohomology group of the F-module M. We write H*(T, M) for the corresponding ordinary cohomology group. Thus jff*(r, Af) = # l ( F , M ) for each integer i > 1. Condition J^.13 The extension class

au £ Ext2r(l,Cs,f)

= H2(T,CS,})

of the sequence (4.11)(c) is the pullback via diagram (4.9) of the canonical class in # 2 ( F , C ) as defined in Tate (1966). Before stating the third condition we must make some preliminary remarks. Since Ys is Z-torsion free one has Ext\(Ys, Js) = 0 for q > 0. Hence the spectral sequence

H'(T,Exti(Xs,Js))

=*

degenerates, and in particular Exfr(Ys,Js)

=

H\Y,Homz{Ys,Js)).

But if $0 denotes a set of representatives for the F-orbits of S then

and hence by Shapiro's Lemma Extl(Ys,Js)

Si ®H2(TV,JS).

(4.14)

ves0

Condition J^.15 By means of the identification (4.14) the extension class

a2J

eExt2r(Ys,JSj)

of the sequence (4.11)(b) is the pullback via diagram (4.9) of the class

4 Additive-multiplicative Galois structures

115

where here, z* is the map induced on cohomologies by the natural injection Nv <—> JSj and each av is the canonical class in H2(TV,N*) as defined in Tate (1966). One now uses diagram (4.11) to define the invariants tt(N/K,i). For this we first note that any finitely generated Z[F]-module X of finite projective dimension defines in a natural way a class (^QI[r] £ C7(Z[F]). Indeed, by standard module theoretic results, for any such module X there exist finitely generated locally-free Z[F]-modules Xx and X2 such that the sequence 0 —>XX —>X2 —>X —> 0

(4.16)

is exact and one defines

By Schanuel's Lemma this class (^02rri is indeed independent of the choices of locally-free Z[F]-modules X\ and X2 satisfying (4.16). Definition J^.ll For any Galois extension N/K of number fields one defines

for i = 1,2 and 3. The main theorem of this section is then Theorem 1 (Chinburg (1985)) Assume the above notations. Then the classes Q(N/K,i) for i = 1,2 and 3 depend only upon the extension N/K. In particular they do not depend upon the choice of the diagram (4.11), the Wv for v £ Soo(N), F, So or S satisfying the conditions discussed above. Clearly, as a direct consequence of diagram (4.11), for any Galois extension of number fields N/K one has the equality

Theorems Sl(N/K,2) = Sl(N/K,l) + fl(N/K,3). Finally we briefly remark on the function field case. Thus N/K is now a finite Galois extension of global function fields with F = Gal(N/K). Let p be the characteristic of K. Let 5 ^ = SOO(N) be any finite non-empty F-stable set of places of N which are split over K and which have sufficiently divisible residue field degrees over Z/pZ. With such a choice of places Soo , we let ON

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denote the ring of elements of N which are regular at all places outside S^ , and we define OK = K n ON . Then the elements tl(N/K,i) of Cl(l[T]) for i = 1,2 and 3 can be defined exactly as in the number field case, and are again only dependent upon the extension N/K. (The assumption that the residue field degrees of the places in S^ are sufficiently divisible is needed to ensure that the invariants O(iV/if, 2) and Q,(N/K,3) do not depend on the choice of the free 0 K [r]-module F). 4.2 The theorems and conjectures Having defined in §4.1 the invariants Q,(N/K, i), in this section we shall consider connections with other aspects of the arithmetic of N/K. In particular we shall discuss the conjectural relationship between il(N/Kyi) for each i = 1,2 and 3 and the root number class introduced in §1. Conjecture 3 (Chinburg) If N/K is a finite Galois extension of global fields then (i) Sl(N/K,3) (ii) tl(N/K,2) (iii) n(N/K,l)

= WN/K. = WNIK. = 0.

Note that, because of Theorem 2, any two of the conjectures 3(i), 3(ii) and 3(iii) imply the third. Remark J^.18 Suppose x is a complex valued character of F. If N/K is an extension of number fields recall that Z/(s,x) denotes the extended Artin Lfunction of x a s discussed in §1; if N/K is an extension of global function fields we let L(s, x) denote the usual Artin L-function of x- The (Artin) root number of x is a complex constant of absolute value 1 which is defined by the functional equation (1.6) (which is also valid in the global function field case). In §1 we discussed how, for tame Galois extensions of number fields, one can define an element WN/K G C7(Z[F]) using the root numbers associated to the symplectic characters x °f T- More generally in Frohlich (1978) it is shown how to define a class WN/K coming from the symplectic root numbers of N/K without the tameness hypothesis. Frohlich's definition of WN/K is recounted in Chinburg (1989a) and this definition applies equally well to function fields. The following result connects Q,{N/K,2) and Conjecture 3(ii) to the classical theory of the Galois structure of the ring of integers of tame Galois extensions of number fields as discussed in §1. The proof of this result given in Chinburg (1989c) is a modification of the proof given in Chinburg (1985) for number fields.

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Theorem 4 (Chinburg (1989c)) Let N/K denote a finite tame Galois extension of global fields. Let F = Gal(N/K). If F is any free rank one 0#[F]-submodule of ON then the quotient O^/F is a finite Z[r]-module of finite projective dimension and Q(N/K,2) = (ON/F)m

.

But if in particular N/K is a tame extension of number fields then (i r ) I r r i = 0 and hence Sl(N/K,2) = (ON)z[r]. One can therefore regard Sl{N/K,2) as generalising the class (<9j\r)zrr] ^° wildly ramified Galois extensions as well as to function fields. Conjecture 3(ii) thus conjecturally generalises Theorem 2 of§l. To state what is known concerning Conjecture 3 we must recall some further definitions. Let D(1[T]) be the kernel subgroup of Cl(l[T]) as introduced in §1. The argument which Frohlich gives for number fields in Frohlich (1983) (Proposition III 3.1(i)) implies that WN/K G D(z[F]) for all tame extensions N/K of global fields. A wildly ramified extension N/K of number fields for which WN/K £ D(Z[T]) is constructed in Chinburg (1983a) (Proposition 7.1). We will be concerned below with whether the equalities in Conjecture 3 hold modulo D(Z[F]). An important complementary test of Conjecture 3 is provided by quaternion N/K, i.e., those for which F = Gal(N/K) is isomorphic to the quaternion group H8. For quaternion N/K, C7(Z[F]) has order two and equals JD(Z[F]). Quaternion N/K are the extensions of smallest degree in which WN/K can be non-trivial. The following table lists results about Q,(N/K, 2) and Ct(N/K, 3) with later results appearing below earlier ones in each box. The results together with Theorem 2 give all that is known in each case about Q,(N/K, 1). The Strong Stark Conjecture is stated in Conjecture 5 below. We will now briefly discuss some of the ideas involved in the proofs of the results in Table I. Let N/K be a tame Galois extension of number fields. By Theorem 4 the invariant Q(N/K,2) equals the stable isomorphism class (ON)Zrry But as we have noted WN/K € D(Z[F]) and hence the assertion tl(N/K,2)

= WN,K modulo D(l[T))

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Table I. Results on Galois structure INVARIANT tl(N/K,2) CONJECTURE = w N/K GENERAL Frohlich (1977): True modulo RESULTS D(1[T]) it N/K is tame. (number fields). Taylor (1981): True if N/K is tame. GENERAL Chinburg (1989c): True RESULTS modulo D(1[T]) if (function fields). N/K is tame. QUATERNION Frohlich (1977): True if N/K CASE is tame. (number fields). Kim (1989) (1990): True if K = Q and N is not totally ramified over 2. QUATERNION Chinburg (1989c): True if N/K CASE is tame. (Junction fields).

invariants tl{N/K,3)

= wN/K Chinburg (1983a): Implied modulo D(l[T]) by the Strong Stark Conjecture. Bae (1987): True modulo D(1[T]). Chinburg (1989a): True for some infinite families of quaternion iV/Q.

for tame extensions of number fields is equivalent to (ON)Z^ G D(Z[F]), which is precisely the assertion of Corollary 7' of §1. Similarly the claim that fi,(N/K,2) = WN/K is, in this context, equivalent to Taylor's Theorem (Theorem 2 of §1). In Chinburg (1989c) Frohlich's methods and results are carried over to tame Galois extensions of function fields in the following way. In the function field case we always assume that S^ has the properties stated at the end of §4.1. We let 0N}00 be the ring of elements of N which are regular at the places in ^oo and define OKoQ = K f] ON)OO . (Recall that, in this context, ON denotes the ring of elements of N which are regular at all places outside 5oo, and that OK = K D ON). Let K0T(l[T]) be the Grothendieck group of finite Z[F]-modules of finite projective dimension. In Chinburg (1989c) it is shown that there is an F as in Lemma 4.5 such that F = j3OK\T] for some /? G ON for which ON)OQ C /3OK,OO[F]. One then checks that the class

*(N/K)

= (oN//3oK[r}) -

(po^iryo^)

in K0T(l[T]) depends only on N/K, not on the choice offtor of S^ satisfying the above conditions. Further, the image of $(N/K) under the natural map of KOT(1[T]) to C7(Z[F]) is (l(N/K,2). Finally, one can describe Q(N/K,2) in terms of Galois Gauss sums using Frohlich's Horn-description of K0T(l[T]). This leads to the tame function field results stated in Table I. To carry out the proofs, it is useful to rewrite $(N/K) in terms of Chern class (degree)

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invariants of the kind studied in Chapman (1989). (Added 28 August 1990: A sequel to Chinburg (1989c) will contain a proof that Sl(N/K, 2) = WN/K if N/K is a tame finite Galois extension of function fields.) Kim proved the Theorem indicated in Table I by first obtaining a formula of the following kind for all finite Galois extensions N/K of number fields. (A similar formula holds also for function fields). Let Swiid(K) be the set of finite places of K which are wildly ramified in the extension N/K. One can find a projective Z[F]-submodule O'N of ON such that the index [ON : O'N]oK is finite and supported on Swild(K); such O'N are specified by giving their localisations at each finite place w of K. For each such place w we let v(w) be a place of N over w and let Tv(w) be the decomposition group of v(w). Induction of modules from Tv(w) to F gives rise to a homomorphism ^

: C1(2[T<W)})

—

Kim proves that fl(N/K,2)

= (O'N)1[r] +

£ w£Swild(K)

Indl

Slw{O'N)

(4.19)

where 0,w(O'N) is a correction factor in Cl(l[Tv^]) which depends on w and O'N. If Cl(2\rv(w-)]) = 0 for w G Swiid(K) then Kim's formula simplifies to Q,(N/K,2) = (O'N)Z^ ; the case in which Swiid(K) is empty is just Theorem 4 for number fields. If il(N/K,2) = (OfN)zrr^ then one can try to show that (^iv)zrri = ^N/K using Frohlich's additive methods. This is the technique used in Kim (1989) to prove the result stated in Table I in case K = Q and 7V/Q is a quaternion extension having at least two places over 2. If there is just one place above 2 then the analysis is much more difficult (Kim (1990)) because the correction term Slw(O'N) must also be computed when w is the place of Q determined by the prime 2. The Strong Stark Conjecture involved in the upper right of Table I is the multiplicative counterpart of Frohlich's determination of the ideals generated by Galois Gauss sums in terms of resolvents (this is Theorem 5 of §1 - recall also Remark 3.18). This parallel was developed in Chinburg (1983a) and was an important motivation for the definition of fi,(N/K, 3). We will now briefly recall Stark's conjecture and discuss its implications for Cl(N/K,3). Let N/K be a finite Galois extension of global fields of group F. Let S be a finite F-stable set of places of N (containing the set Soo(N) in case N is a number field). For any complex valued character x of F we write Ls(s,x)

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for the Artin L-function attached to x a n d truncated by removing the Euler factors corresponding to places in S. We let 1^(0, x) denote the first nonzero coefficient in the Taylor expansion at s = 0 of Ls(s,x)- For a fixed F-embedding 0 —-> Xs - U Us (4.20) (see (3.5)) we write R{x->l) f° r the regulator associated to % and to the character x i n Tate (1984). Thus i?(x> 0 is a non-zero complex number. Recall however that this is not a regulator in the sense of §3 (but see the remarks following Remark 3.12). Define a complex number A(x,i) by

Conjecture 5 (Stark) For each complex valued character x of F and embedding i as in (4.20) both of the following conditions are satisfied: (a) (Tate's formulation, Tate (1984)) For each element a of the automorphism group Aut(C/Q) one has

In particular therefore A(x?0 G Q(x)(b) (The Strong Stark Conjecture, Chinburg (1983a)) The fractional £>Q(x)ideal generated by A(x,i) is an Euler characteristic ideal constructed from the Galois cohomologies of Xs and UsA conjecture equivalent to (b) had been formulated earlier in Lichtenbaum (1975) in terms of etale cohomology groups. In Tate (1984) both (a) and (b) of Conjecture (6) are proved for rational valued characters x- Results equivalent to Tate's were proved for this case in Bienenfeld and Lichtenbaum (1986) (for more details see also Lichtenbaum (1975) and Chinburg (1983b)). The Strong Stark Conjecture in the function field case was proved in Bae (1987) using the cohomological interpretation of L-functions; this implies Bae's result in Table I. For a discussion of the Strong Stark Conjecture for Dirichlet characters see Solomon (1987). In Chinburg (1989a) it is shown that the Strong Stark Conjecture reduces the proof that Q,(N/K, 3) = WN/K to certain congruences for the (conjecturally algebraic) numbers A(x,«) • Because of Tate's result above, no conjectural assumptions are needed if all of the representations of F have rational character, e.g., if N/K is a quaternion extension.

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In Chinburg (1989a) complex quaternion extensions N of K = Q are considered. The above congruences then concern the non-zero integer i5oo(^)(0,x(A^/Q)) modulo powers of two, where x(N/Q) denotes the twodimensional irreducible symplectic character of Gal(N/Q,). Using Shintani's formulae these congruences are proved in Chinburg (1989a) for certain infinite families of quaternion iV/Q. This leads to the result stated in Table I that fi(iV/Q, 3) = VFjv/Q for infinitely many such N/Q. A sharper method for proving the required congruences results from the use of Hilbert modular forms and the ^-expansion principle, as in Chinburg (1989b) and Schmidt (1987). Finally we note an interesting consequence of Table I concerning the existence of certain distinguished subgroups of the unit group UN of O^ in case N/K is an extension of number fields. For this we let G0(Z[r]) denote the Grothendieck group of the category TQT of all finitely generated Z[F]modules. For each element X of TQv we let (X) denote its class in Go^F]). By means of the forgetful functor from the category CJ-? of all finitely generated locally-free Z[F]-modules to TQT there is a natural map > G 0 (Z[r]) to defined for each element X of C T? by

In Curtis and Reiner (1987) it is shown that fr(D(Z[T})) = 0.

(4.21)

Proposition 6 (Chinburg, Queyrut) Let N/K denote a finite Galois extension of number fields of Galois group F. The statement that /r(O(A^/if, 3)) = fr(WN/K), which is implied by the Strong Stark conjecture, is equivalent to the existence of subgroups £ of finite index in UN for which both of the following conditions are satisfied: (4.22) (i) There exists an isomorphism of Z[F]-modules

s^xN. (ii) In G0(Z[T]) (UN/S) = (C/w) where here CIN is the ideal classgroup of TV considered as an element of

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Remark J[.23 The subgroups £ thus predicted by this Proposition are not explicitly described. It is interesting therefore to ask whether the so-called 'special units' of Thaine, Rubin and of Kolyvagin can play any role here (see Rubin (1987) or Thaine (1988)). For example, suppose that V has prime order and that K = Q. The homomorphism / r is then an isomorphism by a theorem of Reiner. In Chinburg (1983a) (Theorem 3.3) it is shown that the Gras Conjecture concerning cyclotomic units implies that f2(7V/Q, 3) = WN/Q . For a precise statement of this conjecture see §5 of Gras and Gras (1977). By work of Greenberg (1977), the Gras Conjecture at primes p ^ 2 is a consequence of the Main Conjecture of Iwasawa theory over Q together with the vanishing of the cyclotomic /^-invariants associated to the abelian extensions of Q. The Main Conjecture over Q was proved for p ^ 2 in Mazur and Wiles (1984), while the vanishing of the cyclotomic //-invariants associated to abelian K over Q was proved in Ferrero and Washington (1979). More recently Wiles has given a proof of the Main Conjecture over Q, as it concerns zeroes of distinguished polynomials, for p — 2 (see Wiles (1989)). In Chinburg (1989d), Chinburg indicates how, by combining this work of Wiles with that of Greenberg (1977) and of Gillard (1979), one can complete the proof of the Gras Conjecture for p = 2. We will now sketch this argument of Chinburg. (Note that in particular this will complete the proof that fl(iV/Q,3) = WJ^/Q if iV/Q is a Galois extension of prime degree). Suppose then that iV/Q is a real abelian extension of odd degree, and that p = 2. We adopt the notation of §5 of Greenberg (1977), except that we here replace Greenberg's K with N and we define the cyclotomic unit group CN as in Gillard (1979), rather than as in Greenberg (1977). The formulation of the Gras Conjecture on page 152 of Greenberg (1977) then agrees with that of Gras and Gras (1977) when p = 2. We let L (respectively M'Q, respectively Mo) denote the maximal abelian pro-p-extension of N which is unramified (respectively unramified outside of p, respectively unramified outside of p and infinity) so that N Q L C MJ g Mo . By considering complex conjugations, one sees that when p = 2, Gal(M0IM'Q) is isomorphic to (Z/2Z)[A] with A = Ga/(iV/Q). Greenberg's arguments show that in order to prove the Gras Conjecture it is in fact sufficient to prove that e^{U'N/'CN) and e^Gal(MQ/N) have the same order, where here e^ is the idempotent of ZP[A] corresponding to any non-trivial irreducible representation ^ of A over Qp, U'N is the product of the principal units in the completions of A above p, and CN is the intersection of U*N with the closure of CN . Because of Theorem 6.2 of Wiles (1989), the arguments on pages 154-5

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123

and 150-2 of Greenberg (1977) express the order of eyGal(M0/N) in terms of the values of p-adic L-functions at s = 1. But taking into account the fact that Ga/(MO/Mo) = (Z/2Z)[A] when p = 2, this expresses the order of e^Ga/(Mo/iV) in terms of p-adic L-function values. The resulting expression now agrees with the formula for the order of e*(U'N/CN) as obtained in Lemma 5 of Gillard (1979) (see also the footnote to page 3 of Gillard (1979)), which therefore completes the proof. To end this section we shall show how to prove Proposition 6. Firstly, using an isomorphism (3.5) it is always possible to find a subgroup £ of finite index in UN and satisfying condition (4.22)(i). For any such subgroup £ one has (S) = (XN).

(4.24)

By unpublished work of Queyrut (1983), the root number class WN/K lies in the kernel of the map / r . Hence the defining sequence (4.11)(a) implies that the equality fr(Q,(N/K,3)) = fr{WN/K) is equivalent to the following equality in G0(l[T]): (Us) - (Xs) = 0. On the other hand, if Is (respectively Vs) denotes the group of (^-fractional ideals (respectively principal CV-fractional ideals) which have support only involving primes in Sf then one has natural exact sequences

0 —> X

N

—> X

5

—+ Div*(S f) —> 0

and 0 —> Div z(Ss)

—> I

5

—> 0

(4.25)(iv)

where the sequence (4.25)(i) is a consequence of the assumption (4.2). The relations in Gro(z[r]) produced by (4.24) and (4.25) give the following equalities in G0(Z\T\): (Us) - (Xs) = (UN) - (XN) - (ClN) =

(UN/S)-(ClN).

But since the left hand side of (4.26)(i) is 0 if and only if

the statement of Proposition 6 follows.

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5 EXPLICIT GALOIS MODULES Let N/F denote a finite abelian extension of number fields of Galois group F. In this final paragraph we return to consider the structure of ON as a Galois module. As outlined in §1 there is a comprehensive theory for the case in which N/F is at most tamely ramified and ON is considered as a Z[r]-module. However we are now concerned with the case of wildly ramified extensions: in this case even today there are only partial results dealing with certain special classes of extensions. The best of these results is Leopoldt's theorem (to be described in §5.1) which, for any absolutely abelian extension, completely describes the structure of ON as a module over its associated order *4(7V/Q) (recall that in general A(N/F) denotes the full set of elements of F[T] that induce endomorphisms of ON). In this paragraph we shall describe a technique which, for example, provides a closely analogous result for a class of extensions arising from the torsion points of an elliptic curve with complex multiplication. It also gives a result dealing with rings of integers of ray class fields of an imaginary quadratic field, which may be considered as an integral contribution to the famous 'Jugendtraum' of Kronecker. 5.1 Leopoldt's theorem Let iV/Q be a finite abelian extension of Galois group F. For each x € F* 1^

be the corresponding primitive idempotent in QC[F], and let F(x) denote the conductor of x- Thus ^F(x) is a positive integer, and we can regard x a s a group homomorphism (Z/^x)^)* —> (Qc)*« We decompose ^F(x) in^>° 'tame' and 'wild' parts by setting

Hx) = Hx)F4x) where PWHX)

the product being taken over all rational primes p such that p \ !F{x) but We define an equivalence relation on F* with characters x a n d ^ equivalent if and only if ^ ( x ) = ^ ( ^ ) , and to each equivalence class $ we associate the idempotent

5 Explicit Galois modules

125

Since the fig-conjugates of any given character <j> all have the same conductor (and hence lie in the same equivalence class), the idempotents e$ each belong to Q[r]. We also define the conductor and the kernel of an equivalence class by JT($) = l.c.m.{f(x)

: X € $}

and fcer($) =

p | ker().

To each character <j> we now associate a Gauss sum as follows: let $ be the equivalence class containing and set

=

£

#*)(6w)' >

where, for any positive integer n, £n denotes the primitive nth root of unity e27rt/n. Although these Gauss sums are in general imprimitive , with ), they do not all vanish since (J r ($)/^ r (<^), F()) = 1. For each equivalence class $ we define

We now have the notation to state Theorem 1 (Leopoldt (1959)) A(N/q) and

We note that the statement of this theorem comprises three elements: (i) an explicit description of the associated order (ii) the assertion that ON is a free w4(Ar/Q)-module; (iii) an explicit generator of ON over A(N/($). The construction of the generator involves the roots of unity £?($), as well as the character values: thus to obtain the generator we have used the values of

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the transcendental function e2*tx at division points of its period. Below we will give a result (Theorem 3) which provides the same three elements in a rather different context, the generator being constructed from the values of a certain elliptic function at division points of its period lattice. 5.2 Hopf orders and the generalised Noether criterion In general if we do not know that ON is free over A(N/F) then we can at least ask whether it is locally-free. Over the integral group ring C?p[r] one has the following criterion: Proposition 5.1 (Noether criterion) If N/F is a finite Galois extension of number fields (not necessarily abelian) with Galois group F, then ON is locally-free over OF[T] if and only if traceNiF{ON)

= OF .

Remark 5.2 This criterion is equivalent to the result of Theorem 1 of §1. In the case that F is abelian, there is a generalisation of Proposition 5.1, due to Childs and Hurley, in which OF[T] can be replaced by certain other (9F-orders in F[T], namely the Hopf orders. Let A : F[T] —*F[T]® FF[T] be the F-linear map given by A(7) = 7 ® 7 for each 7 £ F. Then A is an algebra homomorphism and is called the comultiplication of F[T]. Now any Op-order A in F[T] is necessarily C^-projective and so A ®oF A can be regarded as an O^-submodule of F[T] ®F F[T]. We call A a Hopf order if A(^4) Q A ®oF A. (This definition is somewhat simpler than the abstract definition of a Hopf algebra, since the augmentation map e : A —* O F and the antipode map 5 : A —• A, determined by 6(7) = 1 and 5(7) = 7" 1 for each 7 E F , are automatically inherited from the Hopf algebra structure of

The condition that an order A in F[T] be a Hopf order is a very strong one: if, for example, F has odd order then the only Hopf order in the rational group algebra Q[F] is the integral group ring Z[F]. More Hopf orders are obtained however if we allow ramification in the base field at primes dividing the order of F. In particular, if F contains enough roots of unity, then the (unique) maximal order M of F[T] is split (i.e., has a OF-basis of idempotents), and is therefore a Hopf order since A is an algebra homomorphism and M ®Op M. is the unique maximal order in F[T] ®F F[T].

5 Explicit Galois modules

127

Let S = E 7 €r7 be the trace element of F[T], To any Hopf order A in F[T] we associate a fractional OF-ideal %A of OF by the rule ^ r (= A In fact iA is an integral ideal, and has the following property: Proposition 5.3 (Generalised Noether Criterion) Let N/F be a finite abelian extension of number fields of Galois group F. Let A be any 0 F -order of F[T] which is Hopf, and such that ON is an .4-module. Then ON is a locally-free ,4-module if, and only if, traceN/F(ON) = iA(N/F). Proof We must show that, for each prime p of F, the OF)P-ordeT ON®OFOF& is free over Ap if and only if (ON ®oF OFyP)Y, = (iA)p (where here OFp denotes the completion of OF at p). We write M = ON ®Op OF}P and set (iA)p = tOFtP. Thus

Now, if M is free over ^4P, say on the generator m, then OFiP = Mr = m(Arp) = m{rlV)OFtP

= r

and so (Af)E = (iA)p . Conversely suppose that (Af )E = {iA)p. Then x.Y* = t for some x E M. For m G M we define \(m)

= r1S76rx.(m7)®7-1 G

M®OFpFp[T}.

In fact A(M) g Af ® -4P since A(ra) is obtained by applying to m ® 1 G Af ® FP[F] the element (id® Sr)A(t"1E) of ^4P ® ^4P and then multiplying by x ® 1. Moreover A : Af —> Af ® Ap is a homomorphism of .Ap-modules, where Ap acts on Af ® Ap by multiplication in the second factor, and is split by the homomorphism m ® a i—> 772a. It follows that Af is projective over Ap and so, since we are working over the local ring OFp, this implies that Af is free over Ap.

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5.3 The map * We now use an elliptic curve E to generate Hopf orders. This will enable us to study the integral Galois module structure of fields arising from the division points of E. Thus let E be an elliptic curve defined over a number field F with complex multiplication by the full ring of integers OK of a quadratic imaginary field K. We assume that E has everywhere good reduction. Fix a non-zero element ?r £ OK and let G = E[ir] be the group of 7r-division points of E. Replacing JP by a larger field if necessary we may assume that G is contained in the group of F-rational points E(F) of E. We consider the two F-algebras A = F[G] (the group algebra), and B = Map(G, F) (the algebra of functions G i—* F with pointwise addition and multiplication). There is a non-degenerate F-linear pairing < . , . > : B x A —> F given by =

f{g) for e a c h

feB^geG.

Now let Q be the group scheme over OF determined by G: thus as a functor on the category of commutative F-algebras, Q associates to each algebra R the group Q(R) of 7r-division points in the group E(R) of irrational points of E. In particular, since G Q E(F), we have G(F) = G?, and hence Q has generic fibre Q/F = Spec(B), the constant group scheme of G over F. Since E has good reduction, Q is finite and flat over OF- We may therefore identify Q with Spec(B) for some OF-order B in B. The fact that the scheme Q is a group scheme means that B is a Hopf order. Let A C A be the OFdual to B with respect to the pairing < ., . >. Thus A is a Hopf order in A, representing the Cartier dual QD of the group scheme Q, and we have a non-degenerate (9F-linear pairing <.,.>:

B x A —> OF .

Now let Q be an element of E(F) and set GQ = {Q'€E(Q°) :*Q' = Q}.

5 Explicit Galois modules

129

Our aim is essentially to study as a Galois module the ring of integers of the field F(Q') obtained by adjoining to F the coordinates of Q' for any Q' G GQ . We have to make two adjustments, however, to the module to be studied. The first problem is that the degree of the extension F(Q')/F depends on the point Q. In order to deal with all Q G E(F) uniformly, we replace F(Q') by the Galois algebra FQ =

Here Q,F acts in the natural way on both GQ and Qc so that FQ is a product of field extensions, one for each Galois orbit of points Q', and has F-dimension equal to the order of G. Let OQ denote the integral closure of OF in FQ. The group G acts on FQ by translations on GQ, and so we may consider OQ as an CVfGJ-module. The fields occuring in FQ may however be wildly ramified over F , so that OQ will not in general be locally-free over C?F[G]. Thus we try to study OQ as an A-module, and here we need to make the second adjustment since OQ does not necessarily admit A. We define OQ to be the largest A-module in OQ. Thus OQ = {x G OQ : xA Q OQ }

(and also 6Q * HomoF[G]{AOQ) via the map taking x G OQ to the homomorphism a i—> xa for each a € A). The module OQ is clearly a lattice in FQ, and, since A is a Hopf order, OQ is in fact an order in FQ: for if x, y G OQ and a £ A then using Sweedler's notational convention

one has (xy)a = so that xy G OQ ; and 1 E OQ since e(A) g OF. Moreover, one has

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Cassou-Nogues et al. - L-functions and Galois modules

Proposition 54

(Taylor (1988)) 6Q is locally-free over A.

We can therefore consider the class (OQ) of OQ in the classgroup Cl(A) of locally-free A-modules, and can investigate how this class varies with Q. More precisely, we have a map

* : E(F) —> C1(A) given by


Ext(gD,gm)

— H\gD,gm)

= Pic(gD)

(Waterhouse (1971) §0), we obtain a homomorphism

The map $ may be regarded as a special case of this homomorphism. If we take Q = TTP for some P G E(F), then GQ Q E(F) and so the class (OQ) is trivial. Thus TTE(F) g fcer($). Under suitable hypotheses we can say much more about Let wK denote the number of roots of unity in K, and let E(F)tor torsion subgroup of E(F). Then we have: Theorem 2

be the

(Srivastav and Taylor (1990)) If (7r,u;K) = 1 then E(F)tor

g

Remark 5.1 This is essentially a result about the rings of integers of the ray-class fields of K and so may be considered as an integral contribution to the famous 'Jugendtraum' of Kronecker.

5 Explicit Galois modules

131

Outline of proof Let Q G E(F)tor. We take an auxiliary prime / which splits in K/q, say LOK = ft . Let F' = F(E[P]), and let F'Q and 6'Q be the analogues of FQ and OQ respectively, but defined here with respect to the basefield F'. Thus OfQ is locally-free over A! = A ®OF OFt and we have a group homomorphism #' : E(F') —> Cl(A'). As in Theorem 3 below, one can construct a function / on E, defined over F , which has restriction fQ to GQ such that 6'F = fQA!. Thus W{Q) = 1. To descend to JF, we use the commutativity of the diagram E{F')

-^

[Res

[Tr E(F)

-^

Cl(A)

where Tr is here the trace map with respect to the addition on E, and Res is the homomorphism of classgroups given by restriction of scalars. Since Q € E(F) one has Tr(Q) = [F1 : F]Q and hence (V(Q))lF':F] = 1. By the theory of complex multiplication, [F' : F] \ I2(l — I) 2 , and so, varying /, we have that ^(Q) is annihilated by h.c.f.{l2{l - I) 2 : / splits in K } = h.c.f.{{l - I) 2 : / splits in K } the last equality being by a standard lemma of class field theory. But we already know that ^(TTQ) = 1, and so since (TT^WK) = 1 we deduce that

Remark 5.8(i) In general there appear to be many non-trivial classes of Cl(A) in the image of \P. Remark 5.8(ii) By means of this result, we obtain some implications of the conjecture of Birch and Swinnerton-Dyer for Galois module structure. Indeed, if the Hasse-Weil L-function L(E/F, s) of E does not vanish at s = 1 then, conjecturally, E(F) has rank 0 so that E(F) = E(F)tor, and therefore $ = 0. Moreover, if we enlarge the field of definition of E to some extension N of F, and set AN = A ®oF ON-, then we can still say something about the map $N : E(N) —> Cl(AN) even if L(E/N} 1) = 0; if 6Q)N denotes the analogue of OQ but here defined with respect to the field JV, then the

132

Cassou-Nogues et al. - L-functions and Galois modules

commutativity of the diagram E(N)

- ^

[Res

[Tr E(F)

Cl(AN)

^

Cl(A)

implies that (OQ,N) = 1 in Cl(A), so that OQ,N — AN as an A-module (although not necessarily as an ^4^-module). Thus, as with the tame theory of §1 we find that L-functions dominate the Galois module structure of rings of integers. Remark 5.8(iii) We have seen that < 7rE(F),E(F)tor > Q ker(V), but whether there is equality here remains an open question. Recent work of A. Agboola shows that equality holds in the analogous situation for abelian varieties over function fields. 5.4 Explicit results We keep the notation and hypotheses of the previous section and impose some further assumptions. We suppose that 2 splits in i^/Q, say 2OK = l^ • Let TT be a prime element of OK (possibly a rational prime) with IT = ±1 modulo 4, and set q = TT.TT. We assume that E[4] g E(F) (in addition to the previous assumption that E[K] Q E(F)), and that the ramification index of ?r in F/K is q — 1. Fix an element Q of E[TC]. We will give an explicit description of the Hopf order A and of the structure of OQ as an *4-module. Let 5, T and ii be points of E(F) with annihilators precisely 1,1 and 4 respectively (so £[2] = {0, Sj T, 2R} ). Let D be a meromorphic elliptic function on E, defined over F , with divisor (D) = (0) + (2R) - (S) - (T); by the Abel-Jacobi theorem such a D exists and is unique up to multiplication by a non-zero constant in F. This function D was in fact introduced by Weber in his work on complex multiplication. To describe the order A it is sufficient to specify its local completions Ap at each prime ideal p of OpThe result is the following

5 Explicit Galois modules

133

Theorem 3 (essentially from Taylor (1985)) The order A is given by

Ap =

OFiP[G],

ifptx,

where, for 0 < i < # — 2,

and

,

_

D(Q

U

where here Q' is any element of GQ. Here in fact FQ is a field (not just a Galois algebra), and OQ = OpQ is its full ring of integers. Thus we have obtained, for the extension FQ/F, the three elements of Theorem 1, namely an explicit description of the associated order ,4, the freeness of OQ over A, and an explicit Galois generator.

5.5 Heegner points In Theorems 2 and 3 we considered elliptic curves of a special type, namely those that admitted complex multiplication, and obtained a Galois module result for a field extension arising from an arbitrary F-rational point Q. In this section we apply similar methods to elliptic curves from another special family, namely those parameterised by a modular curve, and to certain special points Q, namely Heegner points. We now take E to be an elliptic curve defined over Q (but not necessarily admitting complex multiplication), and we suppose that £ is a Weil curve of Eisenstein type, with prime conductor TV. Thus there is a map <j> : XQ{N) —> E , defined over Q, where X0(N) is the modular curve corresponding to the congruence subgroup T0(N) of SX2(Z). Set m = (12,iV--l) and suppose that (A(Nz)/A(z))m lies in the function field fy(E) (and not just in Q(X0(iV))). The curve X0(N) has two cusps, at 0 and at oo, and these map to the points 0 and S on E, where 0 is the identity of the group law, and S is a torsion point. Let p be an odd prime number dividing the order of 5*. Then p also divides N — 1. Let K be an imaginary quadratic field in which N splits, say NOK = ', and let F = HK(fip), where HK is the Hilbert class field of K and /xp

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Cassou-Nogues et al. - L-functions and Galois modules

is the group of pth roots of unity in Qc. Then E\p] Q E(F), and taking 7T = p, we obtain a group scheme Q over OF corresponding to G = E[TT] as before. In fact Q splits as a product Qi x Q2 of its etale and connected parts, corresponding to a splitting G\ x G2 of G, and the Hopf order A in F[G] representing QD is given by A =

OF[Gl]®M,

where M. is the split maximal OF-order in F[G2]> Thus we obtain a map Cl{A) —• C/(O F[G!]) x Cl{M)\ and we then write and

6 : C1(A) — for the corresponding projection maps. Now let P be a Heegner point on X0(N), given by a triple (OK,Af, [V]) for some ideal class [V] of OK (see §2 of Gross (1983)). Thus P corresponds to an isogeny of degree N from an elliptic curve with j-invariant j(V) to one with j-'mvariant j(V.W), both curves having complex multiplication by OKLet Qp>] = (P) G E(F). Then OQ[V] is locally-free over ^4 as before and so we obtain a class $(Q[
- V(Q[v'])) = 1 for all ideal classes [P] D

REFERENCES S. Bae (1987) 'The conjecture of Lichtenbaum and Chinburg over function fields', Math. Ann., 285 417-45.

References

135

E. Bayer-Fluckiger and H. W. Lenstra (Jr.) (1989) 'Forms in odd degree extensions and self-dual normal bases', to appear in Amer. J. Math. M. Bienenfeld and S. Lichtenbaum (1986) 'Values of zeta- and L-functions at zero', to appear in Amer. J. Math. D. Burns (1989) 'Factorisability and the arithmetic of wildly ramified Galois extensions', Seminaire de Theorie des Nombres de Bordeaux, 1 59-66. D. Burns (1990) 'Canonical factorisability and a variant of Martinet's conjecture', to appear in J. London Math. Soc. Ph. Cassou-Nogues and M. J. Taylor (1983a) 'Constante de l'equation fonctionelle de la fonction L d'Artin d'une representation symplectique et moderee', Ann. Inst. Fourier, 33 1-17. Ph. Cassou-Nogues and M. J. Taylor (1983b) 'Local root numbers and Hermitian Galois module structure of rings of integers', Math. Ann., 263 251-61. Ph. Cassou-Nogues and M. J. Taylor (1989) 'The trace form and Swan modules', to appear. R. Chapman (1989) 'L-functions and Galois module structure in real cyclotomic function fields', to appear. T. Chinburg (1983a) 'On the Galois structure of algebraic integers and 5-units', Invent. Math., 74 321-49. T. Chinburg (1983b) 'Derivatives of L-functions at s = 0 (after Stark, Tate, Bienenfeld and Lichtenbaum)', Compositio Mathematica, 48 119-27.

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T. Chinburg (1985) 'Exact sequences and Galois module structures', Annals of Maths., 121 351-76. T. Chinburg (1989a) 'The analytic theory of multiplicative Galois structure', Memoirs of the A.M.S. Vol. 77. T. Chinburg (1989b) 'A quaternionic L-value congruence', J. of the Fac. of Sciences of the University of Tokyo, Sec. IA, 36 no. 3 765-87. T. Chinburg (1989c) 'Galois structure invariants of global fields', to appear. T. Chinburg (1989d) Letter to D. Burns, December 1st, 1989. C. Curtis and I. Reiner (1987) Methods of Representation Theory, Vol.11. Wiley, New York. B. Erez and J. Morales (1989) 'The hermitian structure of rings of integers in odd degree abelian extensions', preprint, University of Geneva. B. Ferrero and L. Washington (1979) 'The Iwasawa invariant fip vanishes for abelian number fields', Ann. of Math., 109 377-95. A. Frohlich (1976) 'Arithmetic and Galois module structure for tame extensions', J. reine u. angew. Math., 286/287 380-440. A. Frohlich (1977) 'Galois module structure', in A. Frohlich (ed.) Algebraic Number Fields, Proc. Durham Symp. 1975, Academic Press, London 133-91. A. Frohlich (1978) 'Some problems of Galois module structure for wild extensions', Proc. London Math. Soc, 27 193-212.

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A. Frohlich (1983) 'Galois module structure of algebraic integers' Springer-Verlag, Heidelberg, New York, Tokyo. A. Frohlich (1984) 'Classgroups and Hermitian modules', Progress in Maths, Volume 84, Birkhauser, Boston, Basel, Stuttgart. A. Frohlich (1988) 'Module defect and factorisability', Illinois J. Math., 32 407-21. A. Frohlich (1989) 'L-values at zero and multiplicative Galois module structure (also Galois Gauss sums and additive Galois module structure)', J. reine u. angew. Math., 397 42-99. R. Gillard (1979) 'Unites cyclotomiques, unites semi-locales et Zrextensions IF, Ann. Inst. Fourier, 29 fasc. 4, 1-15. G. Gras and M. N. Gras (1977) 'Calcul du nombre de classes et des unites des extensions abeliennes reeles de Q', Bull. Sci. Math, de France, 2e serie, 101 97-129. R. Greenberg (1977) 'On p-adic L-functions and cyclotomic fields IF, Nagoya Math. J., 67 139— 58. B. H. Gross (1984) 'Heegner points on XQ(N)\ in R. A. Rankin (ed.) Modular Forms, Proc. Durham Symp. 1983, Ellis Horwood Ltd. S. Kim (1989) 'A generalisation of Frohlich's theorem to wildly ramified quaternion extensions of Q', to appear in the 111. J. of Math. S. Kim (1990) to appear.

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H. W. Leopoldt (1959) 'Uber die hauptordnung der ganzen elemente eines abelschen zahlkorpers', J. reine u. angew. Math., 201 119-49. S. Lichtenbaum (1975) 'Values of L-functions at s = 0', Asterisque, 24-5 133-8. J. Martinet (1977) 'Character theory and Artin L-functions', in A. Frohlich (ed.) Algebraic Number Fields, Proc. Durham Symp. 1975, Academic Press London 1-87. B. Mazur and A. Wiles (1984) 'Class fields of abelian extensions of Q', Invent. Math., 76 179-330. J. Queyrut (1983) Letter to T. Chinburg. K. Rubin (1987) 'Global units and ideal class groups', Invent. Math., 89 511-526. T. Schmidt (1987) 'Quaternionic L-value congruences', Ph.D. thesis, Univ. of Pennsylvania. D. Solomon (1987) 'Lichtenbaum's conjecture for Dirichlet characters', Ph.D. thesis, Brown University. A. Srivastav and M. J. Taylor (1990) 'Elliptic curves with complex multiplication and Galois module structure', Invent. Math., 99 165-84. J. Tate (1950) 'Fourier analysis in number fields and Hecke's zeta-Functions', Ph.D. thesis, Princeton University. (Also in J. W. S. Cassels and A. Frohlich (eds.) Algebraic Number Theory, Proc. Brighton Symp. 1965, Academic Press, London (1967)). J. Tate (1966) 'The cohomology groups of tori in finite Galois extensions of number fields', Nagoya Math. J., 27 709-19.

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J. Tate (1984) 'Les conjectures de Stark sur les fonctions L d'Artin en s = 0', Notes d'un cours a Orsay redigees par D. Bernardi et N. Schappacher, Progress in Maths, Vol 47, Birkhauser, Boston-Basel-Stuttgart. M. J. Taylor (1981) 'On Frohlich's conjecture for rings of integers of tame extensions', Invent. Math., 63 41-79. M. J. Taylor (1985) 'Formal groups and the Galois module structure of local rings of integers', J. reine u. angew. Math., 358 97-103. M. J. Taylor (1988) 'Mordell-Weil groups and the Galois module structure of rings of integers', Illinois J. Math., 32 428-52. M. J. Taylor (1989) 'Rings of integers and trace forms for extensions of odd degree', Math. Zeit. 202 (3). F. Thaine (1988) 'On the ideal class groups of real abelian number fields', Ann. of Math., 128 1-18. W. C. Waterhouse (1971) 'Principal homogeneous spaces and group scheme extensions', Trans. Amer. Math. Soc, 153 181-9. A. Wiles (1990) 'The Iwasawa conjecture for totally real fields', Ann. of Math, 131 493-540.

Coates: Motivic p-adic L-functions

141

Motivic p-adic L-functions John Coates Introduction. The connexions between special values of Lfunctions and arithmetic is an ancient and mysterious theme in number theory, which can be traced through the work of Dirichlet, Kummer, Minkowski, Siegel, Tamagawa, Weil, Birch and Swinnerton-Dyer, Iwasawa, ... . Recently, Bloch and Kato [1], using ideas which rely heavily on the work of Fontaine, have succeeded in formulating a very general version of the classical Tamagawa number conjecture for linear algebraic groups for arbitrary motives over the rational field Q, which seems to contain as special cases all earlier conjectures about these questions. Needless to say, only a very modest amount of progress has been made so far towards proving the BlochKato conjecture for specific motives over Q (essentially, the only cases where it can be established at present are for the Tate motives, and certain motives arising from elliptic curves with complex multiplication). In all the cases where proofs are known, the conjecture is established for each prime p separately, and the deepest part of the argument involves ideas from Iwasawa theory. Specifically, one must use a version for the motive of the so called 'main conjecture1 of Iwasawa theory, which has now been completely proven for the above motives (apart from the troublesome primes 2 and 3 in the case of elliptic curves with complex multiplication), thanks to the beautiful work of Mazur, Wiles, Thaine, Kolyvagin and Rubin (see the article by Rubin in this volume). It does at least make sense to try to formulate the 'main conjecture' for arbitrary motives over Q, although one should have no illusions about the difficulty of proving it. The formulation of this 'main conjecture1 involves, on the one hand, p-adic Iwasawa modules which are built out of the representations of the Galois group of Q given by the p-adic realisations of the motive (see the article by Greenberg in this volume for a discussion of the case when p is ordinary), and on the other hand, p-adic avatars of the complex L-function of the motive, which are built out of the critical special values of the complex L-function. The aim of

142

Coates: Motivic p-adic L-functions

the present article is to give a detailed conjectural description of these motivic p-adic L-functions, at least for primes p for which the motive has good ordinary reduction. Nearly everything which is contained in this paper is already given in the earlier articles by B. Perrin-Riou and myself ([3], [4], [10]). However, the assertions made about holomorphy in these earlier papers were too strong, and I have, I hope, corrected these here, as well as giving somewhat fuller versions of some of the crucial arguments about modifications of the Euler factors at both finite and infinite primes. 1. Notation and normalization. Let Q denote the field of rational numbers and C (resp. R) the field of complex numbers (resp. real numbers). Throughout, p will signify an arbitrary prime number (we do not exclude p=2), and we write Z p , Q p , C p for the ring of p-adic integers, the field of p-adic numbers, and the completion of an algebraic closure of the field of p-adic numbers. Let U denote the group of units of Zp . Let A denote the algebraic closure of Q in C . We fix, once and for all, an embedding j : A -> C p

(1)

which we will often not make explicit in our formulae. Let Qat> be the maximal abelian extension of Q in A. If K/F is a Galois extension of fields, we write G(K/F) for the Galois group of K over F. For brevity, we put G = G(A/Q) , G*b=G(Qab/Q) . We use the embedding (1) to identify complex and p-adic characters of finite order of G ab . For each integer m > 1, let \im denote the group of m-th roots of unity in A. Let S be the group of all p-power roots of unity, and put P = Q(S) ,

H = Q(H)+ ,

J = G(H/Q) .

(2)

(here the + denotes the maximal real subfield). We write \|/:G(P/Q) -> U

(3)

Coates: Motivic p-adic L-functions

143

for the isomorphism given by the action of this Galois group on W, i.e. C a = C v ( a ) for all C in S and a in Q. We also put X = Homcont(J,Cp*) .

(4)

As far as the sign of the reciprocity law map is concerned, we must stress that we adopt throughout the geometric convention of [5], rather than the more classical arithmetic convention. Specifically, this convention is as follows. For each finite prime q, let Frobq denote the arithmetic Frobenius, i.e. it operates on the algebraic closure of the field with q elements by sending x to x9 . Let C denote the idele class group of Q. Let Xq be any idele whose q -th component is a local parameter at q, and all of whose other components are equal to 1. Then we choose the sign of the reciprocity map r: C -> G ab

(5)

such that r(xq) is an element of G ab which acts on the algebraic closure of the residue field at q via the inverse of Frobq . Let y: C -> C* be a continuous homomorphism. The complex L-function of y is then defined, as usual, by the Euler product L(y, s) = II (1 - 7(xq)/qs) -1 ,

(6)

where the product is taken over all finite primes q which are not ramified for y, and Xq is as above. Similarly, if S is any finite set of primes of Q , we write Ls(y , s) for the function obtained by omitting from (6) all Euler factors at primes which lie in S. Now let § : G ab -» A* be any character of finite order. We define its associated idele class character (7)

via the formula <|>R = <> | ° r. Thus, if q is a finite prime which does not divide the conductor of <(>, we have <|)(Frobq-l) .

(8)

144

Coates: Motivic p-adic L-functions

This last formula explains our choice of the sign of the reciprocity m a p (4), because it s h o w s that the complex L-function (6) attached to <|>R coincides with the motivic L-function attached to <> | (see §4). 2. p-adic p s e u d o - m e a s u r e s . The aim of this section is to give a slight generalization of the notion of a p-adic p s e u d o - m e a s u r e which is given in [13]. Let O be the ring of integers of some finite extension of Q p , and let I be any profinite abelian group (in the rest of the paper, w e take I = J). For simplicity, let X also denote the g r o u p of continuous h o m o m o r p h i s m s from I to C p *.The O - I w a s a w a algebra 3 of I is defined to be the projective limit of the g r o u p rings O [ I / H ] , w h e r e H r u n s over the o p e n s u b g r o u p s of I. It is a compact algebra, which contains O[I] as a d e n s e sub-algebra. The elements of 3 are called integral measures on I (with values in O). This terminology is justified because, if \i is in 3 , and f is any continuous function from I to C p ,we can define the integral

by passage to the limit from the case w h e n f is locally constant. In this latter case, if H is an open subgroup of I such that f is locally constant m o d u l o H, and if the image of p. in O [ I / H ] is equal to Zp.(s)s, then the value of the above integral is equal to Zp.(s)f(s), where, in both sums, s r u n s over I / H . W e shall n e e d the following generalization of the notion of an integral m e a s u r e o n I, in o r d e r to take into account possible poles of o u r

p-adic L-functions. Let Q ( 3 ) be the ring of

quotients of 3 , i.e. the ring of all quotients <x/p , where a and p belong to 3 a n d p is not a divisor of 0. A n element \i of Q ( 3 ) is said to be a measure if there exists a non-zero element d of O such that d|i belongs to 3 . W e say that an element |i of Q ( 3 ) is a pseudo-measure

if there

exists a non-zero element d of O , a finite subset S of X, a n d nonnegative integers n(£) (£ e S), such that, for all choices of elements in I for % running over S, w e have >H

(9)

Coates: Motivic p-adic L-functions

145

belongs to the Iwasawa algebra 3. It is dear that the pseudo-measures form a subring of Q(3). Suppose now that n is a pseudo-measure. Let ty be any element of X which is distinct from all § in S. For each \ in S, choose a(£) in I such that <|>(a(£)) * £(a(£)). We then define the integral of <>| against \i by the formula Ji 0 dji = d-i n ( ^ € S) <$
(10)

where A, denotes the integral measure (9). It is immediately verified that this definition is independent of all choices. Also, if X given by (9) is an integral measure, we say that the pseudo-measure n has a pole at each £ in S of order ^ n(£); the minimal value of n(£) such that the expression (9) lies in 3 is called the exact order of the pole of \i at £. Finally, there is an important involution on the ring of pseudomeasures on I, which we denote by \i -» n> #. This involution is given on O [I] by the O -linear map which sends a to a"1 for all a in I, and it extends by continuity to 3. It plainly extends to Q(I), and preserves the subring of pseudo-measures. 3. The cyclotomic theory. This section will be devoted to a brief account of the p-adic analogue of the Riemann zeta function. Recalling that X is given by (4), we write X aig for the subgroup of X consisting of all characters of the form £ =Vnx

(neZ),

(11)

where % is any character of finite order of G(P/Q), and y given by (3) is the p-adic cyclotomic character. Let Too denote the element of G given by complex conjugation. We are assuming that (11) is a character of the Galois group J, and this is clearly equivalent to the assertion that X(Too) = ( - l ) n

.

(12)

The following is the basic existence theorem for the p-adic analogue of the Riemann zeta function. Fix an integer m < 0 and a character <>| of finite order of G(P/Q), which satisfy

146

Coates: Motivic p-adic L-functions

.

(13)

The reason for this condition will become apparent later (in the notation and terminology explained later, we want the motive Q(m) twisted by <>| to have weight > 0 and to be critical at s = 0). Let O be the ring of integers of the field obtained by adjoining the values of <>| to Q p , and let 3 be the O - Iwasawa algebra of J. We remark that formula (15) below shows that, in the following theorem, the right hand side of (14) belongs to the field A of algebraic numbers, and so can be viewed as lying in C p via the embedding (1). Theorem 1. There exists a unique pseudo-measure [i = |i(m,<|>) on the Galois group J satisfying :- (i). For all a in J, (\|/1"m(|)-l(a) - a)|i belongs to the Iwasawa algebra 3 ; (ii). If £ given by (11) is any element of Xaig such that m+n < 0, then Jj § dn = LT(G5R, n+m),

(14)

where T = {p}, and 05 = (|)% . Moreover, [i has a pole of order 1 at yl-nty-i. We sketch what is essentially Iwasawa's proof of the existence of \i. Put r=4 or r=p, according as p is even or odd, and put r k = rp k for all k > 0. For each p-adic unit u, write [u] k for its class in the group of relatively prime classes of integers modulo r k . The partial zeta function C(u,r k ;s) = Zw-s

(R(s) > 1)

,

where the sum is over all positive integers in [u] k , has an analytic continuation over the whole complex plane, apart from a simple pole at s=l. For each non-negative integer t, we have C(u, rk; -t) = - rkt Bt+i ({u}k /r k )/(t+l) ,

(15)

where {u}k denotes the unique representative in Z of [u] k ,which lies between 0 and r k ; here Bt+i(x) denotes the (t+l)-th Bernoulli polynomial, which is defined by the expansion

Coates: Motivic p-adic L-functions

147

In particular, we have Bi(x) = x - 1 / 2 , Bt+i(x) = xt+1 - (t+l)/2 xt + ... .

(16)

For t fixed, let p e denote the largest power of p occurring in the denominators of the coefficients of Bt+i (x)/(t+l). One deduces immediately from (15) and (16) that, for all integers k > 0 and all p-adic units u, we have C(u, r k + e ; -t) = t ut+V((t+l)r k+e ) + u* £(u, r k + e ; 0) mod rk .

(17)

If v is also a p-adic unit, we define 5 t (u, v; rk) = vt+l C(u, r k ; -t) - £(uv, r k ; -t). Then we claim that, for all integers k > 0, we have 8t (u, v; rk) s (uv)* 8o(u, v; rk) mod rk

.

(18)

Note that (17) immediately implies the weaker version of (18), in which the first two rk's appearing in (18) are replaced by r k+e . But it is easy to see that this weaker congruence implies (18), when it is combined with the additional identity 2C(z,r h ;s) =C(u,r k ;s) ,

(19)

where h is any integer > k, and z runs over any set of representatives in U of those classes modulo rh which map to the class of u modulo rk . Note that one obvious consequence of (18) is that 8t(u, v; rk) is integral at p for all t > 0, because this is plainly true for t = 0 from the explicit formula (16). We can now construct the pseudo-measure |n. For each u in U, let a(u) denote the unique element of G(P/Q) such that y(a(u)) = u, and let T(U) denote the restriction of o(u) to N. Let P k be the field obtained by adjoining the group of rk -th roots of unity to Q , and let N k be the maximal real subfield of P k . We write a k (u) (resp. xk(u)) for the restriction of a(u) to P k (resp. to N k ). Write V k for any set of representatives in U of the group of relatively prime residue classes

148

Coates: Motivic p-adic L-functions

modulo rk . We assume in what follows that k is so large that the conductor of p divides rk . For each p-adic unit v, define the following element of the O - group ring of the Galois group of Nk over Q

where the sum is over all u in Vk . The identity (19) shows that, as k varies, the A,k(v) define an element X(y) of the Iwasawa algebra 3 . Put 9 = yl-mty~l# By virtue of (13), 0 is a character of the Galois group J. If v is not of finite order in U, it is easy to see that 0(T(V)) - x(v) is not a zero divisor in 3 . For any such v, we define ji = Mv).(0(x(v)) - T(v)H . It is readily verified that \i is a pseudo-measure on J, which is independent of the choice of v of infinite order in U, and which satisfies assertion (i) of the above theorem. Assume now that k is so large that rk is also divisible by the conductor of %. To prove (ii), we note that, by definition, the integral of § = \|/n% against the measure A,(v) is the p-adic limit as k -> «> of the expression 4> ©© of the expression (Ka(v))-1 v n S(U) 5-n-m(u, v; rk) G5(G(U))-1 . But, again using (19), we see that this last quantity has a value independent of k, which is given by (0(x(v)) - §(T(V)) LT(G3R, n+m) . Assertion (ii) is now plain. We omit the proof of the final statement of the theorem, which is a well known consequence of the von-StaudtClausen theorem giving the exact power of p occuring in the denominators of the k-th Bernoulli numbers, where k runs over the positive even integers which are divisible by p-1.

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4. Complex L-functions. Motives arise in nature as direct summands of the cohomology of a given dimension of a smooth projective algebraic variety defined over Q. However, we shall simply view motives in the naive sense, as being defined by a collection of realisations, satisfying certain axioms. Moreover, since we must consider the twists of our motives by arbitrary characters of finite order of G ab , it is technically necessary to consider motives over Q, with coefficients in some finite extension K of Q. A detailed account of such motives and their realizations is given in §2 of [6], and we only briefly recall some of the key definitions here. Let Z(K) denote the set of embeddings of K in the complex field C . We identify the C-algebras K®C (unless indicated to the contrary, all tensor products will be understood to be taken over Q) and C^O®v i a K®C = C«K) : u®w -> (w. a(u))a .

(20)

In addition, for each prime number 1, we put

where X runs over the primes of K dividing 1, and K% denotes the completion of K at ^. By a homogeneous motive M over Q, with coefficients in K, of weight w(M) and dimension d(M), we mean a collection of Betti H B ( M ) , de Rham H D R ( M ) , and 1-adic Hi(M) (one for each prime 1) realisations, which are, respectively, free modules over K, K, and Ki , all of the same rank d(M). Moreover, these realisations are endowed with the following additional structure :- (i). H B ( M ) admits an involution F<x> ; (ii). The global Galois group G has a continuous action on Hi(M) for each prime 1, and there is an isomorphism gl: HB(M)®KKi -> Hi(M) which transforms the involution Foo into the complex conjugation; (iii). There is a decreasing exhaustive filtration {FkHDR(M) : k G Z} on the de Rham realisation; (iv). There is a Hodge decomposition into Cvector spaces

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HB(M)®C = 0 HM(M) ,

(21)

where i,j run over a finite set of indices satisfying i+j=w(M), and where Foo maps HM(M) to HM(M); (v). There is a Goo = G(C/R) isomorphism of C - vector spaces (which also commutes with the action of K) goo:HB(M)®C ->HDR (M)®C

(22)

where complex conjugation acts on the space on the right via its action on C, and on the space on the left via Foo on HB(M) and via its natural action on C; (vi). Finally, for all k e Z, we have goo (0(i>k) HM(M)) = F^HDR(M)

.

(23)

The first basic example of such a motive M is the Tate motive Q(m), for any m in Z, which is of weight -2m and dimension 1. Let V\([0 be the tensor product over Z\ with Q\ of the projective limit of the Galois modules |iin of l n - th roots of unity, and let Vidi)®™ be the m-th tensor power of Vi(p,). Then the realisations of M = Q(m) are given by HB(M)

= K , HDR(M) = K , Hi(M)

=

The involution Foo is (-l) m , and the action of G is the natural one. The Hodge decomposition is specified by taking H"m'"m = K®C , and the k-th term in the filtration of the de Rham cohomology is either K or 0, according as k < -m or k > -m. The isomorphism (22) is given by goo(l®l) = If M is any such motive, we can construct the following motives from M :- (i). The twists M(n) for any n in Z ; the realisations of M(n) are the tensor products of the corresponding realisations of M and Q(n); (ii). The dual motive M A ; the realisations of M A are the dual vector spaces of the realisations of M. We briefly recall the standard definitions and conjectures for the complex L-function attached to such a motive M. Put

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151

For simplicity, we assume that, when w(M) is even, Foo acts on H k ' k , where k = w(M)/2, via a scalar (this will be automatically implied by our assumption made later that M is critical at s=0). As is explained in §2 of [6], the fact that HB(M)®C is a free K®C - module, when with the identification (20), yields a decomposition HB(M)®C = 0 HB (a, M), where H B (a, M) = H B (M)®(K/ a)C; here a runs over Z(K) and the tensor product on the right is taken by regarding C as a K-algebra via a. Each H B (a, M) admits a Hodge decomposition H B (G, M) = 0 H)/k(a, M) , and we let h(j,k) = C -dimension of HJ' k(a, M). This notation is justified, since it is shown in [6] that these dimensions are independent of a e Z(K). The Euler factor at <», which is also shown in [6] to be independent of the choice of a, is then defined by

where U runs over the direct summands of H B (a, M) of either the form (i) U = HJ/k(a)0Hk'J(a) with j < k, or (ii) U = H k ' k (a), (where we have abbreviated HJ'k(o,M) by HJ'k(a)) and Loo(U, s) is given explicitly by :- (a). In case (i), Loo(U, s) = IT(j < k) T C (s-j)h<j>k>; (b). In case (ii) when Foo acts on H k ' k (a) via (-l) k , then Loo(U,s) = r R (s-k) h (k,k); (c). In case (ii) when Foo acts on H k ' k (a) via (-l) k+ * ,then Loo(U, s) = rR(s+l-k)fc. By contrast, the Euler factors at finite primes do depend on the choice of a in Z(K). If q is a finite prime, let Iq denote the inertial subgroup in G of some fixed prime of A lying above q. Then the Euler factor at q is given by M,s) = (aZq)(M,q-s)/ where Zq(M, X) = det (1 - Frobq"1 .X I

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and where X is any prime K not lying above q. We have imposed the standard hypothesis that Zq(M, X) is a rational function in X , with coefficients in K , which are independent of the choice of the prime X. The complex L-function of M is then defined by the Euler product

where v runs over all primes of Q , including v = «>. We also write L(a,M, s) for this Euler product with the infinite Euler factor omitted. Note that we have A(a, M(n), s) = A(a, M, s+n) for all n e Z . We assume that there exists a finite set of primes S = S(M) such that (i) for each prime X, and each q which is not in S and which does not lie below X, the inertia group Iq operates trivially on H^(M), and (ii) for q not in S, the reciprocal complex roots of (aZq )(M, X)"1 have absolute value equal to qw(M)/2# Under additional hypotheses, one can then define the conductor of M and the global e-factor e(o, M, s) (see [14]). Here is the standard conjecture about the analytic continuation and functional equation of this L-function. Conjecture A (Complex Version). A(a, M, s) has a meromorphic continuation over the whole complex plane to a function of order < 1, and satisfies the functional equation A(c, M, s) = e(a, M, s)A(a, MA(1), - s) .

(24)

It is also conjectured that A(a, M, s) is entire if w(M) is odd, and that the only possible pole which can occur, if w(M) is even, is at s = l+w(M)/2 . In this latter case, the order of the pole is conjectured to be the K\- dimension of the subspace of H^(M(w(M)/2)) which is fixed by the global Galois group G, for any prime X of K. As is explained in [5] and [14], the global e-factor e(a, M, s) has a decomposition into local factors, which we shall see plays an important role in the non-archimedean theory. Let 0 denote the adele group of Q . Fix, once and for all, the Haar measure dx = II dx v on 0,

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153

where dxoo is the usual Haar measure on R , and, for each prime q, is the Haar measure on Qq which gives Zq volume 1. To define the local e-factors, we must choose a complex character of the adele class group 0 / Q , and there are inescapably two natural choices. For the rest of this paper, p will denote i or -i, where i has its usual meaning as a complex number. Let Tip denote the character of 0 / Q with components T|p,oo(x) = exp(2rcpx), and, for each finite prime q, Tip,q(x) = exp(-2rcpx), where we have identified Qq /Zq with the q-primary subgroup of Q/Z . For each place v of Q,, let e v (a, M, p, s) denote Deligne's local e-factor for the relative to the various choices just described (we have suppressed the the fixed measure dx v in the notation, and we simply write p instead of the additive character Tip). Then we have e(a, M, s) = n v e v (a, M, p, s) ,

(25)

where the product is taken over all primes v of Q, including v=<» . Note also that we have the fundamental relation e v (a, M, p , s) e v (a, MA(1), - p , -s) = 1

.

(26)

It is fundamental for the non-archimedean theory that we also consider the twists of our motive M by characters of finite order of G ab, and we now briefly recall the definition of these twists . Let <> | : G ab -» A* be a character of finite order, and assume that the values of <|) lie in K. Following [6], §6, we can attach to <> | a motive [())] of dimension 1 and weight 0 over Q , with coefficients in K. Let V(<|>) be the vector space of dimension 1 over K, on which G acts via <|>. We then define HB(()>) to be the underlying space of V(<)>), with the action of Foo given by <|)(Too), where too is complex conjugation. The de Rham realisation is given by HDR(<|>) = (V(<|))®A)G, where the global Galois group G acts both on V(<|>) via <|), and on A in the natural fashion (we endow the de Rham realisation with the trivial filtration for which F k is 0 for k > 0, and the whole space for k < 0). The comparison isomorphism g/oo:HB(<|))®C -> HDR«>)®C

(27)

is obtained by noting that HDR((|>) provides a Q - structure for HB(<(>)®A, and then extending scalars from A to C. For each finite prime A, of K,

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we take the A,-adic realisation to be H^((|)) to be a vector space of dimension 1 over the completion K^ of K at X, on which G acts via <> |. For each embedding a: K -> C, we can apply the above motivic recipe for attaching a complex L-function L(a, <(>, s) to the motive [<))], and, in view of (8), we see that L(a, <j>, s) coincides with the L-function L((<|)a)R, s) defined by (6) - indeed, our sign of the reciprocity map was chosen to assure this. Now let M be a motive over Q , with coefficients in K, as above. The twist M(<|>) is then defined to be the motive over Q , with coefficients in K, whose realisations are the tensor products over K of the realisations of M with the corresponding realisations of [(()]. 5. Critical points and the period conjecture. Our goal in this section is to give a modified version of Deligne's period conjecture of [6], which seems essential for problems of p-adic interpolation. We shall be concerned with the following question. Let M be a fixed motive over Q , with coefficients in some finite extension K of Q , and consider twists of M of the form W = M(n)((|>) , with <> | (Too) = (-Dn ,

(28)

where n ranges over Z , and (> | over the characters of finite order of G ab with values in K. How does the Deligne period c+(W) vary with n and <()? It turns out that the naive answer to this question is not precise enough for problems of p-adic interpolation, and our aim will be to use the properties of the complex L-function to give a finer answer, at least when both M and W are critical at s = 0. We begin by briefly explaining the naive answer to the above question, which does not depend on any assumptions about M or W being critical at s = 0. In fact, the techniques of [6] reduce this to a problem of linear algebra (see [6] for the background material, which we do not repeat in detail here). We suppose always that K contains | . We assume that Foo acts on Hk'k(M) by a scalar. In §2 of the values of <> [6], Deligne attaches to W a period c+(W) in (K®C)*, which is well defined up to multiplication by an element of K*. Let p denote a choice of either +i or -i in C. Let f(<|)) denote the conductor of <|>, so that (> factors through the Galois group, which we denote by A((|>), of the

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155

field generated over Q by the group of f(<|>)-th roots of unity. Following §6 of [6], we define the element 8p(<|)) e (K®C)* by 8p(<|>) = Z(x6 A(*» (|)-1(x)®(exp(-2icp/f(^))^

.

(29)

If a = + or -, let HB(M) a denote the subspace of H B ( M ) on which Foo acts via the sign a, and let d a (M) denote its K- dimension. L e m m a 2. Let W be the motive given by (28). Then, up to multiplication by an element of K*, c +(W) coincides with C+(M)((27ri)n 5p(<|>))
(30)

Proof. Let T = M(n), and put e = <|>(Too). Then (28) implies (see [6], p. 329) that = (27ri)*d+(M) C+(M) .

(31)

Now W = T((|))/ and (28) gives immediately , H D R ( W ) + = H D R ( T ) £ ® K HDR(<|>) .

By definition, isomorphism

c + (W) is the determinant of the

comparison

gR,~+ : HB(W)+ ® C -> H D R ( W ) + ® C , computed relative to K-bases of the two sides (which each have cardinality equal to d + (M)). Now a K-basis of the left hand side is given by {oci®l®l}, where [a{] is a K-basis of H B ( T ) £ . Similarly, a K-basis of the right hand side is given by {Pi®g®l}, where g is any non-zero element of HDR(<|>) and (pi) is a K-basis of H D R ( T ) S . Thus c+(W) coincides, up to multiplication by an element of K*, with coincides with tfCRg-d^M) . The assertion of the lemma now follows from (31), and the fact that, as remarked in §6 of [6], we can take g=5-p(<|>-1)/f(<|>), whence g"1 = 5p(<|)). Recall that an integer s = n is said to be critical for M if both the infinite Euler factors Loo(a,M, s) and Loo(a,MA(l), -s) are holomorphic at s = n. The following lemma (due to Bloch, Deligne, Scholl, ...) gives

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several useful equivalent forms of this definition. As before, we write h(j,k) = C-dimension of HJ*(G) = HJ*(a, M) (see (24) and (25)), where a € S(K). We recall that both the infinite Euler factors, and these dimensions, are independent of the choice of a. Lemma 3. The following three assertions are equivalent for M:- (i). M is critical at s = 0; (ii). If j < k and h(j,k) * 0, then j < 0 and k > 0, and, in addition, if h(k,k) * 0, then Foo acts on Hk/k(a) by +1 if k < 0 and by -1 if k > 0; (iii). The map hoo:HB(M)+®R -» ( H D R ( M ) / F O H D R ( M ) ) ® R

,

(32)

induced from (22) is an isomorphism. Proof. The equivalence of (i) and (ii) follows from the explicit formulae for the infinite Euler factors given above, and we do not give the details. Assume now that (ii) is valid. It follows that d+(M) = E( j
(33)

It follows from (23) and these formulae that the two sides of (32) have the same R - dimension, and so it suffices to prove (32) is injective. Again by (23), hoo will certainly be injective if (HB(M)+®C) n (ee<j*o>Hi*(a)) = 0 .

(34)

Let a C - basis of Hi'k(o) be given by {ei(a,j,k): i = l,...,h(j,k)}. Then a C basis of H B ( W ) + ® C is given by the set {ei(o,j,k) + Foo(ei(a,j,k)): j < 0 < k , i = l,...,hi(j,k), oeS(K)}, together with the set {ei(a,k,k): i = l,...,hi(k,k), aeZ(K)} if k = w(W)/2 is < 0. Hence any non-zero element of H B ( W ) + ® C will have a non-zero projection on at least one of the subspaces HJ'k(a) with j < 0. This proves (34), and so also (iii). Conversely, assume (iii) holds. The equality of dimensions on the two sides of (32) shows that (33) is then valid. But, if j < k, then the space HJ'k © Hk'j contributes h(j,k) to both d+(M) and d"(M), and so it follows from (33) that we must have j < 0 < k. If H k ' k * 0, we also conclude from (33) that Foo acts on Hk>k by +1 if k < 0, and by -1 if k > 0. This establishes (ii), and completes the proof of the lemma.

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157

Fix an embedding a in E(K). If v is any place of Q , we define Rv(a, M, p, s) = Lv(a, M, s)/(e v (a, M, p, s)Lv(a,MA(l)/ - s)).

(35)

As is already noted in [6] when v is non-archimedean (see Remark 5.2.1, p. 329), this ratio tends to be better behaved that the individual factors defining it. We shall exploit this fact in what follows. Clearly, we have R v (c, M, p, s) = R v(c, MA(1), - p, - sH. It is therefore natural to ask whether one can define canonical new factors Ev(a, M, p, s) such that Rv(a, M, p, s) = Ev(a, M, p, s)/E v (a, MA(1), - p, - s)

(36)

(of course, this last equation cannot characterize the factors E v(a, M, p, s)). In fact this is the case, as we shall subsequently explain. Note one immediate consequence of such a construction. Let S be any finite set of primes of Q . Define the modified L-function A(S)(a, M, p, s) = II(v E S) Ev(d, M, p, s). II Lv(a, M, p, s), where the latter product is over primes v not in S. Then we have the following modified form of the functional equation (24) A(S)(a, M, p, s) = (II(v * S) £v(a, M, p, s)) A(S)(a, MA(1), - p, - s). (37) We now give the definition of the E v - factors when v = °°. For s in C, put p-s = exp( - prcs/2), Tc, p(s) = p~s Tc (s). We also recall that the Euler factor at <» is independent of the choice of the embedding a. Similarly, the e - factor at «> is independent of the choice of a (see, for example, the explicit formulae given on p. 329 of [6]), so that we may drop the a from our notation in this case. We then define Eoo(M, p, s) = n Eoo(U, p, s),

(38)

where U runs over the direct summands of the Hodge decomposition,

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and Eoo(U, p, s) is given explicitly by :(a) If U = HJ*(a)0HM(a) with j0, then Eoo(U, p, s) = 1 ; (c) If U = Hk,k( a ) with k<0, then Eoo(U, p, s) = Roo(U, p , s). From the table of values of the £oo(U, p, s) given on p. 329 of [6], w e deduce easily that (36) is valid in case (a), and it is plainly true in cases (b) and (c). If u and v are complex numbers, w e write u - v if there exists y in Q* such that u = yv. The integer r(M) defined by

plays an important role in what follows, thanks to the next crucial lemma. Lemma 4. Assume that M is critical at s = 0. Then Eoo(M,p,0) ~ (2rcp)r(M) ,

(39)

where the rational number implicit in the ~ is independent of the choice of p. Corollary. Assume that M is critical at s = 0, and that W given by (28) is also critical at s = 0. Then Eoo(W,p,0) ~ Eoo(M,p,0)(27cp)-nd+(M)

.

(40)

To deduce the corollary, w e first note that hM(j/k) = hw(j-n, k-n), and that d+(M) = d + (W), because of our hypothesis that <|>(O = (-D n . As M is critical at s = 0, Lemma 3 shows that d + (M) is given by (33). On the other hand, since W is critical at s = 0, (ii) of Lemma 3 shows that j-n < 0 if and only if j < 0. Hence r(W) = r(M) - nd + (M), as required.

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159

We now turn to the proof of Lemma 4. We prove the lemma by considering the three cases (a), (b), (c) for U above, and verifying (39) for each U, where r(U) = jh(j,k) in case (a), r(U) = 0 in case (b), and r(U) = kh(k,k) in case (c). This suffices since clearly r(M) = X(u) r(U). If s is in Z , we shall make use of the following classical properties of the Tfunction (see [6], p. 330) :r c (s) ~ (2rc)-s (s>0),

TR(s) ~ (2TC)(1-S)/2 (S odd),

TR(S) ~ (27c)-s/2 (s>0 and even). Suppose we are in case (a). Then EooOJ, p, 0) = (pj rc(-j))h(i'k> - (27ip)r(U) , as required. In case (b), (39) is plainly valid. Suppose finally that we are in case (c) (the one delicate case). Put h =h(k,k). There are two possibilities, according as k is even or odd. (i). Assume k is even, so that Foo acts on U by (-l) k . Then EooOJ, p, 0) = 1 by the table on p. 329 of [6], and we have , 0) = rR(-k)** ~ (2rc)rh/2, Loo(UA(l),0) = r R (j+2)h ~

whence (39) is again plain. The reader should also note that the unknown non-zero rational number implicit in (39) is independent of the choice of p.This completes the proof of Lemma 4. We can now give the modified form of the period conjecture. Recall that we identify K®C with CZ
,

(41)

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whose components are well defined up to multiplication by a system of numbers a(oc) (a e Z(K)), for any a in K*. For each choice of p = i or -i, we now put Qp(M) = (Qp(o, M)) = c+(M)(27cp)r(M) .

(42)

If <|) is a character of finite order of G a b with values in K, we also recall that 8p(c()) is given by (29), and its image under the isomorphism (20) is 5p(<|>) = (8 p (a, <(>)), where

8 p (a, <|>) = S(T e A((t>» (WHtexpWrcp/fty)))* .

(43)

L e m m a 5. Assume that M is critical at s = 0, a n d that W given b y (28) is also critical at s = 0. Then, for each a € E(K), the quantity A(oo)(c, W, p) Q p (a, M H 8p(G, (|))-<*+(M)

(44)

does n o t d e p e n d on the choice of p = i or -i. Proof. It is plain that 8- P (G, <|>) = <|>°(Too) 8p (a, <)>).

(45)

The assertion of the lemma n o w follows from (28), Lemma 4, a n d the fact noted earlier that r(W) = r(M) - nd+(M). Period Conjecture. Assume that M is critical at s = 0, a n d that W given by (28) is also critical at s = 0. Then there exists a e K such that the expression (44) is of the form a ( a ) for all o e L(K). Indeed, b y Lemmas 3 a n d 4, w e see that this conjecture is equivalent to Conjecture 2.8 of [6], applied to the motive W. 6. Modification of the Euler factor at p. W e again let M be any m o t i v e over Q with coefficients in K. For this section, w e d r o p the assumption that M is critical at s = 0, as it will not be needed. Let p b e a n y p r i m e n u m b e r - t h e only restriction placed o n p is given b y H y p o t h e s i s I(p) b e l o w . O u r a i m in this section is to define a modification of the Euler factor at p , which is analogous to that already

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161

given for the Euler factor at ©o. Throughout, a will denote any element of 2(K), and p will again denote i or -L Let Gp denote the absolute Galois group of Q p , and let Ip (resp. Wp) be the subgroup of Gp given by the inertial subgroup (resp. the Weil group). We fix an element O of G p, whose image in Gp/I p is the geometric Frobenius (i.e. the inverse of Frob p ). For each s € C, let c»s : W p -> C* be the homomorphism which is trivial on Ip , and which satisfies co s (0) = p"s . We also fix a prime number 1 * p, and a non-zero homomorphism ti: I p -> Z\. Let X denote a prime of K above 1, and K\ the completion at X. Now write W p' for the Weil - Deligne group of Qp . Recall that the representations of W p' are defined as follows (see [5], §8). Let V be a finite dimensional vector space over K\ . Then a representation of W p ' in V is a pair 0 = (y, N), where (i) y : W p -» GL(V) is a homomorphism, whose kernel contains an open subgroup of Ip , and (ii) N is a nilpotent endomorphism of V such that yio)Ny(oy1

=coi(a)N

for all a in Wp .

Given such a representation 0 , we can define the dual representation 0A = (yA^ JSJA^ w here yA is the contragredient representation. Writing I = Ip for brevity, we define V N = Ker (N) , Z p (0, X) = det( 1 - 7(0) X I V N * J) Y1. Let a : K% -» C denote a fixed extension of the embedding a in £(K). We then put L p (a, 0 , s) = ( a Z p ) ( 0 , p-s). Write e p (a, y®cos, p) for Deligne's e-factor attached to the representation y®cos of W p on the complex vector space V®(K3L,
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behaved. In particular, it does not change if we replace the representation 0 = (y, N) by 0* = (y, 0). Finally, we recall (see [5], §8.5) that, for each representation 0 = (y, N) of Wp', one can define a new representation 0 S S = (*fs , N) called the O-semisimplification of 0 , which has the property that y88 is a semisimple representation of the ordinary Weil group W p . Again, Rp(a, 0, p, s) does not change if we replace 0 by Now let us return to the A,-adic representation of W p given by its natural action on H^(M). By Grothendieck's theorem, this A,-adic representation gives rise to a unique representation 0 = (y, N) of the Weil - Deligne group Wp' (see [5], §8). Lemma 6. There exists a representation 0 ' = (y1, N1) of the WeilDeligne group in H^(M), which satisfies :- (i). N1 = 0; (ii). If we extend scalars from K^ to C via the embedding a, then y1 is a semisimple complex representation of W p ; and (iii). We have R p (a, M, p, s) = Rp(a, 0', p, s).

(46)

Proof. By the construction of the representation 0 via Grothendieck's theorem, we have that (46) is valid with 0 replaced by 0'. On the other hand, it was remarked above that R p (a, 0 , p, s) does not change if we replace 0 by ©i = (y, 0), and subseqently ©i by ©' = @i ss . It is plain that this choice of ©' satisfies the assertions of the lemma. We can now define the factors E p(a, M, p, s) satisfying (36). Let Y : W p -> GL(Y), where Y = HJL(M) ®(Kh a) C

(47)

be the semisimple complex representation of the Weil group given by Lemma 6. Let Y = © U be the decomposition of Y into irreducible complex representations of W p . For each such representation U, we can define the expression R p (a, U, p, s) by the formula (35) with M replaced by U, and, in view of (46), we have R p (o, M, p, s) = ri(u) R p(a, U, p, s).

(48)

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163

Each U occurring in this decomposition is irreducible, and hence is known (see [5], §4.10) to be of the form £u ® cos(u) / where s(U) is some complex number, and £u is a complex representation of the Weil group such that £uCWp) is a finite group. Consequently, the inverse roots of the polynomial det (1 - O. X IU) (note that we do not take the subspace of U fixed by I, but rather the whole of U) are all of the form a root of unity times one fixed root. Thus, assuming that these inverse roots are algebraic numbers, and viewing them as lying inside Cp via the embedding (1), we can unambiguously define ordp(U) to be ordp(ot) for any inverse root a of this polynomial; here ordp denotes the order valuation of C p , normalized so that ordp(p) = 1. Note also that ordp(U) is independent of the choice of <E>, since the image of I in GL(U) is a finite group. Clearly, we have ordp(U A(l)) = - ordp(U) - 1 , so that it is natural to impose the following hypothesis :Hypothesis I(p). For each U occurring in the above decomposition, we have ordp(U)* -1/2. For the rest of the paper, we assume that Hypothesis I(p) is valid for our motive M. We then define :(a).

If ordp(U) > -1 / 2 , then Ep(a, U, p, s) = 1;

(b).

If ordp(U) < -1 / 2 , then Ep(a, U, p, s) = Rp(a, U, p , s).

Note that the case (a) holds for U if and only if case (b) holds for U A(1), because of Hypothesis I(p). Thus, putting Ep(a, M, p, s) = n ( U ) Ep(a, U, p, s), it follows from (48) that the equation (36) is valid, as required. We now explicitly calculate the the E p - factors in some simple cases. Let d p (a,M) denote the number of inverse roots a of the polynomial (aZp(M, X))-l = a det (1 - O. X I H^(M)I)

(49)

which satisfy ordp (a) < 0 (by hypothesis, the coefficients of this polynomial are algebraic numbers in C, and we view these as lying in

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Cp via the embedding (1)). As usual, we say that M has good reduction at p if the inertia group I = I p acts trivially on H^(M) for any prime A, of K not dividing p. Lemma 7. Assume that M has good reduction at p. Let p (resp. a) run over all inverse roots, counted with multiplicity, of the polynomial (49) such that ordp(p) > -1/2 (resp. ordp(a) < -1/2). Then Ep(a, M, p, s)/L p (a, M, s) = n ( p ) (1 - Pp-s). n ( a ) (1 - crlps-l). Moreover, if ty is a non-trivial character of finite order of G(P/Q ), we have E p (c, M(())), p, s)/L p (c, M(<|>), s) = (5p(o, <> | ) c^)*)' <¥*' M) (n ( a ) a)-M*>, where 5p(a, <|>) is the Gauss sum given by (43), and c(<|>) = ph(4>) is the conductor of <|>. Proof. The first assertion is immediate from the definitions because £p(a, U, p, s) = 1, since U is an unramified representation of W p . To prove the second, we note that our hypotheses that M has good reduction at p and that p actually divides the conductor of <|>, imply easily that Ep(a, U((l>), p, s)/L p (c, U(<|)), s) is equal to 1 or Epic, U(<|>), p, s)"1, according as ord p(U) is > -1/2 or < 1/2. Let U be such that ord p (U) < -1/2. As U is an unramified representation of the Weil group, a standard formula (see (3.4.6) on p.15 of [14]) shows that e p (a, U((|>), p, s) = e p (a, <|>, p, s)dim(U). ( d e t UXOWO)) . But (det U)(O) = the product of the inverse roots of det (1- OX |U), and it is well known and readily verified that ep(, P, s) = 8p(a, <|>). The second assertion now follows.

Coates: Motivic p-adic L-functions

165

7. p - adic L-functions. We now take N to be any fixed motive over Q with coefficients in Q itself. Subsequently, we shall take the motive M considered in the earlier sections to be the extension of scalars of N to a variable finite extension K of Q. Our aim is to propose a definition of the p-adic L-function of N. While it is very probable that such p-adic analogues exist for all primes p, we can only make precise conjectures at present when N has good ordinary reduction at p. The definition of good reduction at p is given at the end of the previous section. The ordinarity hypothesis is the following condition on the p-adic realisation V = H p (N) of N as a representation for the local Galois group G p of the algebraic closure of Q p over Q p . There exists a decreasing filtration F m V of V (with F m V = V (resp. 0) for m sufficiently small (resp. large)) of Q p -subspaces, which are stable under the action of G p , such that, for all m in Z, G p acts on Frny/Fm+1V via \|/ m ; here \\r is the p-adic cyclotomic character (3). We suppose henceforth that N has good ordinary reduction at p. The same is then easily seen to be true for the motive N A (1). We shall require two additional hypotheses, which are known in many cases, but which we must impose as axioms because of our naive definition of motives. It is well known (see, for example, [9], §6) that our hypothesis that V is ordinary at p implies that it is of HodgeTate type. We recall that this means the following. For each n in Z, let Cp(n) be the 1-dimensional vector space over C p , on which G p acts via the normal action twisted by \|/n. Then there is an isomorphism of G p modules V® Cp = 0(i e z ) Cp(-i)h«) , where h(i) = dim Fty/F 1 * 1 V;

(50)

here the tensor product on the left is over Q p , and where G p acts on this tensor product in the natural fashion, i.e. a(u®v) = a(u)®a(v) for all a in Gp . The fist condition we impose is that the integers h(i) appearing in (50) are related to the complex Hodge numbers by h(i) = h(i, w(N) - i) for all i in Z ,

(51)

where we recall that w(N) is the weight of the motive N. In fact, (51)

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has been proven by Faltings [7] when N is of the form Hk(X)(n), for a smooth projective variety X over Q . As before, let Z p (N, X)-l = det (1 - O. X I Hi(N)), where 1 is any prime distinct from p. Let d = d(N), and let ai,..., ad be the inverse roots in C p of this polynomial, taken with multiplicity. Our second assumption is that, for each i in Z, the number of these inverse roots, counted with multiplicity, which satisfy ordp(oc) = i is equal to the complex Hodge number h(i, w(N) - i). Note, in particular, that this implies that, in the ordinary case, the p-adic order of these inverse roots is an integer, and so Hypothesis I(p) is automatically valid. I understand that the second assumption is known to be true, by the work of Fontaine and Messing (see [8]), when N has good ordinary reduction and is of the form Hk(X)(n), for a smooth projective variety X over Q. We assume from now on that these additional hypotheses are valid. This implies their validity for N A (1). Lemma 8. Assume that N is critical at s = 0. Let a run over the inverse roots of the polynomial Z p (N, X)"1. The number of these a, counted with multiplicity, satisfying ordp(cc) < 0 is equal to d + (N). In other words, we have d p (N) = d + (N). Proof. This is plain from (33) and the second additional assumption made above. Recall that X is the group of all continuous homomorphisms from the Galois group J = G(H/Q) to C p *. We write X aig for the subgroup of X consisting of all \ of the form (11). For such a %, we take K to be the finite extension of Q generated by the values of %, and let M be the motive over Q , which is given by extending the scalars of N to K in the obvious fashion. We then define N(£) = M(n)(x).

(52)

Note that our fixed choice of the embedding (1), together with the fact that we take A to be the algebraic closure of Q in C, implies that K is

Coates: Motivic p-adic L-functions

167

actually given with a canonical embedding i : K -> C In the following, it is convenient and harmless to systematically omit the embedding i from the notation, and simply write A(N(£), s) instead of A(i, N(£), s), etc. We next consider a question which is important for the study the poles of both complex and p-adic L-functions. Let Y p (N) be the subspace of H p (N) given by Yp(N) = Hp(N)G(A/H)

.

(53)

Since G(A/H) is a normal subgroup of G, it is plain that Y p(N) is stable under the action of G, and so provides an abelian p-adic representation ofG. Lemma 9. Endowing Y p (N) ® Cp with the linear action of G (i.e. a(u®b) = a(u)® b for a in G), it breaks up as a direct sum of G-modules Yp(N) ® Cp = 0<§€ B(N)) £#>

,

(54)

where B(N) is some finite subset of Xaig , and the e(£) are integers > 1. Moreover, each £ e B(N) is of the form £ = \|/n%, where n = - w(M)/2 and % is a character of finite order of G(P/Q). Proof. Since the representation factors through the Galois group J, and the decomposition group of p in J is equal to J, it suffices to establish (54) as an isomorphism of G p - modules. Now, viewed as a G p module, Y p (N) is of Hodge-Tate type, because it is easily seen that a sub-representation of a Hodge-Tate representation is again HodgeTate. Thus Yp(N) is an abelian p-adic Hodge-Tate representation of G p . By a theorem of Tate ([11], §111 - 7), this implies that Yp (N) is locally algebraic (note that in [11], it is necessary to assume that the restriction of the representation to the inertia group is semisimple, but it is pointed out in [12], §2 that this condition is automatically true). Since p is totally ramified in the fixed field of the kernel of the representation on Y p(N), it follows that, as a G p-module, Yp(N) is a direct sum of simple locally algebraic abelian representations. Extending scalars to C p , we conclude that it is a direct sum of locally algebraic

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Coates: Motivic p-adic L-functions

characters of G p , which factor through J. But such characters are precisely the elements of Xaig , and so (54) follows. The final assertion is a consequence of the fact that, for any good prime q * p, the reciprocal complex roots of det(l - Frobq^.xl Y p(N)) must have complex absolute value equal to qw(N)/2/ because of our hypothesis that N has weight w(N). It is conjectured that the integers e(£) occurring in the decomposition (54) are related to the poles of the complex L-functions by e(£) = order of pole of L(M(£-l), s) at s = 1.

(55)

Moreover, for ^ e B(N), the function L(M(^ -1), s) should be holomorphic at all points s # 1, and, for £ in Xaig but not in B(N), this function should be entire. Assume from now on that our motive N is critical at s=0. We then consider variable twists of N of the form N(£), with £ in X a ig, which are also critical at s = 0 (infinitely many such £ clearly exist, since we can, in particular, take 2; to be any character of finite order of J). Let c+(N) be the Deligne period of N (it is well defined up to multiplication by a non-zero element of Q). As earlier, let r(N) = Z(j < o) jh(j,k). We assume the strong form of the Period Conjecture which is explained in §5. As always, let p denote i or -i. Conjecture A (p-adic version). Assume that N is critical at s = 0, and let p be a good ordinary prime for N. For each choice of the Deligne period c+(N), there exists a unique pseudo-measure n(c+(N)) on J as follows: for all £ in Xaig such that (i) N(£) is also critical at s = 0, and (ii) £"* does not belong to B(N) and £ does not belong to B(NA(1)), we have j

, p, 0)/(c+(N)(27cp)r(N)) ,

(56)

where S = {<», p}, and A(S)(N(£), p, s) is the modified L-function defined in §5, for the standard embedding i : K -» C .

Coates: Motivic p-adic L-functions

169

Remarks. 1. Using Lemmas 5, 7, and 8, we see easily that the right hand side of (56) is independent of the choice of p, and hence so is the pseudomeasure n(c+(N)). 2. Using the Period Conjecture of §5, and taking into account all the embeddings of K in C , one can show that the pseudo-measure |i(c+(N)) takes values in Q p . 3. The following conjecture about the possible poles of the pseudomeasure |x(c+(N)) should replace that proposed in our earlier papers [3], [4]. Our previous conjecture was too strong because it failed to take into account possible twisting by the characters of finite order of the Galois group J. Holomorphy Conjecture (p-adic version). Let B(N) and B(NA(1)) be the subsets of Xaig occurring in the decomposition (54) for N and NA(1), respectively. Then there exists a non-zero b in Z p such that b 11(5 e B(N)) ( ^ ( o © ) - o(S))e® II(T| e B(NA(D) (TI(O(TI)) - o(i\)Y*<0 *i(c+(M)) belongs to the Iwasawa algebra 3 = Zp[[ J ]], for all choices of a(£) and oil]) in J.

In parallel with (55), it seems reasonable to conjecture the even stronger assertion, that |i(c+(N)) will have poles of exact order e(£) at £-! for £ in B(N), and of exact order e(r\) at each T| in B(NA(1)). 4. If N = Q(m), where m is an odd negative integer, then N is critical at s = 0, and it is easily seen that Theorem 1 (with <>| = 1) shows that both Conjecture A and the Holomorphy Conjecture above do indeed hold for this motive. For further examples, see [4]. The pseudo-measure n(c+(N)) satisfies a simple p-adic analogue of the functional equation of the complex L-function. Recall that |i -> |i # is the involution of the ring of pseudo-measures on J, which is induced by sending a in J to cr1. The conductor of N is an integral ideal of Z , which is prime to p because N has good reduction at p, and we write G(N) for its Artin symbol in J.

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Put e(N, 0). This number does depend on the choice of the periods c +(N), and c+(NA(l)), but the arguments of §5 show that it is independent of the choice of p. Moreover, the arguments of [6], §5 prove that it lies in Q. p - adic functional equation. We have ^i(c+(N)) = y(N). ^(C+(N A (1)) # . G(M)# ,

(57)

where a(M) denotes the Artin symbol of the conductor of M in the Galois group J. Proof. It suffices to show that, for any % in X a i g satisfying the conditions set out in Conjecture A above, the integrals of ^ against both sides of (57) are equal. This follows immediately by combining (56), the modified functional equation (37), and the well known formula that, for q * °°,p, we have eq(N(£), p, 0) = eq(N, p, 0) ^Frobq- 1 )^), where q a ^ is the power of q in the conductor of M; this latter formula is valid because % is unramified at q. In view of (57), we see that y(N) plays the role of a global p-adic e-factor. We only make one observation here about its properties. Lemma 10. Assume that w(N) is odd. Then Y(N) = e(N, 0). (2TC)0 + w(N))d+(N). c+(N)/c+(NA(l)).

(58)

In particular, if N = NA(1), and if we take c+(N) = c+(NA(l)), then 7(N) = e(N, 0).

(59)

Proof. We note that (59) follows immediately from (58), since N = NA(1) implies that w(N) = -1. To prove (58), we observe that, because w(N) is odd, the only terms in the Hodge - decomposition (21) are the HM(N) with i * j, and hence the explicit formula for the infinite efactors, given in [6], §5, shows that

Coates: Motivic p-adic L-functions

171

£oo (N, p, 0) = pWN), where b(N) = Eg < 0) (w(N) - 2j + l)h(j, w(N) - j). In view of (33), we conclude that £oo (N, p, 0) = pd + w(N))d+(N). (_Dr(N).

(60)

On the other hand, because w(N) is odd, it is readily verified that r(N) - r(NA(l)) = (1+ w(N))d+(N). Since the right hand side of this last formula is even, (58) now follows from (60) and the definition of y(N). We conclude by remarking that (59) is exactly what would be predicted by the main conjecture and algebraic arguments involving Iwasawa modules (see Proposition 1 of Greenberg's article in this volume).

References 1. Bloch, S., Kato, K., L-functions and Tamagawa number of motives, to appear. 2. Bloch, S., Kato, K., p-adic etale cohomology, Publ. Math. I.H.E.S. No. 63 (1986), 107-152. 3. Coates, J., Perrin-Riou, B., On p-adic L-functions attached to motives over Q , Algebraic Number Theory, Advanced Stud. Pure Math. 17 (1989), p. 23-54. 4. Coates, J., On p-adic L-functions, S6minaire Bourbaki, Exp. 701, Asterisque 177-178 (1989), p. 33-59. 5. Deligne, P., Les constantes des equations fonctionelles des fonctions L, Antwerp II, Springer L. N. 349 (1973), p. 501-595. 6. Deligne, P., Valeurs de fonctions L et periodes d'int£grales, Proc. Symp. Pure Math. A.M.S. 33 (1979), Vol. 2, p. 313-346. 7. Faltings, G., p-adic Hodge theory, Journal A.M.S. 1 (1988), p. 255-299.

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Coates: Motivic p-adic L-functions

8. Fontaine, J-M., Messing, W., p-adic periods and p-adic 6tale cohomology, Contemp. Math. A.M.S. 67 (1987), p. 179-207. 9. Greenberg, R., Iwasawa theory for p-adic representations, Algebraic Number Theory, Advanced Stud. Pure Math. 17 (1989), p. 97-137. 10. Perrin - Riou, B., Representations p-adiques, p£riodes et fonctions L p-adiques, to appear. 11. Serre, J-P., Abelian l-adic representations and elliptic Benjamin, New York, 1968.

curves,

12. Serre, J-P., Groupes alg£briques associ£es aux modules de Hodge Tate, Ast<§risque 65 (1979), p. 155-188. 13. Serre, J-P., Sur le residu de la fonction z§ta p-adique d'un corps de nombres, C.R.A.S. Paris 287 (1978), p. 183-188. 14. Tate, J., Number Theoretic Background, Proc. Symp. Pure Math. A.M.S. 33 (1979), Vol. 2, p. 3-26.

Emmanuel College, Cambridge CB2 3AP, England.

The Beilinson conjectures CHRISTOPHER DENINGER AND ANTHONY J. SCHOLL*

Introduction

The Beilinson conjectures describe the leading coefficients of L-series of varieties over number fields up to rational factors in terms of generalized regulators. We begin with a short but almost selfcontained introduction to this circle of ideas. This is possible by using Bloch's description of Beilinson's motivic cohomology and regulator map in terms of higher Chow groups and generalized cycle maps. Here we follow [B13] rather closely. We will then sketch how much of the known evidence in favour of these conjectures — to the left of the central point — can be obtained in a uniform way. The basic construction is Beilinson's Eisenstein symbol which will be explained in some detail. Finally in an appendix a map is constructed from higher Chow theory to a suitable Ext-group in the category of mixed motives as defined by Deligne and Jannsen. This smooths the way towards an interpretation of Beilinson's conjectures in terms of a Deligne conjecture for critical mixed motives [Sc2]. It also explains how work of Harder [Ha2] and Anderson fits into the picture. For further preliminary reading on the Beilinson conjectures, one should consult the Bourbaki seminar of Soule [Sol], the survey article by Ramakrishnan [Ra2] and the introductory article by Schneider [Sch]. For the full story see the book [RSS] and of course Beilinson's original paper [Bel]. Here one will also find the conjectures for the central and near-central points, which for brevity we have omitted here.

Partially funded by NSF grant DMS-8610730

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Deninger & Scholl - The Beilinson conjectures

1. Motivic cohomology Motivic cohomology is a kind of universal cohomology theory for algebraic varieties. There are two constructions both generalizing ideas from algebraic topology. The first one is due to Beilinson [Bel]. He defines motivic cohomology as a suitable graded piece of the 7-filtration on Quillen's algebraic if-groups tensored with Q. This is analogous to the introduction of singular cohomology as a suitable graded piece of topological K-theory by Atiyah in [At] 3.2.7. For smooth varieties there is a second more elementary construction which is due to Bloch [B12,3,4]. It is modeled on singular cohomology: instead of continuous maps from the n-simplex to a topological space one considers algebraic correspondences from the algebraic n-simplex A n = A n to the variety. We proceed with the details: Let fcbea field and set for n > 0 A n = Specfc[T 0 ,...,T n ]/(STi-l). There are face maps (1.1)

di:A n <->A n +i

for

0
which in coordinates are given by di (to,..., tn) = (to,..., ti-1,0, ti..., tn). Let X be an equidimensional scheme over k. A face of X x A m is the image of some X x A m /, m! < m under a composition of face maps induced by (1.1) ft:XxAB<^XxAn+i. We denote by zq(X,n) the free abelian group generated by the irreducible codimension q subvarieties of X X A n meeting all faces properly. Here subvarieties Y\, Yi C X x A n of codimensions c\, c c\ +C2 on X. Observe that zq(X1n) is a subgroup of correspondences from A n to X. Using the differential n+l t=0

one obtains a complex of abelian groups zq(X, •). If X is a smooth quasiprojective variety over k setting

1. Motivic cohomolgy

175

we define: (1.2)

HpM(

for any ring A. By one of the main results of Bloch these groups coincide for A = Q with the groups ff^(X,Q(g)) defined by Beilinson using algebraic If-theory. Using either definition the following formal properties can be proved: (1.3) Theorem. (1) H'M{ ,Q(*)) is a contravariant functor from the category of smooth quasiprojective varieties over k into the category of bigraded Q-vecior spaces. For proper maps f :X —*Y of pure codimension c = dimy—dimX we also have covariant functoriality with a shift of degrees

(2) There is a cup product which is contravariant functorial, associative and graded commutative with respect to •. (3) There are functorial isomorphisms compatible with the product structure (4) Fj w (X,Q(l)) = r(X,0*)(8)Q functorially (5) Let i : Y <—> X be a closed immersion (of smooth varieties) of codimension c with open complement j :U = X — Y c-> X. Then there is a functorial long exact localization sequence

(6) If 7T : X' —> X is a finite galois covering with group G we have = \G\ id and 7r*7r* = X^(TeGcr*- ^n Particular

TT^TT*

is an isomorphism, i.e. H'M(

,Q(*)) has galois descent.

For zero dimensional -X" over Q the motivic cohomology groups are known by the work of Borel [Bol,Bo2] on algebraic if-theory of number fields. A proof of the following result which does not make use of algebraic iiC-theory seems to be out of reach.

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Deninger & Scholl - The Beilinson conjectures

(1.4) Theorem. Let k be a number Geld, X = Speck. Then dim®H1M(X,Q(q) = r1+r2 = T2

ifq>l

is odd

ifq>l

is even

where r i , r 2 denote the numbers of real resp. complex places of k. (2)

H'M(XM9)) = 0

for

p?l.

Observe that for X as in the theorem we have:

In view of the class number formula which involves a regulator formed with the units of k we see that for arithmetic purposes the groups HpM(X,ty{qj) may have to be replaced by smaller ones: If X is a variety over Q we set:

M

iorq>P.

M p

(1.7) H'M(X,Q(q))z=lm(H'M(X,
M{X,<}(9))

*or q
Here X is a proper regular model of X over SpecZ which is supposed to exist. The groups HvM(X,ty(q)) are either defined by the above construction which also works over SpecZ or by using the if-theory of X. The "motivic cohomology groups of an integral model" HPM(X\ty(q))i are independent of X. It is a conjecture that (1.6) holds if the definition in (1.7) is extended to q>p.

2. Deligne cohomology and regulator map

177

2. Deligne cohomology and regulator map The definition of Deligne cohomology which is about to follow may seem rather unmotivated at first. We refer to (2.9) below where a conceptual interpretation of these groups as Ext's in a category of mixed Hodge structures is described. (2.1) For a subring A of C we set A(q) = (2m)qh C C. Let X be a smooth projective variety over C and consider the following complex of sheaves on the analytic manifold X a n :

in degrees 0 to q. We set

Apart from the Deligne cohomology groups we need the singular (Betti) cohomology groups of Xan

and the de Rham groups HPDR(X) = (2.2) If X is smooth projective over R there is an antiholomorphic involution FOQ on (-X"c)aiu the infinite Frobenius. We set

where the superscript + denotes the fixed module under F^ = F£QO(complex conjugation on the coefficients). The groups HPB(X, A(#)) are defined similarly if 1/2 G A. Observe that under the comparison isomorphism

the de Rham conjugation corresponds to F^ and hence

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Deninger & Scholl - The Beilinson conjectures

(2.3) Recall that if u : A —> B' is a morphism of complexes of sheaves the cone of u is the complex

with the differentials

There are quasi-isomorphisms on

where Q,^q = (0 -> > 0-> Q,q -+ ft*+1 ->•••) and u is the difference of the obvious embeddings. For a smooth projective variety X over R or C we thus obtain a long exact sequence

Recall here the definition of the Hodge filtration (for X/C say): F*H*DR(X) = and observe that by GAGA and the degeneration of the Hodge spectral sequence we have

Now assume that X is a variety over R. Using Hodge theory we obtain for q > | + 1 exact sequences (B)

0->F*HpDR(X) ^^( 0 -»fl» (X, R(«))-* JJ£ B (X)/F»

-tH!?1 (X, R(q)) -»0.

For a smooth projective variety X over Q these define Q-structures

on detif^ (Xo8,R(^)). Here det W denotes the highest exterior power of a finite dimensional vector space and v is the dual.

2. Deligne cohomology and regulator map

179

(2.4) For smooth quasiprojective varieties X over C the above definition of Deligne cohomology leads to vector spaces which are in general infinite dimensional. A more sophisticated definition imposing growth conditions at infinity remedies this defect. By resolution of singularities there exists an open immersion

of X into a smooth, projective variety X over C such that the complement D = X — X is a divisor with only normal crossings. Consider the natural maps of complexes of sheaves Q^(D) —> j*£l'x on X a n and R(#) —* Cl'x o n -X"an. Choose injective resolutions (q)

x

J'

and set Rj*R(q)=j*I'

and RjM'x =J*J •

We get induced maps on X a n 0^9{D)^RjMx

and

and using the difference of these we can form R(q)v = Cone(Q^ The Deligne cohomology groups

are independent of the choice of resolutions and compactification. As before we can define D-cohomology of varieties over R. For X over R or C there is still a long exact sequence (2.3.1) where now q p F H DR(X) is the Deligne Hodge filtration on HPDR(X). Observe that by the degeneration of the logarithmic Hodge spectral sequence

Assertions (1), (2), (5), (6) of theorem (1.3) have their counterparts for Deligne cohomology. The analogue of assertion (4) is the formula (2.4.1)

H],(X,R(1)) = {g€H°(X

180

Deninger & Scholl - The Beilinson conjectures

which follows immediately from the definition. The typical element of this group should be thought of as an R-linear combination of logarithms of regular invertible functions on X. (2.5) In the proofs of the Beilinson conjectures a more explicit description of P-cohomology in terms of C°°-differential forms is used. Let A' be the de Rham complex of real valued C°°-forms and let TT^ : C —> R(fc), Kk(z) = \{z + (—l)kz) be the natural projection. There is a quasi isomorphism u

J

R(q)v := on Xan induced by the projection

and by the composition:

In particular For p = q we obtain by a straightforward computation f v G H^X^A'-1 (2.5.1)

®R(p-

^ ( X , i

In case p = 1 we find

Under this isomorphism the section g of (2.4.1) is mapped to

&] in Hj)(X)R(a)) resp. jEf|>(X,R(6)) with associated forms u;a, u?^ is represented by One checks that ua ACJ& is associated to

2. Deligne cohomology and regulator map

181

We also note that in this description the boundary map in (2.4.1) is given by

Observe that 7i>_i[u;] = 0 and hence [a;] G H%(X,R(p)). (2.6) The final ingredient in the formulation of the Beilinson conjectures is the regulator map. This is a co- and contravariant functorial homomorphism

rv:HM(X,Q(*))-+Hv(X,R(*)) for smooth quasiprojective varieties X over R or C which commutes with cup products. If motivic cohomology is described in terms of if-theory rx> is a generalized Chern character. In the description of H'M given in section 1 rx> becomes a generalized cycle map (see (2.8)). There is a commutative diagram:

^ (2.6.1)

||

O*(X)®Q

/

HP(X,R(1)) ll*o

-» LeT(X,A°)

log 11 \

f0 = ^ H a, withj

log-smg. at infinity

J

If X is a smooth quasiprojective variety over Q the regulator map is defined by composition:

W : HM(X,q(*)) - ^ HM(Xm,q(*))

"^Hv(Xu,R(*))-

(2.7) The formal properties of motivic and Deligne cohomology and of the regulator map which we have mentioned up to now are sufficient for an understanding of the proofs of Beilinson's conjectures in the cases sketched in section 4. For Bloch's actual construction of the regulator map as a generalized cycle class map in (2.8) however more properties of Deligne cohomology are required. We list them briefly: (2.7.1) There are relative D-cohomology groups H%> Y(X,R(q)) for smooth, quasi-protective X over R and C and arbitrary closed subschemes 7 of J . These fit into a co- and contravariant functorial long exact sequence

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Deninger & Scholl - The Beilinson conjectures

(2.7.2) If Y C X has pure codimension q there is a contravariant functorial cycle class [Y] in H^Y(X, R(q)). Moreover weak purity holds: in other words, H!piY(X,R(q)) = 0 for p < 2q. (2.7.3) Homotopy: H^(X xf\\R(q))

=

H^(X,R(q)).

(2.7.4) For Y C X of pure codimension there are complexes of R-vector spaces D'Y(X,q) which are contravariant functorial with respect to cartesian diagrams Y' *-+ X1

i

i

Y

<-> X

and such that (X,q)) Remarks.

functorially.

(1) The relative £>-cohomology groups are defined by

where R(q)v,x and R(q)v,X-Y are the Deligne complexes on X and X — Y computed with respect to compatible compactifications. The long exact sequence is then an immediate consequence. (2) For the complexes D'(X\q) = D'^X\q) we can choose:

the limit over all coverings U of X a n of the associated simple complex to the Cech complex with coefficients in R(q). Moreover res

DY{X,q) := Cone(D'(X,q) —>D\X- Y,q))[-l\. (2.8) We now proceed to the construction [B13] of the regulator map for smooth quasiprojective varieties X over R or C. Consider the cohomological double complex D-(X*,q) = D-(XxA-*,q) non-zero for • > 0, * < 0 with *-differential: d= E ( - l ) ^ * : D\Xa,q) -» D\Xa+1 ,q). i=0

2. Deligne cohomology and regulator map

183

Similarly another double complex is defined -Dsupp(-X'*^)=

Kj?1

For technical reasons we truncate these complexes (non-trivially) in large negative *-degree: D

(supp)( X *' ?) = ^ " ^ ( s u p p ) ( X * ' «)

where iV > > 0 is an even integer. Consider the spectral sequence

E?h = Hb(D'(Xa,q))

=* if

a

+b(sB'(X*,q))

where s denotes the associated simple complex of a double complex. Because of the homotopy axiom

Ef'h = H!j>(X,R(q))

for

-N
b>0

and E"' = 0 for all other a, b. Moreover d"' = 0 except for a even, — N < a < 0 and 6 > 0 in which case dj' = id. Hence we obtain isomorphisms

In the spectral sequence El* = Hb(Dsapp(Xa,q))

=* £ a + 6 = H a+b(sBsnpp(X*,

q))

we have El'b=

lim

^

for — iV < a < 0, 6 > 0 and E*' = 0 otherwise. The cycle map induces a natural map of complexes

and hence for all p a map

Due to weak purity the groups E^ there are natural maps

are zero for b < 2q and all r > 1. Hence

184

Deninger & Scholl - The Beilinson conjectures

Choosing — N

nat

p

H v{X,R{q)) =

l -

H"(sD-(X*,q)).

It is independent of N. Similarly a regulator (or cycle) map into continuous etale cohomology [Jal] can be constructed. (2.9) We now sketch how the notions introduced fit into the philosophy of motives. More details will be given in the appendix. Assume X is smooth, projective over R. Let MH® be the abelian category of R-mixed Hodge structures with the action of a real Frobenius. According to Beilinson ([Be3], see also [Ca]) there is a natural isomorphism

One would like to give a similar interpretation to the motivie cohomology groups as Ext-groups in a suitable abelian category of "mixed motives". The ultimate definition of such a category remains to be found. However via realizations (^-adic, Betti, ... ) working definitions have been found for M M Q and MMi the categories of mixed motives over Q resp. Z, see [De3,Ja2,Sc2]. It is shown in the appendix that for smooth, projective varieties X over Q there are natural maps (conjecturally isomorphisms) for , H*(X)(q)) and one hopes that the image offir^{"1(-X",Q(^))zis precisely Ext^ M a (Q(0), Hp(X)(q)). Moreover there is a commutative diagram

where HB maps a mixed motive to the Betti realization over R endowed with its mixed R-Hodge structure.

S. The conjectures

185

3. The conjectures Recall the definition of the i-th. L-series of a smooth projective variety X over Q by the following Euler product:

Here we have set

where £ is a prime different from p, Ip is the inertia group in GQ P and Frp is the inverse of a Frobenius element in GQ P . For primes p where X has good reduction the polynomial Pp(Hl(X),t) has coefficients in Q independent of £. The product of the Pp(H%(X),p~s) extended over the good primes converges absolutely in the usual topology for Re s > ^ + 1. Conjectures

[Se]: - The polynomials Pp(Hl(X),t) lie in Q[£] for allp, and are independent of ^, and nonvanishing for \t\
and L(Hl{X),i +

Concerning the special values of these L-functions there is the following conjecture. (3.1) Conjecture. Assume n> | + 1. Then: (3.1.1) rp®R:if^ 1 (X 7 Q(n))z®R^ J ff^ fl (X|R,R(n)) is an isomorphism. (3.1.2) rv(det ^ 1 ( X , Q ( n ) ) z ) = L{H\X),n)Viin

in det

H*\Xu,R(n))

with T>ifn as defined in (2.3.2). If the above hypothesis on the L-iunction of Hl(X) are satisfied assertion (3.1.2) is equivalent to: (3.1.3)

rv(detHi^1(X,q(n))j)

= L(Hi(X),i +

l-nyBi,n

in det if^ +1 (X re ,R(n)) where L(iP(X),A;)* denotes the leading coefficient at s = k in the Taylor development of the i-series [Ja3]. The following result on the order of vanishing follows from a straightforward calculation and the expected functional equation[Sch]: (3 1 4)

dilf^1(X,Q(n))z

assuming (3.1.1).

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Deninger & Scholl - The Beilinson conjectures

Observe that the conjectures determine the special values of the L-series up to a non-vanishing rational number. Equation (3.1.3) is the original proposal by Beilinson. The version (3.1.2) is a reformulation due to Deligne. It requires less information about the L-series to make sense. For the remaining values of n: the right central point n = | + 1 and the central point n = - ^ the conjectures have to be modified ([Bel], Conjecture 3.7 et seq.). Since we don't deal with examples for these cases we skip the formulation. A uniform approach is possible in the framework of mixed motives: The Beilinson conjectures are seen to be equivalent to a Deligne conjecture for critical mixed motives [Sc2]. An integral refinement of the conjectures has been proposed by Bloch and Kato [BIK] using their philosophy of Tamagawa measures for motives. Essentially the only case where (3.1.1) is known is for X the spectrum of a number field. In this case the result is due to Borel with a different definition of the regulator map. For a comparison of the regulator maps see [Bel,Rap]. For a proof of Borel's result the if-theoretical approach to motivic cohomology is essential. In a number of cases to be treated in section 5 and 6 the following weakened version of the conjectures can be proved. (3.2) Conjecture. Assume n > | + l. Then (3.1.1) and (3.1.2) (or (3.1.3)) hold with £T^ 1 (X, Q(n))z replaced by a suitable Q-subspace. Thus motivic cohomology as we have defined it should at least be large enough so that a sensible regulator can be formed having the expected relation to the L-values. (3.3) Generalization to Chow motives. For some well known Lseries the above framework is too restrictive. For example the Dirichlet L-functions of algebraic number theory are not covered. This is remedied by extending the above notions and conjectures to the category of Chow motives, which should be thought of as generalised varieties. Fix a number field T/Q - the field of coefficients. Let V* be the category of smooth projective varieties over a field k. Consider the category Ck(T) with objects TX for each object X in Vk and morphisms Hom(TX, TY) = CHdim Y(X x k Y) ® T. For a : TX\ —» TX2 and b: TX2 —* TX3 composition is defined by intersecting cycles: where pij : X\ x X2 x X3 —* X{ x Xj are the projections. Sending X to TX and a morphism / to its graph / defines a covariant functor from

S. The conjectures

187

Vk —»Cfc(T). The category of effective Chow motives M^(T) is obtained byadding images of projectors to Ck(T). Objects axe pairs M = (TX,p) where p £ End(TX), p2 =p and morphisms are the obvious ones. Setting

the cohomologies and conjecture (3.1) factorize over MQ(T). They determine special values of T®C-valued L-series L(Hl(M),s) up to numbers in T*. See [Bel,Ja3,Kl,Ma] for more details. Remarks, The category A 4 Q ( T ) is not abelian. If instead of Chow theory one considers cycles modulo homological equivalence one obtains what is essentially Grothendieck's category of (effective) motives. Standard conjectures on algebraic cycles would imply that it is an abelian semisimple category. Nowadays these motives are called pure in contrast to more general "mixed" motives which should come e.g. from the H% of singular varieties. The category of these mixed motives MMQ was already alluded to in section (2.9). As yet there is no Grothendieck style definition for MMQ using cycles but only a definition via realizations. As an example of a Chow motive let us construct the motive Mx of a Dirichlet character x of a number field k: Via class field theory we may view x as a one-dimensional representation of the absolute galois group Gk of k with values in a number field T:

We may assume that T is generated over Q by the values of %• Choose a finite abelian extension F/k such that x factorizes over G — G&\(F/k) and set Mx = e x (TSpec(F)) in Mt(T) where e x is the idempotent:

e x ^GE x l ^ I UGG

1

in T[G).

Observe that Mx is independent of the choice of F. We end this section with a short discussion of known cases for the conjectures. (3.1) is known for X = Specif i^/Q a number field [Bo2] and for the motives Mx attached to Dirichlet characters of k = Q or k = K an imaginary quadratic field [Bel,Den2]. In section 5 and 6 we will deduce the evidence for the weak conjecture (3.2) from the theory of Beilinson's

188

Deninger & Scholl - The Beilinson conjectures

Eisenstein symbol map (section 4). The logical dependencies in our approach are depicted in a diagram: modular curves/Q; Shimura curves/Q [Ral]

T

modular curves [Bel,Be2,SSl] _,. , . L . E a s t e r n symbol

1

\

^. . , , , , , . c. Dinchlet characters of imaginary J i.- n u TT\ oi quadratic fields L[Den2]J ^

I

~. . , , , , L Dinchlet characters r ,rh r-rfc -« T^

OMI

of Q [Bel,Den2,N] L

modular forms of > 2 [Sc2]

-

J

we ight

algebraic Hecke characters °r. . , t. of imaginaryJ quadratic , , ° _ S^^. , r fields a [Denl,Den2]

i

CM elliptic curves over , % . , r ni .

number fields 01 bnimura

. ., r D h n type [Bll,Denl]

4. Kuga-Sato varieties and the Eisenstein symbol The Eisenstein symbol is a certain "universal" construction of elements of motivic cohomology of an elliptic curve, or more generally self-products of an elliptic curve. It has its origins in the work of Bloch [Bll] on K2 of elliptic curves but was constructed in generality by Beilinson [Be2]. For a constant elliptic curve a slightly refined construction is made in [Denl]. 4.1 We first introduce the modular and Kuga-Sato varieties. In what follows n will be an integer > 3. Let Mn be the modular curve of level n, parameterising elliptic curves E together with level n structure (Z/n) —• E[n]. Thus the set of complex points M n (C) is the disjoint union of <j)(n) copies of Y(ri)\H, the quotient of the upper half-plane by the principal congruence subgroup F(n) C SX2(Z). The assumption n > 3 assures that there is a universal family of elliptic curves: 7r:Xn-*Mn. Write Mn for the usual compactification of M n , and M£° = Mn — Mn for the cusps of Mn (a sum of copies of SpecQ(£n)). Then we can consider the minimal (regular) model of Xn over Mn:

4- Kuga-Sato varieties and the Eisenstein symbol

189

whose restriction to Mn is just ?r. For each cusp s G M£°, the fibre TT~1(S) is a Neron polygon with n sides. Write Xn C Xn for the connected component of the smooth part (Neron model) of Xn. Then 7^~1(s) fl Xn is (non-canonically) isomorphic to the multiplicative group G m . (4.2) For / > 0_write Xln, Xln, Xlnj.ov the /-fold fibre product of X n (resp. X n , X n ) over M n . The variety Xln has singularities for / > 2; we shall consider these in 5.2 below. Since Xln is a group scheme over M n , in addition to the obvious projections Pi :Xln->Xn

for 1 < % < I

onto the factors, there is a further projection Po = ~Pi

Pi: Xn —> X n .

These (/+1) projections pi allow us to regard Xln as a closed subscheme of XJ*1. This gives an action of the symmetric group §/+i on Xln, permuting the projections po> • • • Pi- The same construction works also for Xln. (4.3) From the localisation sequence (1.3.5) for the pair (Xln,Xln) we have:

Consider the eigenspaces for the sign character sgn ;+1 of §/+i. Under the involution a : x i-> x~x of G m , the motivic cohomology (1.3.4) of Grn decomposes:—

(7=4-1

cr=-l

Using this it is not hard to see that (4.3.2)

H

Here Q[M^°] denotes the set of Q-valued functions on the closed points of M£°. By composing (4.3.1) and (4.3.2) we therefore obtain a "residue map" in motivic cohomology: (4.3.3) Beilinson's key result is then:—

ff£1(*^^

190

Deninger & Scholl - The Beilinson conjectures

(4.4) Theorem ([Be2], Theorem 3.1.7). R e s ^ is surjective for l> 1. (4.5) Remarks. This theorem can be viewed as a generalisation of the theorem of Manin and Drinfeld, which is the case / = 0. For then X% = Mn and (4.3.1) comes from the exact sequence 0

> O*(Mn)

> O*(Mn)

Div

> Z[Jlf~]

c

> PicM n

by tensorising with Q. Here Res^ = Div is the divisor map, and c maps a divisor supported on the cusps to its class in PicM n . According to the Manin-Drinfeld theorem, the divisors of degree zero

Z[Mn°°]0 ^ ker {z[Mn°°] ^ Z} are torsion in PicM n , or equivalently

For / > 0 the picture should be even better. Firstly, there is no restriction to divisors of degree zero. Secondly, the general Beilinson conjectures would imply that Res^ is actually an isomorphism. (To see this one examines carefully the exact sequence (4.3.1).) This makes Beilinson's proof of (4.4) philosophically reasonable—he constructs a totally explicit left inverse to R e s ^ , the Eisenstein symbol map

whose construction we now describe. (4.6) Let Un C Xn be the complement of the n2 sections of order dividing n, and write

K= n

i P-\un)cx n.

We first construct symbols on U]l as follows. Start with any invertible functions go, ... g\ £ O*(Un). Then (4.6.1)

4* Kuga-Sato varieties and the Eisenstein symbol

191

To get an element of HlJ^1(Xln,^(l + l)) we apply three projectors: —U]l is stable under the symmetric group S/+i, and we take the sgneigenspace; —The group of sections of finite order (Z/n) 2 ' acts on UJl by translations, and we project onto the subspace of invariants; —For an integer m > 1, there is a multiplication map

defined as follows: consider the diagram

u" .

j

u"mn | xm]

Here j denotes the inclusion map, and the multiplication [xm] is a Galois etale covering with group (JL/rn)21. By (1.3.6) we have

[m"1]

V

\ ^

l|[xm]*

whence there is a map [m *] as indicated. Denote by a subscript / the maximal quotient of if^(f/^,Q(*)) on which [m"1] is multiplication by m~', for every ra > 1. (In fact it suffices to consider only one m > 1.) (4.7) Theorem. The restriction from Xln to U]l induces an isomorphism

Applying this to the elements (4.6.1) projected to the right hand group gives a map (4.7.1)

0'+10*(C

(4.8) Lemma. The divisor map O*(J7 n )®Q-^QK^/n) 2 ] 0 is surjective. Proof. Let s : Mn —> Xn be a section of order dividing n, and let e : Mn —> Xn be the unit section. We have to show that O(s — e) is torsion in

192

Deninger & Scholl - The Beilinson conjectures

PicX n . It certainly is torsion in the relative Picard group Pic(X n /M n ), so for some N>1 and some line bundle C on Mn we have O(s — e)®N ~ TT*£. Hence C = e*ir*C ~ e*O(s - e)®N = e*O(-e)®N = Af®N where Afe is the normal bundle of the unit section. Hence C ~LO%,AJ , and u>®12 is trivial (a nowhere-vanishing section being the discriminant A). We now have a diagram:

I •&

It is not hard to show that the map (4.7.1) factors through the dotted arrow as shown. The isomorphism d is given by a = n (0)—

^2

(x).

This defines a composite map EM as indicated, which is "almost" the Eisenstein symbol. (4.9) At this point we want to describe the composite of the map ElM just constructed with the regulator map. Let us restrict attention to the component of M n (C) containing the cusp at infinity, and write r for the variable on the complex upper half-plane. The corresponding component of Xln(C) then can be described as the quotient

where the actions of F(n) and I?1 are given by: a

b\

,

v

,ar + b

z\

z\

.

Let j3 6 Q[(Z/n) 2 ] 0 . In terms of the description (2.5.1) of Deligne cohomology by differential forms, rvElM{fi) € H£1(XlJR,R(l + 1)) is represented by (4.9.1)

V

^ o t ' e z ( c ^ + C2V+1(cir + c 2 )'-J+ lV • A dzj A dzj+i A • • • A dzi)sgni + (dr, df term)

4- Kuga-Sato varieties and the Eisenstein symbol

193

where for c = (ci, C2) G (Z/rc)2

(The omitted terms in (4.9.1) involving c?r, df vanish in the applications of §§5, 6.) See 4.12 below for remarks concerning the proof of this formula. (4.10) To pass from ElM to £lM we first recall that the set of closed points of M£° is canonically isomorphic to

GL2(I/n)/Q

± * x ).

The definition of the residue map (4.3.2) involves choosing for each s E M£° an isomorphism of the fibre of X at s with G m (see 4.1 above), and the two such isomorphisms are interchanged by —1 G GL2(l/n). If we replace HQM(M£°,Q(0)) by the (non-canonically) isomorphic space y(~) defined as

then the map R e s ^ becomes Gi2(Z/n)-equivariant. (4,11) Now consider the family of maps

n-1

=

^2 x,y=O

u

(where -B/+2 are Bernoulli polynomials). It is fairly elementary to prove that Xln is surjective. (These maps are the finite level analogues of the horospherical map r of [Be2], paragraph following 3.1.6.) One now proves that (up to a non-zero constant factor) the diagram

(4.11.1)

is commutative, and that ElM factors through Xln. Thus there is a map

194

Deninger & Scholl - The Beilinson conjectures

satisfying £lMo\ln = ElM

and

RealMo£lM=id.

This proves Theorem 4.4. (4.12) We finally say some words about the commutativity of (4.11.1), on which the theorem rests. Beilinson's original proof uses the fact (from BorePs theorem) that the regulator map

is injective. From this we see that one need only check the commutativity of the analogue of (4.11.1) in Deligne cohomology. To do this Beilinson explicitly calculates rx>oElM, by integrating along the fibres of the projection Xln(C) -> Mn(C)—see [Be2] §3.3 for details. (The resulting formula we gave as (4.9.1) above.) An alternative proof [SS2] is by direct computation of R e s ^ o ^ ^ using the Neron model of Xln. In this approach, the formula (4.9.1) is obtained as a consequence of the commutativity of (4.11.1). In fact, the analogue of ^ in Deligne cohomology is an isomorphism

(by consideration of the Hodge numbers) whose inverse is given by real analytic Eisenstein series.

5. L-functions of modular forms.

195

5. L-functions of modular forms. In this section we sketch how, mildly generalising the results of Beilinson, the Eisenstein symbol can be used to exhibit a relation between special values of L-functions of cusp forms of weight > 2 and higher regulators. 5.1 Let k > 0 be an integer, and / a classical cusp form of weight k + 2, which we assume to be a newform on some TQ(N) with character \f- F°r simplicity we shall assume that the field generated by the Fourier coefficients of / is Q. As is well known [Del], attached to / is a strictly compatible system of ^-adic representations {Vi?(/)}, whose associated L-function is the Hecke jL-series L(f,s). Moreover Vi{f) is a subspace of the parabolic cohomology (5.1.1)

Hl{Mn®q,*$ymkRl
for suitable n. (Recall that <j) denotes the inclusion Mn <—> Mn.) In Lemma 7 of [Del] a canonical resolution of singularities of X* is constructed, which we denote by X*, and it is shown that Vj?(/) is a constituent of if^t+1(X^(8) 5.2 Theorem [Scl]. There exists a projector Hf in the ring of algebraic correspondences on X* modulo homological equivalence such that for every prime £

Remarks.

(1) In fact n^ annihilates Hl for i^k + 1.

(2) The pair V(f) = [X*,H/] is a motive in the sense of Grothendieck (cf. 3.3 above); by the above remark and the theorem, the ^-adic representations of V(f) are {Vi(f)}. The Betti realisation of V(/) is given by the singular parabolic cohomology groups (Eichler-Shimura). It has Hodge type (fc + l,0) + (0,fc+l) and the (fc + 1,0) part is spanned by the differential form on X* ojf = 27rif(r)drAdz1A' (3) A construction of V(f) as a motive defined by absolute Hodge cycles was given by Jannsen ([Ja2], §1; see also [Scha], V.I.I). (4) For the purposes of testing Beilinson's conjectures, one would like V(f) to be a Chow motive (3.3). In general this seems rather difficult to establish. However, one can consider in place of Vt(f) the whole parabolic

196

Deninger & Scholl - The Beilinson conjectures

cohomology group (5.1.1) of level n. There is then a Chow motive with this group for its ^-adic realisation. (See step (i) below.) (5) One may also consider, for p prime to the level of / , the p-adic realisation V^(/), which is a crystalline representation of Ga^Qp/Q^) [Fa,FM]. A consequence of 5.2 is that the characteristic polynomial of Frobenius on A the associated filtered module is the Hecke polynomial t2 — «p (5.3) Sketch of the construction. For k = 0 the theorem amounts to the decomposition of the Jacobian of Mn under the action of the Hecke algebra, and is classical. In this case the problem (4) does not arise. In the case k > 0 there are two steps: (i) The use of automorphisms: acting on X^ one has the following groups of automorphisms and characters: —(Z/n) 2fe , the translations by sections of finite order; —ji2, inversions in the components of the fibres; —Sjt, the symmetric group. These generate a group V of automorphisms of X*, and this extends to a group of automorphisms of X*. There is a unique character of F which restricts to the trivial character on (Z/n) , the product character on /i*, and the_sign character of §*. This defines a projector II in the group algebra Q[AutX*]. By explicit calculation of the cohomology of the boundary of X* one shows that II cuts out the parabolic cohomology (5.1.1). (ii) To pass to the individual T^(/)'s one projects using an idempotent in the Hecke algebra (which is semisimple as an algebra of correspondences modulo homological equivalence). (5.4) The integers s — 1,..., k + 1 are critical for L(/,s). At these points the Beilinson conjectures reduce to the conjunction of Deligne's conjecture (already proved in [De2]) and the vanishing of n [iJ^ 2 (X£,Q(r)) z ] for 1 < r < k +1, r ^ fc/2 (for which there is at present no evidence). At s = — I < 0 the L-function has a simple zero, and the conjectures predict a relation between Ll(f,-l) and a regulator coming from #j^ 2 (X*,Q(fc + / + 2)). The target for this regulator is the Deligne cohomology group

and its n/-component is the space (if#(F(/))<8)QR(fc + Z + l)) , which is one-dimensional.

5. L-functions of modular forms.

197

(5.5) Theorem. There is a subspace Vn C #^J 2 (X*,Q(fc + 1 + 2)) such that

(5.6) Remarks. (1) For k = 0 (the case of modular curves) this was proved by Beilinson [Bel,Be2,SSl]. The main ideas for the general case can already be found there. The case k = 1, / = 0 was also considered by Ramakrishnan (unpublished). Full details for the general case will appear in [Sc3]. (2) Recall that for the correct formalism of Beilinson's conjecture it is necessary to consider "motivic cohomology over Z" (cf. 1.7 above). Although in general we cannot prove that Vn C i?j^2(X*,Q(fc + Z + 2))z, we have the following: (i) Standard conjectures on the K-theory of varieties over finite fields would imply that Hkj+\1 *,Q(jfc + / + 2)) I = Hhj+2(1 *,Q(fc + / + 2)) except in the case k = I = 0. (ii) For curves these conjectures are known [Hal]. Thus for k = 0 the only obstruction to integrality occurs when / = 0; in this case it is known (see [SSI], §7) that Vn C #£<(M n ,Q(2)) z . (iii) For k > 0 one can at least show that Vn contains enough elements which are integral away from primes dividing n, using a modification of a trick of Soule [Sol]. (5.7) Construction ofVn.

Consider the diagram

I= SpecQ(Cn)

where p, q are the projections onto the first k and last / factors of the fibre product, respectively. We define two subspaces

(where the projector II is as in (5.3.1) above) as follows:

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Deninger & Scholl - The Beilinson conjectures

(Note that the Eisenstein symbol and Borel's theorem give a plentiful supply of elements of lin and Vn.) Let a be the restriction (which is in fact an inclusion) and write We then define (cf. 4.10 above) •Pn= n\n'

\Jpln,n..(Qn>).

(5.8) Calculation of the regulator. At this point we should observe that the assumption that the Fourier coefficients of / are rational simplifies the calculation somewhat; in particular, we need not distinguish between / and its complex conjugate. There is a nondegenerate pairing (Poincare duality)

<, >: HB(V(f)) x HB(V(f)) - Q(-* -1) and one has to prove that

(5.8.1)

< rvCPn),uf >= L'(f, -l).c+(V(f)(k +1 +1)) • Q.

Here c + is Deligne's period [De2]. To calculate the left hand side we pull back to X*,+/ for suitable n', and use the description (2.5) of the cup-product in Deligne cohomology. One obtains an integral of the form

(5 8 2)

--

(djs*

In this expression S^ is the image of an Eisenstein symbol in Deligne cohomology, and E\+i is a (variable) weight k -f- 2 holomorphic Eisenstein series. This is a standard Rankin-Selberg integral and can be calculated explicitly. The Eisenstein series Ei+2 is a linear combination of Eisenstein series Ex, for various Dirichlet characters x with x(~l) = (—1)Z> a n d the integral becomes a linear combination of terms which, up to a finite number of Euler factors, are of the form At this stage one applies Shimura's algebraicity results on the twisted Lfunctions i ( / ® x , k + 1) (which are critical values) and the functional equation for L(f,s). In this way it can be shown that the left hand side of (5.8.1) is contained in the right hand side. The final step is to prove the equality— that is, to find suitable Eisenstein symbols for which the integral (5.8.2) is non-zero. For this one has to analyse the bad Euler factors carefully, and it is essential to work adelically. See [Be2], §4 or [SSI], §§2,4,5,6 for further details.

6. L-functions of algebraic Hecke characters

199

6. ^-functions of algebraic Hecke characters In this section we describe a construction involving the Eisenstein symbol which will give us elements in the motivic cohomology of motives attached to Hecke characters of imaginary quadratic number fields. The regulators of these elements have the expected relation to special values of Hecke L-series. As a corollary one obtains results on Beilinson's conjectures for CM elliptic curves of Shimura type and for Dirichlet characters of Q and of imaginary number field. Full details are contained in [Denl,2]. (6.1) Consider an algebraic Hecke character e: IK/K* —> C* of weight w of an imaginary quadratic field K. We wish to understand the special values L(e, n) for n > -j + 1 of the corresponding L-series in terms of Beilinson's conjectures. In fact one can treat the L-values of all conjugates of e simultaneously. Thus it is better to take a slightly different point of view and to look at the associated CM character

Here T/K is a number field and there exist integers a, b with a + b = w such that (x) = xaxb

for all x in K* CI K .

From <j> we obtain an L-series taking values in T®C = C Hom ( TjC ) by setting L(,s) = {L((f> via the embedding a of T. For critical n Beilinson's conjectures reduce to the Deligne conjecture, which for i(<^,n) is proved in [GS1,2] and in much greater generality in

[Bla].

For non-critical n we first have to find a Chow motive (3.3) with coefficients in T whose L-series equals L(,s). Note that if <j> is a Dirichlet character x oi K—i.e. if a = b = 0—we can take the motive Mx constructed in (3.3). For the general case one needs the theory of CM elliptic curves of Shimura type [GS1]. These are elliptic curves E with CM by OK which are defined over an abelian extension F of K such that the extension F(Etoxs)/K is abelian as well. One checks that eo = [22x0], e2 = [0xi£] and e\ = 1—eo—e2 are pairwise orthogonal projectors of the motive QJ5 in -M^(Q). The motive hi(E) — ei(QJS) in A<J(Q), viewed as a motive in AlJ-(Q), will be called M. (6.2) Proposition. For w>l and possibly alter enlarging the Held T there exists an elliptic curve as above such that M®w contains a direct factor M with End(M) = T and

200

Deninger & Scholl - The Beilinson conjectures In the last equation M^ is viewed as a motive in

M~K{T)

via [De2] 2.1.

Note that it is sufficient to treat Hecke characters of positive weight since multiplication of <j> by the norm just results in a shift by one of s in the L-series. For the same reason we may assume that a, b> 0. (6.3) Theorem. Assume that w > 1, n > -j + 1 and in addition that n is non-critical for M, i.e. n > Max(a, b). Then the L-series L(<j>,s) has a first order zero for s = —I := w + 1 — n and there is an element £ in n

))

such

Lf(l-l)rj

modT*

in the free rank one T®R-module

Here 77 is a T-generator of # Remarks. (1) In general the conjectures involve the motivic cohomology of an integral model. However since E —• Specif has potential good reduction one can show that

for n / f + 1 , using [Sol] 3.1.3, Corollary 2. (2) In [Denl] a refined version of (6.3) is proved for w = 1 where one considers motivic cohomology with almost integral coefficients. This was possible by a careful reexamination of the entire (slightly modified) construction of Beilinson's Eisenstein symbol specialised to a constant elliptic curve. (6.4) Construction of f and calculation of rp(£). For simplicity we shall assume that / > 0. For the finitely many negative / in the theorem a slightly different construction is required. Set k = w + 21 > 0 and fix some integer N > 1. For a choice of a square root of the discriminant dx of K consider the map

and let pr : El+W = El x F Ew -> Ew be the projection. Choose a Galois extension F1 of F such that the iV-torsion points of E' = E ®F F1 are

6. L-functions of algebraic Hecke characters rational over F'. The choice of a level N structure a: (I/N)2 determines a commutative diagram

201

^ E'N on E'

E'

I

SpecF'

I

MN

Using (1.3)(6) we find a canonical map EM independent of a which makes the following diagram commute:

Now consider the following composition KM of maps:

pr

*

For / < 0 the map KM is defined differently [Den2] §2. The required element £ is obtained in the form £ = KM£M{P) f°r suitable N and divisor /3 in Q[EN)°. TO prove that it has the right properties we must first of all calculate explicitly the analogous maps Sv and /Cp in Deligne cohomology. For /Cx> this is easy. For Sv we can use formula (4.9.1) for E\, specialised

202 to to of of

Deninger & Scholl - The Beilinson conjectures

the value of r corresponding to our elliptic curve E. Note that in order derive (4.9.1) Beilinson makes essential use of the compactification MN MN—see [Be2], §3.3. In [Denl] a different method for the calculation £j) is described which only uses analysis on E itself.

Looking at (4.9.1) we see that £v(fi) is a certain linear combination of Eisenstein-Kronecker series. Hence it comes as no surprise that for suitable /? the element 1Cv£v{P) is related to Z/(<^, —/) as specified in the theorem. (6.5) Corollary. (1) Let E/F be a CM elliptic curve of Shimura type as above. Then for n > 2 the weak Beilinson conjecture (3.2) holds for L(H\E),n). (2) Assume that F is Galois over Q and let F~*~ be a real subfield ofF, i.e. F + = Fa f)U for some embedding a of F into C. Then for any elliptic curve E~*~/F+ whose base change to F is of Shimura type the analogue of (1) holds. Remark. (2) generalises the case of CM elliptic curves over Q at n — 2 treated by Bloch [Bll] and Beilinson [Bel]; see also [DW].

(6.6) Dirichlet characters.

Given a character

we can attach to it the motive Mx of (3.3) and the twist M^(l) of a motive Mif, as in (6.2) for = XNK®M/R- Possibly after extension of scalars both motives have the same L-function and should in fact be equal. The Beilinson conjectures for M^(l) follow from the theorem. For Mx one can prove them directly using the map

where E is a CM elliptic curve of Shimura type over an abelian extension of K trivialising % and KM is defined by composition:

KM

pr*

By a very simple argument [Den2] (3.6) one can use the theory over K to prove the Beilinson conjectures for Dirichlet characters of Q as well. The complete results are these:

Appendix: motivic cohomology and extensions

203

(6.7) Theorem. For k = Q or K consider a character

and let L(x,s) = {L(<7Xis)) be its T®C-vaJued L-series. For I > 0 the map

is an isomorphism of free T ®R-modules. For k = Q and x(c) k = K their rank equals one. In this case we have CMX=L'(X,-1)

=

(~~1)'

or

modT*

where CMX € (T®R)*/T* denotes the regulator. Remarks. (1) That rp ® R is an isomorphism follows from the work of Borel [Bol] and Beilinson [Bel], app. to §2; see also [Rap]. (2) For i = Q a different proof of the theorem is given in [Bel] §7, see also [N,E]. Appendix: motivic cohomology and extensions

In this appendix, we outline without proof the construction of extensions of motives attached to elements of motivic cohomology. Details should appear in a future paper by the second author. The underlying idea is certainly not new, aand is implicit in the constructions of [B12]. To motivate it, we consider first the case of ordinary Chow theory (i.e., if^(X,Q(^))). The corresponding extensions appear first in a paper of Deligne ([De2], 4.3). So let X be smooth and projective over Q, and let y be a cycle of codimension q, homologous to zero. Write Y for the support of y. Then there is an exact sequence of mixed motives (in the sense of [Ja2], Chap.l): 0 -> h2q~\X)

-> h2q-\X

- Y) -> h2Yq(X) ^

h2q(X)

The cycle class gives a map cl(y) : Q(—q) —> hy(X), and by hypothesis 7oc/(y) = 0. Hence by pullback we obtain an extension

204

Deninger & Scholl - The Beilinson conjectures

Theorem [Ja2]. The class of the extension Ey depends only on the rational equivalence class ofy. The following diagram commutes:

I £-adic J^-adi

cycle

realisation

H-S

Here H—S denotes the edge homomorphism in the Hochschild-Serre spectral sequence (in continuous etale cohomology [Jal])

E? = Ha (Q/Q, Hb(X, qe)(q)) =» Ha+b(X, qe(q)). There is a similar statement for Deligne cohomology (cf. 2.9 above). We now imitate this construction for higher cycles. In an attempt to make the notation tidier we write A ^ for A n xX, and dA^ for the union of the codimension one faces of A^. By the normalisation theorem, any element of #^~ n (X,Q(g)) = Cff?(X,n)(g)Q may be represented by a cycle y G zq(X,n) with d*(y) = 0 for 0 < i < n. Choosing such a representative y, write Y = supp(y), dY = YndA%, U = A%-Y, and dU = UDdA%. We consider the motive /i2g~1(?7,9Z7) which fits into a long exact sequence (A.I)

h2«-2(U) -+ h2q~2(dU) -+ h2q-\U,dU)

- » h 2 q - \ U ) -+

h2q-\dU).

By purity we have h2q~2(U) = h2q~2(A^) = h2q~2(X). It is also easy to deduce that h?p~2(dU) = h2p~2(dAx) by considering the spectral sequence expressing the cohomology of dA^ in terms of that of its faces. Lemma. There is a decomposition

(A.2)

h\dA$) Z h\X) ® /i^ n+1 (X).

(In fact this decomposition is given by the 1- and sgn-eigenspaces for the action of the symmetric group of degree n.) Thus the sequence (A.I) becomes 0 _> h2q~n~\X)

-+ h2*-\U,dU)

-> h2q-\U)

-+

h2q-\dU).

Appendix: motivic cohomology and extensions

205

This fits into a bigger diagram: 0

0

i

i

tfv-ifydU)

>

I

ker/?

i

Here we have written

The cycle class of y gives a map Q(—q) —> ker^S. From the snake lemma and (A.2) we have a long exact sequence (A.3)

0 -> fc2*-"-1^) - • fc^-^J/,dU)

Since n > 0 the composite map Q(—#) -^ ker/3 -> /i 2g ~ n (X) is zero (by weights), hence by pullback we obtain an extension

Theorem. The class of the extension Ey depends only on the class ofy in )- The following diagram commutes: F^- n (X,Q( g ))

^

cycle

#2'-"(;r,Q*(
-£-adic realisation Hochschild-Serre

.

~ '"

206

Deninger & Scholl - The Beilinson conjectures

The analogous statement 2.9 for Deligne cohomology also holds. Remark. In this construction, we have in the interest of clarity freely used "relative" and "local" motives h\(-)y hl(-^). Lest this trouble the reader, we point out that the extension Ey really belongs to the category M M Q of mixed motives generated by hl(V) for quasi-projective varieties V/Q (see [Ja2], Appendix C2). Indeed, the "relative" motive h^^iU.dU) can be constructed as part of the motive of a suitable singular variety (a mapping cylinder) and the motive ker/? is simply the Tate twist of an Artin motive. Therefore the objects in the exact sequence (A.3) are all motives in MMQ. To construct the arrows we need only work in the various realisations, and there the relative and local cohomology groups are available.

References [At] M. F. Atiyah; /^-theory. Benjamin, 1969 [Bel] A. A. Beilinson; Higher regulators and values of L-functions. J. Soviet Math. 30 (1985), 2036-2070 [Be2] A. A. Beilinson; Higher regulators of modular curves. Applications of algebraic if-theory to algebraic geometry and number theory (Contemporary Mathematics 5 (1986)), 1-34 [Be3] A. A. Beilinson; Notes on absolute Hodge cohomology. Applications of algebraic IlT-theory to algebraic geometry and number theory (Contemporary Mathematics 55 (1986)), 35-68 [Bla] D. Blasius; On the critical values of Hecke ^-series. Ann. Math. 124 (1986), 23-63 [Bll] S. Bloch; Lectures on algebraic cycles. Duke Univ. Math, series 4 (1980) [B12] S. Bloch; Algebraic cycles and higher If-theory. Advances in Math. 61 (1986) 267-304 [B13] S. Bloch; Algebraic cycles and the Beilinson conjectures. Contemporary Mathematics 58 (1), 1986, 65-79 [B14] S. Bloch; Algebraic cycles and higher Jif-theory: correction. Preprint, University of Chicago (1989) [B1K] S. Bloch, K. Kato; ^-functions and Tamagawa numbers of motives. Preprint, 1989

References

207

[Bol] A. Borel; Stable real cohomology of arithmetic groups. Ann. Sci. ENS 7 (1974) 235-272 [Bo2] A. Borel; Cohomologie de SLn et valeurs de fonctions zeta. Ann. Scuola Normale Superiore 7 (1974) 613-616 [Ca] J. Carlson; Extensions of mixed Hodge structures. Journees de geometrie algebrique d'Angers 1979, 107-127 (Sijthoff & Noordhoff, 1980) [Del] P. Deligne; Formes modulaires et representations /-adiques. Sem. Bourbaki, expose 355. Lect. notes in mathematics 179, 139-172 (Springer, 1969) [De2] P. Deligne; Valeurs de fonctions L et periodes d'integrales. Proc. Symp. Pure Math. AMS 33 (1979), 313-346 [De3] P. Deligne; Le groupe fondamentale de la droite projective moins trois points. Galois groups over Q (ed. Y. Ihara, K. Ribet, J.-P. Serre). MSRI publications 16, 1989 [Denl] C. Deninger; Higher regulators and Hecke L-series of imaginary quadratic fields I. Inventiones math. 96 (1989) 1-69 [Den2] C. Deninger; Higher regulators and Hecke L-series of imaginary quadratic fields II. Ann. Math, (to appear) [DW] C. Deninger, K. Wingberg; On the Beilinson conjectures for elliptic curves with complex multiplication. In [RSS]. [E] H. Esnault; On the Loday symbol in the Deligne-Beilinson cohomology. K-theory 3 (1989), 1-28 [Fa] G. Faltings; p-adic representations and crystalline cohomology. To appear [FM] J.-M. Fontaine, W. Messing; p-adic periods and p-adic etale cohomology. Current trends in arithmetical algebraic geometry, ed. K. Ribet (Contemporary Mathematics 67 (1987)), 170-207 [GS1] C. Goldstein, N. Schappacher; Series d'Eisenstein et fonctions L de courbes elliptiques a multiplication complexe. J. Reine angew. Math. 327 (1981), 184-218 [GS2] C. Goldstein, N. Schappacher; Conjecture de Deligne et F-hypothese de Lichtenbaum sur les corps quadratiques imaginaires. C.R.A.S. 296, Ser. I (1983), 615-618 [Hal] G. Harder; Die Kohomologie 5-arithmetischer Gruppen iiber Funktionenkorpern. Inventiones math. 42 (1977), 135-175

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Deninger & Scholl - The Beilinson conjectures

[Ha2] G. Harder; Arithmetische Eigenschaften von Eisensteinklassen, die modulare Konstruktion von gemoschten Motiven und von Erweiterungen endlicher Galoismoduln. Preprint, 1989 [Jal] U. Jannsen; Continuous etale cohomology. Math. Annalen 280 (1988) 207-245 [Ja2] U. Jannsen; Mixed motives and algebraic A^-theory. Lecture notes in math. 1400 (1990) [Ja3] U. Jannsen; Deligne homology, Hodge-X>-conjecture and motives. In [RSS] [Kl] S. Kleiman; Motives. Algebraic Geometry, Oslo 1970 (ed. F. Oort). Walters-Noordhoff, Groningen 1972, 53-82 [Ma] Yu. I. Manin; Correspondences, motives and monoidal correspondences. Math. USSR Sbornik 6 (1968) 439-470 [N] J. Neukirch; The Beilinson conjecture for algebraic number fields. In [RSS] [Ral] D. Ramakrishnan; Higner regulators of quaternionic Shimura curves and values of X-functions. Applications of algebraic if-theory to algebraic geometry and number theory (Contemporary Mathematics 5 (1986)), 377-387 [Ra2] D. Ramakrishnan; Regulators, algebraic cycles, and values of Lfunctions. Algebraic if-theory and algebraic number theory (Contemporary Mathematics 83 (1989)), 183-310 [Rap] M. Rapoport; Comparison of the regulators of Beilinson and of Borel. In [RSS] [RSS] M. Rapoport, N. Schappacher, P. Schneider (ed.); Beilinson's conjectures on special values of L-functions. (Academic Press, 1988) [Scha] N. Schappacher; Periods of Hecke characters. Lecture notes in mathematics 1301 (1988) [SSI] N. Schappacher, A. J. Scholl; Beilinson's theorem on modular curves. In [RSS] [SS2] N. Schappacher, A. J. Scholl; The boundary of the Eisenstein symbol. Preprint, 1990 [Sch] P. Schneider; Introduction to the Beilinson conjectures. In [RSS] [Scl] A. J. Scholl; Motives for modular forms. Inventiones math. 100 (1990), 419-430 [Sc2] A. J. Scholl; Remarks on special values of L-iunctions. This volume.

References [Sc3]

209

A. J. Scholl; Higher regulators and special values of L-functions of modular forms. In preparation [Se] J.-P. Serre; Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures). Seminaire Delange-Pisot-Poitou 1969/ 70, expose 19 [Sol] C. Soule; Groupes de Chow et if-theorie de varietes sur un corps fini. Math. Annalen 268 (1984), 237-253 [So2] C. Soule; Regulateurs. Seminaire Borbaki, expose 644. Asterisque 133/134 (1986), 237-253

Iwasawa theory for motives RALPH GREENBERG

Let V = {Vt} be a compatible system of Gaelic representations of GQ = Gal (Q/Q). We will think of Vt as the ^-adic homology of a motive M defined over Q. The corresponding L-function is defined by an Euler product

where Ei(T) =

det(I-Frob1!T\(Vl)l.)

for any £ ^ q. Here Frob^- is an arithmetic Frobenius for some place 7} of Q lying over q, I-q denotes the corresponding inertia subgroup of G?Q, and {Viji- is the maximal quotient space of Vt on which I-q acts trivially. We are interested in the values of Ly(s) where s is an integer, but, by taking a Tate twist of M if necessary, we can consider just the value Lv(l). This turns out to be a convenient normalization. The conjectural functional equation for Lv(s) (stated more precisely in [4] and [23]) can be put in the form AsvTv(s)Lv(s)

= wvA*?Tv.(2 - s)Lv.(2 - s)

where V* = {V^} is the compatible system of ^-adic representations defined by VI = HoniQ^V^Q^l)). Ay and Av* are certain positive constants. Tv(s) and IV* (s) are certain products of F-functions. We will assume throughout this paper that Lv(\) is a critical value in the sense of Deligne. In the above notation, this means that neither Tv(s) nor Fv(s) has a pole at s = 1. Then Lv*(l) is also a critical value. To describe 'Iwasawa Theory' for the motive M, we choose a prime p such that Vp is 'ordinary' in a sense we will explain later. This assumption on p seems to be essential in both the analytic side of the theory (the existence of p-adic L-iunctions with certain properties) as well as the algebraic side (the definition of a natural 'Selmer group'). Then we can formulate a 'Main Conjecture' which gives a link between the zeros of the p-adic Z-function and the structure of the

212

Greenberg - Iwasawa theory for motives

Selmer group. The most natural and general context at present where one can give such a conjecture seems to be when Vp is in an analytic family of ordinary p-adic representations as considered for example in Hida [7] and in Mazur and Wiles [16]. In this paper, we will consider just the simplest case: an analytic family of twists {Vp ® (p}. The twists that we allow can be described as follows. Consider the canonical isomorphism ^:Gal(Q0v°)/Q)^ZpX, where /ipOO denotes the p-power roots of unity. Now Zp = //p_x x (1 +pZp) for odd p. (Forp = 2, Z£ = /z 2 x(l+4Z 2 ).) We let K : Gal(Q(/xpoo)/Q) -> l+pZ p (or 1+4Z2 if p ~ 2) denote the projection. We can write Af = WK where UJ can be thought of as a Zp-valued Dirichlet character of conductor p (or 4 if p = 2). The homomorphism K induces an isomorphism K : F —> 1 + pZp (or 1 + 4Z2) where F = Gal (Qoo/Q) and Qoo is the so-called cyclotomic Zp-extension of Q. We have Q^ = Un>iQn where Qn is a cyclic extension of Q of degree pn. Let Cp denote the completion of (jp with its usual absolute value. We will consider twists by the elements of Homcont(F,Cp<). Each such element

, V). It is defined for all but finitely many

1 The analytic theory

213

It will be necessary to fix embeddings crp : Q —> Cp and a^ : Q —> C. A character tp of F of finite order can then be regarded as either Cp-valued or C-valued. For such {@v) for all but finitely many tp G Hom cont (F,C*), where 9V is an element of the field of fractions of the Iwasawa algebra A. (B) If (p has finite order, then Lp(ipj V) = ap o c r ^ 0^,1^(1, (£>)). (C) Lp(tp,V) = UyLp( Gal(Q n /Q). One can also think of A as a formal power series ring: A = Zp[[70 — 1]]. If (p G Homcont(F,Cp<), then ip can be uniquely extended to a continuous Zp-algebra homomorphism (p : A —>• Cp. If A G A, A = /(70 — 1) say, then
214

Greenberg - Iwasawa theory for motives

chosen interpolation factors cv are specified, properties (A) and (B) determine uniquely the function Lp(). The value at s = 1 of the corresponding complex i-function is Lv^{l^(jj"n) = Lv(l — n,u;~n). Some of these values for n ^ 0 may also turn out to be critical. For such n, we should have Xp(/cn, V) = Vp0^ (cnLv{l — n,u;~n)) for a suitable interpolation factor cn. This is also discussed more precisely in Coates article [1]. The p-adic functional equation in (C) should be a direct consequence of the complex functional equation which relates Lv(l,(p) to Lv*(l, tp'1) for all tp of finite order, together with properties (A) and (B). One can give the following alternative statement. The automorphism 7 —> 7" 1 determines an involution of the Zp-algebra A and itsfieldof fractions which we denote by 0 —> 0\ If (p £ Hom cont (r,C p x ), then
2 The algebraic theory

215

(We mention two examples: /zpoo = Q p (l)/Z p (l). Ep<x> = VP(E)/TP(E) for an elliptic curve E/Q.) We prefer to denote FlVp by F+Vp. Let F+A denote its image in A. As before, T = Gal(Qoo/Q). The 'Selmer group' 5i4(Qoo) is the subgroup of T/^ Gal (Q/Qoo), A) = iJ^Qoo, A) defined by certain local triviality conditions. For each place A of Qoo, let 7A denote the inertia group for some place of <J lying over A. (The choice won't matter. For A = TT, the unique prime of Q^ over p, the choice determines the filtration of Vp and hence F+A.) Here is the definition of our Selmer group. o, A) -+ H1 (/,, A/F+ A) x JJ ^ ( A , A)). Let E be a finite set of primes containing p and oo and all primes ramified in the representation of GQ on Vp. Let QE be the maximal extension of Q unramified outside S. Hence Vp is a representation space for Gal(Q E /Q). Also Qoo C QE since only the prime p is ramified in Qoo/Q- Let EQO be the set of primes of Q^ lying over those in E. If A ^ Eoo, then Ix acts trivially on A and so if1 (7A, A) = Hom(7A,A). From this one easily derives the following alternative description: SA(
J[

H^I

Now F acts naturally on 5A(QOO) and so we can regard ^(Qoo) as well as its Pontryagin dual X = ^(Qoo)" = Hom^S^Qoo^Qp/Zp) as A-modules. X is a compact A-module and can be shown to be finitely generated. (See [5].) The following conjecture is crucial. Conjecture 1 5A(Qoo)^ is a torsion A-module. We would then say that 5A(Qoo) is A-cotorsion. If X is any finitely generated torsion A-module, then it is well-known that X is pseudo-isomorphic to a Amodule of the form ©J=iA/(Af-), where A,- £ A, A,- ^ 0. (This means that there exists a A-homomorphism with finite kernel and cokernel.) The Qp-vector space X ®zp Qp would then be finite dimensional. The characteristic ideal of the A-module X is defined as (\x) = IlLi(^f)- The generator A* is only defined up to a factor in A x . One possible choice would be A^- = p** Yleilo ~e)> where e runs over the eigenvalues (counted with their multiplicities) of 70 acting on X ®Zp Qp. (The value of // is related to the structure of the Zptorsion subgroup of X as a A-module.) Notice that if tp G Hom cont (r,Cp), then
216

Greenberg - Iwasawa theory for motives

discussed fully in [5], Sections 3, 4, and 5. The conjecture is equivalent to the assertion that, as a group, SA(QOO) — (%/^-PY X C f° r some e, where C has bounded exponent. We include some examples at the end of this paper where we can actually verify this. We will also need the A-modules tf^Q^A) = AG*<*> and (A*)Gfto° which we write more briefly as A(QOO) and A*(QOO). Here we define A* = V;/T; where Vp* = Horn (VPJ Qp(l)) as before and T; = Horn (Tpy Zp(l)) C V*. Both A(Qoo) and A*(Qoo) are obviously A-cotorsion modules. 3 THE MAIN CONJECTURE We will now describe the conjectural relationship between the analytic and algebraic theories. Define 0jfg = A^^ 1 , where A^ = ^5A(QOO)" a n d SA = (^(Qoo;fK^*rQ d* ^ course, 0jfs is well-defined only up to a factor in A x . We will now denote the 6V occurring in the definition of the p-adic L-function by 0™**. As we mentioned before, it is only well-defined up to a factor in Q x . Main conjecture

0™* = 0Ag • /?, where fi G Qx • A x .

If tp e Homcont (r,C x ) and /? G Qx • A x , then cp(f3) ^ 0. Thus the above conjecture gives an algebraic interpretation of the zeros and poles of the p-adic i-function Lp( 1, i.e., when Vp contains a G^-subrepresentation factoring through F. Thus Lp((p, V) should have no poles when Vp and V* have no such subrepresentations, which of course is usually the case. The truth of the conjecture as formulated above is independent of the choice of lattice Tp. A change in Tp will not affect the ideal (6A) and will affect the characteristic ideal (^5A(QOO)^ only up to multiplication by a power of p (which is computed by Perrin-Riou in [17]). The factor /? compensates for this ambiguity. However, the choice of lattice Tp in Vp should somehow determine Z-lattices in the Betti and DeRham cohomology of M well-enough so that one can specify fly up to a rational factor with numerator and denominator prime to p. It seems that this should be possible to do under very general assumptions by making use of a recent theorem of Fontaine and Lafaille. One could then hopefully make a more precise definition of 0£nal (which one might denote by O™*1) so that the factor /3 in the above conjecture can be taken in A x . Since then (#jjnal) = Lp((p, V).

3 The main conjecture

217

The p-adic functional equation implies that (0£nal )* and OyTx should differ just by a factor in A x . On the algebraic side, obviously (SAy = 8A*. The compatibility of the main conjecture with the functional equation follows from the theorem below, which is proved in [5] by using Tate's local and global duality theorems. Theorem 1 Assume that SA(QOO) is A-cotorsion. Then so is «S'A*(QOO) and we have (XSA.^ = We want to make an observation about the special case where V* = Vp as GQ- represent at ion spaces. Then both A and A* are of the form V^/(a lattice), although they are not necessarily isomorphic. (One might say they are isogenous.) We will assume that Conjecture 1 is valid for A. Then, as we mentioned above, XA/XA* — Pe>u>i where e G Z, u G A x . By Theorem 1, XA*/XA G A*. Hence XA/XlA = peu', where u' G A x . But since i is an automorphism of A, it is clear that e = 0. Now let A = {id,t}. Then i —> XA/XA defines an element aA in if 1 (A,A x ). Assume that p is odd. We then have Ax = Zx x (1 + (70 - 1)A) = //p_iX (a pro-p group). A acts trivially on the constants Zx and hence on fip-i. Thus Jff1(A, A x ) = if^Aj/Zp.i) = Hom(A,/i p _i) = Horn (A, ±1). Let crA be the image of o~A in the last group and let eA = CFA(L). Since aA is equivalent to the cocycle cr'AJ one can change XA by a factor in Ax obtaining another generator (which we will still call A^) of the characteristic ideal of ^(Qoo)^ with the property that A^ = eAXA where eA = ± 1 . Since p is odd, 70 = 7^ for some 72 G F. Let Ao = 71 — 7J. Then Xl0 = —Ao and (70 — 1) = (Ao) as ideals in A. One easily sees that eA = (—l)m° where m0 is the highest power of the irreducible element 70 — 1 (or Ao) which divides A^. If
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has order 4. The cocycle crA reflects the parities of the multiplicities of both <£>s' as zeros of

(f(XA). On the analytic side one can also define a cocycle cr^al which sends t to V For For odd odd p, one obtains just as 0?*/(Of?*)', still assuming that Vp == V*. nal above a sign e£ . By the main conjecture, one should have e^ = SA- The sign in the complex functional equation should often be the same as e™*1, but not always. The existence of trivial zeros of p-adic L-functions has a bearing on this question. (See [9], [14] and also Example 3 below.) We want to mention one more theorem which shows that the main conjecture is compatible with exact sequences. More precisely, assume that we have an exact sequence of GQ-representation spaces over Qp:

o-+v;^v,-+ v;1 -»o. We assume that Vp and hence Vp and Vp are ordinary p-adic representations and that they come from compatible systems of ^-adic representations V, V , and V", say. If Tp~is a Gq-invariant lattice in Vpi then one can easily choose such lattices Tp C Vp\ Tp C Vp so that we get an exact sequence 0 -+ A -> A -f A" -+ 0 where of course A = Vp/Tp, A' = Vpf/T^ and A" = Vp"lT'p'. Assume that Lv(l) is a critical value and that 5A(Qoo) is A-cotorsion. Then the same statement will be true for V and V"'. In defining the p-adic L-functions, we take fty = flv'^lv"' In [6], we intend to give a proof of the following result. Theorem 2 The main conjecture for any two of A, A', A" implies the main conjecture for the third. 4 EXAMPLES Example 1 Our first example is a reformulation of the classical main conjecture of Iwasawa. Let n be a positive, even integer. We consider first the compatible system V — {tyi(n)} of 1-dimensional ^-adic representations. (Q£(n) is the £-adic homology of the motive denoted by Q(—n) in [4].) We have Lv(s) = Uq(l - qnq~sYl = ((s - n). Hence Lv(l) = ((1 - n), which is a critical value. Let

4 Examples

219

Now Vp = %(n) is ordinary for every prime p. For simplicity, we will assume that p is odd. We can take S = {p, 00}. Since n > 0, we have a n d so F V * P = VP A/F+A = 0, where A = %(n)/lp(n) = Qp/Zp(n). The second description of the Selmer group becomes quite simple in this case: ). Let K*> = Q(/Vo), A = Gal(/Co/Qoo), and where M^ denotes the maximal abelian pro-p extension of i^oo contained in QE. Since (|A|,p) = 1 and since Grj^ acts trivially on A, we get isomorphisms

where X^ n denotes the subgroup (a direct summand) of X^ on which A acts by u>n. Thus ^(Qoo) is essentially the Pontryagin dual of the Galois group X£ (but with a twisting by /c"n for the action of F). Iwasawa proved that X^ (the subgroup of Xoo on which complex conjugation in A acts by +1) is a torsion A-module. Since n is even, it follows that *S'^(QOO) is in fact A-cotorsion. If n ^ 0 (mod p — 1), then both ^4(Qoo) and A*(Qoo) are zero. The main conjecture would then relate the characteristic ideal of the A-module X<£ to a Kubota-Leopoldt p-adic L-function. If one unravels the twisting, one finds that this is one of the two forms of Iwasawa's classical main conjecture. (The factor /3 will be in Ax.) It is the version proved by Rubin using Kolyvagin's Euler system formed from cyclotomic units. (See [19].) If n = 0 (mod p — 1), it is rather simple to verify the main conjecture. We then have A(Qoo) = A and A*(Qoo) = 0. Also let M^ denote the maximal abelian pro-p extension of Qoo contained in QE. Then Gal(M^o/Qoo) = X*£ since cvn = CJ°. But one verifies easily that M^ = Q^ by using the fact that Qi is the only cyclic extension of Q of degree p unramified outside E. Hence X£ = 0. Thus one can take 0jfg = 1/(70 - /c"n(7O)). The known properties of the Kubota-Leopoldt p-adic Z-function for the character a;0 imply that 0V differs from 0jfg by a factor /3 G AX. In particular, Lp((p, V) has a pole at (p = K~n. As a concrete illustration, let (p =
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them so that the p-adic functional equation in (C) takes the simple form L

p(

v) = M ^ " 1 * ^ * ) - N o w t h i s t i m e w e n a v e A* = Q P / ^ C 1 - n) a n d F+A* = 0. Thus 5A*(Qoo) C Hom A (X o o ,Q p /Z p (l-n)) and the local triviality condition at TT requires that a cocycle (or homomorphism in this case) become trivial when restricted to the inertia group in Xoo for any prime over TT. Let l^o = Gal^oo/ifoo), where L^ is the maximal abelian unramified pro-p extension of K^. We have -MQco) = Hom A (F oo ,Q p /Z p (l - n)) Again we get essentially the Pontryagin dual of a Galois group Y£ n (slightly twisted). Iwasawa proved that Y^ is A-torsion and so Conjecture 1 is true in this case. The main conjecture is the version of Iwasawa's main conjecture which was proved by Mazur and Wiles in [15]. We would like to give a hint about how the equivalence of the two versions of Iwasawa's main conjecture can be proved. In this case, it is a theorem of Iwasawa. Let Ko = Q(//p). We consider for simplicity just an extension L of Ko such that L/Ko is unramified, L/Q is Galois, and Gal (L/Ko) = A*p~n as a A-module. (Here // p " n is a cyclic group of order p on which A acts by a;1"". Such an isomorphism gives an element of order p in SA*(Qoo)fixedby the action of F.) By class field theory one finds that there exists a nontrivial ideal class c of Ko such that & = 1 and S(c) = d*1'"™ for 6 G A. If a G c, then ap is principal. One can choose a generator a such that the subgroup H of IQ/(K^)P generated by a is contained in (IQ/(IQ)p)wl~n. n Thus H = ii\- as a A-module. Let M = K0(f/a). Then Af/Q is Galois and, by Kummer theory, Gal (M/Ko) = Horn (H, JJLP) = \inv as a A-module. Such an isomorphism gives an element of order p in 5^(Qoo), again fixed by F. Example 2 Let E be an elliptic curve defined over Q with good, ordinary reduction at p. Consider the compatible system of ^-adic representations V = {Vt(E)}, where Vt(E) = Tt(E) Q<, Tt(E) denoting the Tate module for E for the prime i. The corresponding L-function Lv(s) is the Hasse-Weil Lfunction for E over Q. Assume that E is a modular elliptic curve. Then Lv(s) is the Mellin transform of a modular form fE of weight 2. The value Lv(l) is a critical value. If fi^ is the real period, then L V ( 1 ) / ^ E is rational. Also, as we discussed earlier, VP(E) is an ordinary p-adic representation. The Euler factor for p in Lv(s) is of the form (1 — app~*)(l — ^ p p"*), where ap +/3P — ap, apf3p = p. Here ap is the pth Fourier coefficient of fE and, since E has ordinary reduction at p, we have p \ ap. Identifying the numbers a p ,/? p with their images under the embedding crp, one of them (say /3p) is a p-adic unit.

4 Examples

221

Mazur and Swinnerton-Dyer [13] have constructed a p-adic L-function having the properties (A), (B), and (C). We want to state here just the interpolation property for tp = cp0:

Note Lp(v?0, V) = 0 if and only if Lv(l) = 0 because app~l = /3" 1 ^ 1. (The complex absolute value of ap and /3p is In [12], Mazur formulated a conjecture relating the p-adic L-function to essentially the classical Selmer group for E over QQQ. NOW in this case we can take A = VP(E)/TP{E)

* Epoo a n d F+A = ElpOO, where ElpOQ = kei(Epoo

-> £ p O o),

as discussed previously. Also, one can verify that ^(Qoo) and A*(Qoo) are both finite. (Actually, A = A* by the Weil pairing.) The following result shows the equivalence of Mazur's conjecture and the main conjecture stated in this paper. Proposition 2 SA{fyoo) is isomorphic to the p-primary subgroup of the classical Selmer group 5^lass(Qoo) as A-modules. This is a special case of a much more general result for arbitrary abelian varieties defined over number fields which will be proved in [2]. We will sketch here a simple argument that applies to this case. For each prime A of Qoo, let D\ denote a decomposition group in Gal ( ^(DxiE^i))). To prove the proposition, we

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must show that this kernel is zero for £ ^ p and that it is the same as her (H^D^E,-)

-> H^D^E,*))

for A = TT.

For £ j£ py it is quite easy to see that the above kernel is zero. If is any finite extension, then E(F) is an l-adic Lie group. One then has a canonical decomposition E(F) = (E(F)p,prim) x M(F), where we define M(F) = paE(F) for a >• 0. From this, we obtain a decomposition E(Q£) = Epoo x M as G^-modules, where Af = UjpM(F), F running over all finite extensions of Q*. The map Hx{D,Epoo) -» Hl(D,Ep<x> x M) is injective for any subgroup D of Gqt. Let A = 7T. We have the Kummer exact sequence. 0 -> ^((Qoo),)

^

\

l

and also the exact sequence

Im(6) is the kernel occurring in the definition of S^Iass (Qoo)p-prim- Im(e) is the kernel occurring in the definition of 5jJtr(Qa>) = SA(QOO)- Let P E -B((Qoo)x). For t > 1, let Q G £"(QP) be such that p'Q = P . Then S(P ® (1/p*)) i s the cocycle cr defined by cr(^f) = g(Q) — Q for any g G i}*. For g G /*, it is obvious that a(g) G J^oo = F+Ep<*>. Hence a is mapped to a cocycle "a G Hl(D%,Ep<x>) such that a|/7r is trivial. But, as noted before, this means that a is trivial. Hence Im(8) C Im(e). Now T = Gal((Q00)ir/Qp). Both Im{8) and Im{e) are A-submodules of H1(DiriEpoo). We have a surjective A-module homomorphism $ : Im(e)~—> Im(6)~. If one applies Corollary 1 of Proposition 1 in [5] to F+£*poo, one finds that JtjT1(Z),r,i?+.Epoo)" is a torsionfree A-module of rank 1. The same is true for /m(e)", since e has finite kernel. Thus either $ is an isomorphism or /ra(£)^ is a torsion A-module. But Im(6)^ has A-rank > 1 as one can see by considering the maps £((Q n ) T )® (Qp/Zp) —>• (^((Qoo),) ® (QP/Zp))r"forn >1, where Tn = Gca((Q 00 )J(Q n ) I ). The first group is isomorphic to (Qp/Zp)pn and the kernel is easily verified to be finite (using the fact that ^^((Qoo)^) is finite). These remarks show that Im(6)~ has A-rank 1 and that Im{8) = Im(e). Proposition 2 follows. Both Conjecture 1 and the main conjecture have been proven when E is an elliptic curve/Q with complex multiplication. Conjecture 1 is a consequence of a theorem of Rohrlich that we will state below together with a theorem of Rubin [20], giving an annihilator for a certain Selmer group. This result of Rubin is a weak version of the so-called 'two variable main conjecture' which

4 Examples

223

Rubin has now proven completely [21]. The main conjecture for SA(Qoo)j where A = i?poo, follows from this. If E doesn't have complex multiplication, then very little is known. One can prove that the natural map 5^laS8(Q) -> 5^ aM (Q 00 ) r has finite kernel and cokernel. In particular, if both the Mordell-Weil group J£(Q) and the Tate-Shafarevich group III^Q) are finite, then it follows that SA{tyoo)T is finite, where still A = Ep<x>. This easily implies that S^(Qoo) is A-cotorsion. More generally, 5U(Qoo) is A-cotorsion if and only if both rank z (£(Q n )) and corankip(lHjE?(Qn)p_prim) are bounded as n —> oo. (The Zp-corank is just the Zp-rank of the Pontryagin dual.) Of course, conjecturally IIIjg(Qn) is finite for all n (and hence has Zp-corank 0). A theorem of Rohrlich [18] states that Lv(ly oo. The Birch and Swinnerton-Dyer conjecture then implies that rank I(JB(Qn)) is bounded. Thus it is certainly very reasonable to believe that 5A(QOO) is A-cotorsion. The main conjecture would then imply that (po(0v) = 0 precisely when (po(XA) = 0. Now <po(0v) = Lp(
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(Here one needs the fact that i?p<x>(Qoo) = 0. This follows from -Spoo(Qoo)r = ^(Q)p-prim = 0.) One can then prove by a rather standard type of argument that 5Bd~(Q)p.Pri?45BdMI(Qoo)j;prim. The crucial point is that the restriction maps i71(Q^E(Q^)) -> i/"1((Qn)A, E(Qt)) are injective for all n, where £ is any prime, A any prime of Qn over £. For £ = p, this follows from the Tate parametrization E(fyp) = Q* / g | , where qE turns out to have the form qE = pu, u G Z* but ^ (Z*)p. (Its approximate value is given in [14].) From this one sees that the norm map £((<}„)*) —> J5(QP) is surjective for all n > 1. Tate duality then gives the injectivity of the above restriction map. For £ ^ p, the surjectivity of the norm map i?((Qn)A) —> E(Qt) follows from the fact that E has good reduction at £ and that (QW)A/Q£ is unramified. We again get the injectivity we want. Since 5^lass(Qoo)p-Prim = 0, it follows that ^ - ( Q o o ^ p ™ = 0. (Let X = £^(Qoofr. prim . Then X is a finitely generated A-module such that Xj(^0 — 1)X — 0. This implies that X = 0.) The Tate parametrization gives us an exact sequence: 0 -> /vo -> Epoo -^ Qp/Zp -> 0 of G^p-modules, where G®p acts trivially on Qp/Zp. Thus Vp = Tp(-B) ® Qp is clearly ordinary. Let A = J5poo. We have F+A = //pOO. Thus we can define both SA (Qoo) and 5jJtr(Qoo), but this time they could be different because

H^DJI^A/F+A) = H\D«lh,qPllp)

= Qp/Zp. Thus

SA^)IS^[^)

is isomorphic to a subgroup of Qp/Zp. The analogue of Proposition 2 is that 5^lass (Qoo) = S%T (Qoo)- This is also a special case of the general result proved in [2], but one can give a simple, direct proof using Tate's parametrization and Kummer theory for Q^. It follows that 5^(Qoo) = 0. We will show that 5^(Qoo) = Qp/Zp and that F acts trivially. Hence we see that y>0 is a zero of ^s^Q^f- (In fact, it's the only zero. It should be possible to computationally verify that 1. Let C denote any subgroup of ^ ( Q E / Q , A) such that C = Qp/Zp. (It turns out that hi = 1 and so C is uniquely determined.) We will also let C denote its image under the isomorphism ^ ( Q E / Q , A)AJff1(QE/Qoo, yl) r . We will prove that C is contained in 5A(Qoo)r- Our assertion about the precise structure of

4 Examples 5A(QOO)

225

clearly follows from this. Consider the following sequence of maps C Cff^Cfc/Q,A) -> H\G^A) = Hl{G^%llp)

-

H^G^A/F+A)

= Horn (Gal (KjQp),Qp/lp),

where K^ is the maximal abelian pro-p extension of Qp. Local class field theory implies that Gal^oo/Qp) = T?p and that K^ is the compositum of the Zp-extension (Qoo)* of QP and the maximal unramified Zp-extension of Qp. Hence it is clear that Iv C G a l ^ / i f ^ ) . Thus the image of C in Hl(h, A/F+A) is trivial. This is enough to prove that C C SU(Qoo). Example 4 Let f12 denote the unique, normalized cusp form of weight 12, level 1. Let V = V(fi2) = {Vi} denote the corresponding compatible system of £adic representations. Then Lv(s) = Y^=i T(n)n~$ where T(TI) is Ramanujan's r-function. The Euler factor for p is (1 — app~s)(l — flpp~s) where ap +flp= T(P), apf3p = p11. lip f r(p), then, as in Example 2, one of the numbers a p , f3p (say /3P again) will be a p-adic unit (under the fixed embedding ap). With this assumption, Manin [10] has constructed a p-adic L-function Lp(ip, V). The interpolation property for (p =
app-n)Lv(l)/nv

where ftv is chosen so that Lv(l)/Qv G Q. We would like to formulate a more precise main conjecture (where (3 G Ax rather than just G Qx • A x ). Unfortunately in general we don't know how to do this. In this example, we suspect that the simple choice tiv = Lv(l) will work for all primes p such that Vp is ordinary except for p = 691. When p = 691, there are two distinct (jQ-invariant lattices (up to homothety) and two corresponding choices of fV, which we believe should be Lv(l) and pLv(l). The first choice will make p\0$nai in A. The second will make O^1*1 invertible in A. Mazur and Wiles have proved that Vp is ordinary if p \ r(p). (See [24] for a more general result.) The set of primes p such that p\r(p) (e.g., p = 2,3,5,7,2411) should conjecturally be infinite although very sparse. Let Tp be a GQ-invariant lattice in Vp and let A = Vp/Tp. If p \ r(p), then we suspect that (for this specific example) the Selmer group ^(Qoo) should be trivial if flp ^ l(modpZ p ). (This certainly will be false in general for other cusp forms.) Note that f3p = r(p) (mod p2p). Conjecturally there should be infinitely many many p such that r(p) = l(modp). We will consider three s u c h p : p = 11,23, and 691. For p = 11, we will show that SA(QOO) == Qp/^P as a group and that F acts by AC5. Thus Ac"5 is the only zero of ^5yl(Qoof • O n ^ n e analytic side, one has (by

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Greenberg - Iwasawa theory for motives

Manin [11]) that LP(K~5,V) = cLv(-5)(l, w5) = cLv(Q,w5) for some explicit factor c. Now w5 is an odd quadratic Dirichlet character (of conductor 11) and we find that Lv(6,w5) is forced to vanish because of the sign in the functional equation for Lv{s, to5). (The functional equation relates the values at s and 12 — s.) Thus K~5 is also a zero of 0™*x. To prove our result about S'A(QOO)} w e first point out that Wp = Vp ® /c~5 satisfies W* = Wp as G^-representation spaces. To see this, let £ be any prime, £ ^ p. Let on, fit be the eigenvalues of Frobj acting on Vp, where 1 is any prime of (j over £. (All £ ^ p are unramified in the representation on Vp since fu has level 1.) Then at^t = £n. The eigenvalues 7/,£* of Frobj acting on Wp satisfy ft6t

= £UK(£)-10

= £ since K(£)IQ = w(£)-10£10 = £10.

On W*, the eigenvalues of Frobj are £y^1 = 8t and £8jl = 7*. Thus, by the Tchebotarev density theorem, the representations of GQ on Wp and W* have the same trace. Since Vp and hence Wp are known to be irreducible, we must have W* = Wp. Let B = A ® /c""5 = Wp/i2p, where Rp = Tp ® K~\ Proposition 1 implies that we can choose a generator XB of the characteristic ideal of SB{(^B) = e ^ " 1 ^ ) for all (p G Hom cont (r,C^) where e = (—l)n, n = corankZp(5B(Qoo)). But A = B as GQ^-modules and so it is obvious that ^(Qoo) = ^(Qoo) as groups. Once we show that they are isomorphic to Qp/Zp, it follows that n = 1 and that <^0(AB) = 0. It is then clear that V acts trivially on SB (Qoo)- Now one can verify that for the action of F, we have SB(Qoo) = SU(Qoo) ® «~5 and hence T does act on S^Qoo) by

It remains to verify that S^Qoo) = %/ip as a group. Now we have a wellknown congruence f12 = / 2 (modll) where f2 = JE is the weight 2, level 11 cusp form corresponding to the elliptic curve E = X 0 (ll) discussed in Example 3. As a consequence of this congruence together with the fact that Ep is irreducible as a representation space for GQ over Z/pZ, we see that A\p] = Ep as ^Q-modules. Here A\p] denotes the elements of order dividing p in A] Ep is the p-torsion on E. Now Mazur and Wiles prove that for any p such that p f T(P), there is a filtration for the action of GQ P on Vp, 0 C Up C Vp, where dim(Up) = 1, IQp acts trivially on Vp/Up and by Afu on Up. (See [24].) Thus F+Vp = Up. Let p = 11. We let F+A[p] denote F + A fl A\p] (and similarly define F + Ep). This defines a filtration of the C?Qp-modules A\p] and Ep and it is clear that an isomorphism A[p]-^EP must send F+j4[p] to F+Ep. (The inertia group /Q P acts by a;11 and a;1 respectively. We have u>n = a;1. They are nontrivial while the action of IQp on A\p)lF+A\p) and Ep/F+Ep is trivial.) We will

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227

use the notation 5U[p](Qoo) for the subgroup of H1 (Qoo, A\p]) defined by local triviality conditions analogous to those defining S^Qoo), using F+v4[p] in place of F+A. Similarly, we define 5EP(QOO)- The isomorphism A[p] = Ep shows that 5A[p](Qoo) ^ ^(Qoo). The exact sequence 0 —• >l[p] —> A^+ A —• 0 induces a surjective homomorphism ^(Qoo, A\p]) -> ^(Qoo, A)[p]. It is also injective, since if°(Qoo, A) = ^(Qoo) = 0 in this case. (This is true because v4[p](Qoo) = -E>(Qoo) = 0 as noted in Example 3.) It follows that we have an injective map SA\p](Qoo) —> 5U(Qoo)[p]« We will show that this map is an isomorphism. To prove surjectivity, we must verify that an element of ^(Qooj-Afp]) which satisfies the local triviality conditions defining S^Qoo) already satisfies those defining It is enough to verify the injectivity of the maps /f^/*, A[p]) —» for all primes A ^ TT and of the map H\IT,A[p]/F+A[p]) -> For A ^ TT, the kernel of the above map is the cokernel of the map #°(J A , A) A H°(IX, A). This cokernel is zero since H°(IX, A) = A is divisible. As for TT, since A\p]/F+A\p] = (A/F+ A)\p] and H0^, A/F+A) = A/F+A is again divisible, we see that the kernel of the corresponding map is also zero. All of these considerations apply to Ep and Epoo. We conclude that

In Example 3, we proved that 5^pOO(Qoo) = Qp/Zp. Hence 5A(Q00)[p] is cyclic. It follows that either SUCQoo) is a finite cyclic p-group or is isomorphic to Qp/Zp. But Proposition 10 of [5] shows that S^Qoo)" has no nontrivial finite A-submodules and hence is certainly not finite. (One must check the proof of this proposition to see that it applies to p = 11.) It follows that SA(, the reduction of p mod 23. This is equivalent to a congruence / 12 = /i(mod23). Let Wp be a 2-dimensional Qp-vector space

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Greenberg - Iwasawa theory for motives

on which A acts by /), Rp a G^-invariant lattice, and B = Wp/Rp. Then A\p] = B[p] as Gq-modules. Let ^p be any one of the three prime of K lying over p. The corresponding decomposition (and inertia) subgroup A

Xi}? Xo denoting the trivial character. Each W* has dimension 1. Now Wp is not ordinary in the sense defined earlier, but nevertheless it has two possible filtrations as a G«jp-representation space. For our purpose here, we will consider the filtration 0 C W*1 C Wp and we will denote W*1 by F+Wp. Let F+B denote its image in B. Thus we can define Selmer groups SB(QOO) and S£tr(Qoo) Jus^ a s before. The inertia group JQp acts on F+Vp by V 1 1 and hence on F+A\p) = (F+A) n A\p] by a;11, which is a character of GQp of order 2. One easily sees that a;11 = %i and therefore the isomorphism v4[p]A>5[p] must send ir?+A[p] to F+B\p]. The argument that we described for the case p = 11 applies here without change. We find that ^(Qoo)^] = SB(QOO)[P]- It is enough therefore to prove that = QP/ZP. We will actually show that S'B(QOO) — QP/ZP and T acts trivially. The argument is similar to the one in Example 3. We have B/F+B = Qp/Zp on which GQP acts trivially. Let E = {p, oo}. B is a Gal (Qs/Q)-module and corankZp(i71(Qs/Q,,B)) > 1. We can pick a subgroup C = Qp/Zp in Hl(Qs/Q,B). Just as previously, we find that SB(Qoo)/S^*(Qoo) is isomorphic to a subgroup of Qp/Zp and that CIQ^ = Qp/Zp is contained in S'B(QOO)We will show that 5^tr(Qoo) = 0 by proving the equivalent statement that S^tr(Qoo)r = 0. The assertion that S^Qoo) = Qp/Zp (with a trivial action of F) follows. We will consider the subgroup SB (Q) of Hl(fy,B) defined by local triviality conditions analogous to before, except of course involving inertia group in GQ. We have SB(q) = Since (|A|,p) = 1, we have an isomorphism ) -> H\qvlK,B)*

= RomA(G
Here M denotes the maximal abelian pro-p extension of K contained in Q^. Let U = Il
4 Examples

229

denote its closure in U. One finds that U/E = 1* x T, where T is finite of order prime to p. (One can verify this by a calculation with the roots of X3 — X + 1, which are in E.) Also, the class number of K is not divisible by 23. Class field theory then implies that Gal (M/K) = Z*. Each irreducible representation of A occurs with multiplicity 1 in Gal (M/K) and so one finds that Gal(M/KY = Gal(Mp/K) £ l2p for a certain field M,, K C M, C M. Hence we have ^ ( Q E / Q , ^ ) = HornA( Gal ( M p / # ) , £ ) = QP/ZP. (The Zpcorank is therefore exactly 1. The subgroup C referred to before must be all of ^ 1 ( Q E / Q ? B). This fact will simplify our discussion.) Let <7 G - ^ ( Q E / Q , # ) have order p. Then a\K is a surjective A-homomorphism: Ga\(M/K) -> £[p]. Let 5 = £[p], (7

B such that E C ker ((p). Now A acts on B by ^. Regarding U as a representation space for A over Z/pZ (completely reducible since p f |A|), "J9 occurs with multiplicity 2 and 2? is a A-subspace of C/7. Since dimZ/pZ(E) = 2, we clearly get an isomorphism U^/E-^B induced from (p. Let y$\p. Then A

occurs with multiplicity 1. We have a decomposition B = Bxo x BXl as a A-module. The local triviality condition (for cr) defining 5^(Q) is equivalent to requiring that C/

Hl(GQp,B/F+B)

= Hom(GQp,Qp/Zp).

This map must be injective because SB(Q) = 0. We denote its image also by C. Since C = QP/ZP, we can write C = Rom(Gal(Koo/qp)y%/lp) for a certain Zp-extension K^ of Qp. K^ is determined by its group of universal norms UnivNorm (K^/%) = nnArK-n/Qp(/ir^), where Kn is the nth layer in KQQ. If Koo ^ ^c£nr, the unramified Zp-extension of Qp, then UnivNorm (iifoo/Qp) is of the form //p_i • q* where q = pau, a = p*,^ > 0, and w G 1 + pZp. The fact that 5B(Q) = 0 means that / ^ fl / ^ n r = Qp and this implies that q = pu. Now C = JHrl(Qs/Q,JB)-^JHrl(Qs/Qoo, B) r . Let /^ y c l

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denote the cyclotomic Zp-extension of Qp, /^ y c l = (Qoo),. Then S|tr(Qoo)r = 0 if and only if K^ fl K£cl = Qp or equivalently, u <£ (1 + plp)p = 1 + p%. K^ has the following more concrete description. Fix a prime $* of K lying over p and let Ko be the decomposition subfield K^. Then Gal (Mp/K)Xo has Zp-rank 1 and hence corresponds to a Zp-extension of Ko. Completing at ty gives a Zp-extension of Qp which one easily sees is K^. It is an interesting exercise in class field theory to compute q = pu and we will just give the result here. Let e0 be a fundamental unit of Ko, TT0 a generator of the prime ideal of Ko lying below tp. Let ^ b e a generator of Gal (/iT/Q(\/— 23)). Put 6x = g(£o)/g2(£o) and rjx = g(fto)/g2(7ro). Regarding these numbers in K<$, it turns out that 21ogp(u) = 31ogp(7r0)-361ogp(e0) where b = l o g ^ ^ / l o g ^ ) . This determines u and one finds by calculation that u^ l(modp 2 Z p ). We consider the q occurring above as almost an analogue of the Tate period qE occurring in Example 3. Let E be an elliptic curve/Q with split multiplicative reduction at p. Let A = Epoo. Then GQP acts trivially on A/F+A = Qp/Zp. Let E be a finite set of primes containing oo and all primes where E has bad reduction. Assume that 5j l a s s (Q)p.prim is finite. Then one can show that i/^Qs/Q? A) has Zp-corank 1 and hence contains a unique C * Qp/Zp. The image of C in H^G^A/F+A) = Hom(GQp,Qp/Zp) will then be nontrivial (and so isomorphic to Qp/Zp again) and therefore will determine a Zp-extension 7 ^ of Qp. One can show that qE G UnivNorm (/^oo/Qp). If q is a generator of UnivNorm (KQO/%) mod //p_i as described earlier, then qE = qee for some e > 1 and e G ^P-i- Either q or qE would determine the Zp-extension K^. We will now give some examples where ^5A(Qoor is divisible by p in A (i.e., the //-invariant is positive). Such examples will arise whenever p is an irregular prime, p\Bh say, where 0 < k < p — 3, k even. As is well-known, there will then be a congruence mod p between an Eisenstein series and a cusp form (an eigenform for the Hecke operators), both of level 1, weight k. We will consider the cases where the cusp form has coefficients in Q: /i 2 ,p = 691; /i6,P = 3617; fls,p = 43867; / 20 ,p = 283 or 617; / 22 ,p = 131 or 593; and / 2 6 ,p = 657931. (In each case fk is the unique, normalized cusp form of level 1, weight k.) One will have a congruence a^ = crk_1(n)( modp) where a^ is the nth Fourier coefficient of fk, crfc_1(n) = J^d*"1, and p is one of the above primes (which d\n

are the primes dividing Bk). Let V = V(fk) = {Vt} be the compatible system of £-adic representations corresponding to fk. We have a^ = 1 ^ 0(modp) and so (by a theorem of Mazur and Wiles), the p-adic representation Vp is ordinary. Let Tp be a GQ-invariant lattice and A = Vp/Tp. Then the above

4 Examples

231

congruence implies that A\p] is actually reducible as a representation of GQ over Z/pZ and has composition factors /z*"1 and Z/pZ(= //£). We can choose the lattice Tp so that /x*"1 C A\p]. Now JQp acts on F+Vp by Nk~l and on Vp/F+Vp trivially. Hence it is clear that /x*"1 C F+A\p] = A\p] H F+A. Let Wp = QP(A; — 1), as in Example 1 except that now A; — 1 is positive and odd. Thus F+Wp = Wp. We let Rp = lp(k - 1) and B = Wp/Rp = jij" 1 . Therefore, we have B\p] C A\p] as GQ-modules, F+B[p] C F+A[p] as GQpmodules, and we obtain (in essentially the same way as earlier for p = 11 and 23) homomorphisms

The kernels of the above maps are finite. Referring back to our first example, we have that 5B(Qoo) = Hom(X£~\%/lp(k - 1)). But Iwasawa [8] proved that X^ (the subgroup of X^ on which complex conjugation in A acts by — 1) is a A-module of rank (p — l)/2 and, more precisely, each component X^ , where n is odd, has A-rank 1. Thus ^(Qoo) is not A-cotorsion. As we explain in [5] in a much more general context, this is related to the fact that ((s) has a simple zero at 5 = 1 — (k — 1) = —k forced by a pole of the F-factor in its functional equation. Thus SB\P]{Q oo)^ is a A/pA-module of rank > 1. (This rank turns out to in fact be 1.) Consequently, we see that Sx(Qoo)[p]~ also has A/pA-rank > 1. This implies that p divides a generator of the characteristic ideal of SA(Qoo)~We believe that in all of the above cases 5A(QOO) has exponent p and its dual is a free A/pA-module of rank 1. In order to make the main conjecture valid in the precise form (where (3 G A x ), one should take fl^ = Ly(l) when Tp is the lattice chosen above. One can give an interesting criterion for 5A(QOO) to have the above structure in terms of the ideal class group Cl(F) of a certain field F. Let Ko = Q(//p), A = Gal(if o /Q). Let F be the cyclic extension of Ko of degree p which is Galois over Q and such that A acts on Gal (F/Ko) by uk~\ i.e., Gsl(F/K0) = /x*"1. (In the notation of [5], this field is Lwh-i.) Let C = Cl(F)/Cl(Fy. Let g be a generator of G} , etc. The criterion is that the u>°-component of A acting on C9"1 /C^""1^ is trivial. We will not try to explain this here. If the class number hKo of Ko is not divisible by p 2 , then the criterion is more simply that Cg~l = 0. This turns out to imply that Gal (F/KQ) acts trivially on Cl(F)p.prim and that p2 { hF. Unfortunately, it would seem rather difficult to verify these criteria for any of our cases. In [5], we gave the simpler criterion for the case k = 12,p = 691, mistakenly assuming that p2 \ hKo. (We are grateful to David Penman for

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pointing this error out to us.) The assumption that p2 \ hKo is however true for example when p = 131, 283, and 593. REFERENCES 1. J. Coates, 'p-Adic Z-functions for motives', this volume. 2. J. Coates and R. Greenberg, 'Selmer groups for abelian varieties', to appear. 3. J. Coates and B. Perrin-Riou, 'On p-adic L-functions attached to motives over Q', Advanced Studies in Pure Mathematics, 17, (1989), p. 23-54. 4. P. D. Deligne, 'Valeurs de fonctions L et periodes d'integrates', Proc. Symp. Pure Math., 33 (1979), p. 313-46. 5. R. Greenberg, 'Iwasawa theory for p-adic representations', Advanced Studies in Pure Mathematics, 17 (1989), p. 97-137. 6. R. Greenberg, 'Iwasawa theory for p-adic representations IF, to appear. 7. H. Hida, 'Galois representations into ZP[[X]] attached to ordinary cusp forms', Invent, Math., 85 (1986), 545-613. 8. K. Iwasawa, 'On Zrextensions of algebraic number fields', Ann. of Math., 98 (1973), p. 246-326. 9. J. Jones, 'Iwasawa theory at multiplicative primes', thesis (1987), Harvard University. 10. J. Manin, 'Periods of parabolic forms and p-adic Hecke series', Math. Sbornik, 92 (1973), p. 371-93. 11. J. Manin, 'The values of p-adic Hecke series at integer points of the critical strip', Math. Sbornik, 93 (1974), p. 631-7. 12. B. Mazur, 'Rational points of abelian varieties with values in towers of number fields', Invent. Math., 18 (1972), p. 183-266.

References

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13. B. Mazur and P. Swinnerton-Dyer, 'Arithmetic of Weil curves', Invent. Math., 25 (1974), p. 1-61. 14. B. Mazur, J. Tate, and J. Teitelbaum, 'On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer', Invent. Math., 84 (1986), p. 1-48. 15. B. Mazur and A. Wiles, 'Class fields of abelian extensions of Q', Invent. Math., 76 (1984), p. 179-330. 16. B. Mazur and A. Wiles, 'On p-adic analytic families of Galois representations', Compositio Math., 59 (1986), p. 231-64. 17. B. Perrin-Riou, 'Variation de la fonction L p-adique par isogenie', Advanced Studies in Pure Mathematics, 17 (1989), p. 347-58. 18. D. Rohrlich, 'On L-functions of elliptic curves and cyclotomic towers', Invent. Math., 75 (1984), p. 409-23. 19. K. Rubin, 'The main conjecture. Appendix to: Cyclotomic Fields by S. Lang, Grad. Texts in Math., Springer-Verlag (1989). 20. K. Rubin, 'On the main conjecture of Iwasawa theory for imaginary quadratic fields', Invent. Math., 93 (1988), p. 701-13. 21. K. Rubin, 'The "main conjecture" of Iwasawa theory for imaginary quadratic fields', to appear in Invent. Math. 22. P. Schneider, 'Motive Iwasawa theory', Advanced Studies in Pure Mathematics, 17 (1989), p. 421-56. 23. J. P. Serre, 'Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures)', Sem. Delange—Pisot-Poitou, (1969/70), exp. 19. 24. A. Wiles, 'On ordinary A-adic representations associated to modular forms', Invent. Math., 94 (1988), p. 529-73.

Kolyvagin's work on modular elliptic curves BENEDICT H. GROSS

1. Let X0(N) be the modular curve over Q which classifies elliptic curves with a cyclic TV-isogeny. Let K = Q(y/—D) be an imaginary quadratic field of discriminant —J9, where all prime factors of N are split. For simplicity, we assume that D ^ 3,4, so the integers O of K have unit group O* = (±1). Choose an ideal Af of O with O/Af ~ Z/NZ. We consider K, and all other number fields in this paper, as subfields of C. Then the complex tori C/O and C/Af~l define elliptic curves related by a cyclic 7V-isogeny, hence a complex point xx of X0(N). The theory of complex multiplication shows that the point xx is rational over K^ the Hilbert class field of K. Let £ b e a modular elliptic curve of conductor N over Q, and fix a parametrization

E which maps the cusp oo of X0(N) to the origin of E. Once tp has been chosen, there is a unique invariant differential u> on E over Q such that (p*(u>) is the differential Hanqndq/q associated to a normalized (ax = 1) newform on X0(N). Write UJ0 = a*;, where a;0 is a Neron differential on E. It is known that c is an integer, and we may assume that c>\. Let yx = (p(xi) in E(Ki), and define the point yK = r^rKl/K{y1) in E(K). This point is obtained by adding yx to its conjugates, using the group law on E. If AT' is another ideal with O/AP ~ Z/7VZ, and y'K is the corresponding point in E(K), we have y'K — ±yK + (torsion). Hence the canonical height h(yK) is well-defined, independent of the choice of Af* Zagier and I proved the limit formula [GZ; Ch. I, (6.5)]: (1.1) In particular, the point yK has infinite order if and only if L'(E/K,

236

Gross - Kolyvagin's work on modular elliptic curves

By comparing (1.1) with the conjecture of Birch and Swinnerton-Dyer for L(E/K,s), Zagier and I were led to the following [GZ; Ch. V, 2.2]. Conjecture 1.2 Assume that h(yK) ^ 0, or equivalently, that the point yK has infinite order in E(K). Then (1) the group E(K) has rank 1, so the index IK = \E(K) : ~lyK\ is finite, (2) the Tate-Shafarevich group Ul(E/K) is finite; its order is given by

where mp = (E(%) : E°(%)). In (2), note that both the index IK and the integer c depend on the parametrization 9?, but that the ratio IK/C is independent of the parametrization chosen. Since c and the local factors mp are integers, the formula in (2) predicts that the order of Ul(E/K) should always divide (IK)2- This implies, by the existence of the Cassels pairing, that the group Ul(E/K) should always be annihilated by IKKolyvagin has proved a great part of Conjecture 1.2. His main result is the following [Kl, Thm. A]. Theorem 1.3 (Kolyvagin) Assume that the point yK has infinite order in E(K). Then (1) the group E(K) has rank 1, (2) the group Ul(E/K) is finite, of order dividing tE/K • (IK)2In part (2) of this theorem, tE/K is an integer > 1, whose prime factors depend only on the curve E: they consist of 2 and the odd primes p where the Galois group of the extension <$(EP) is smaller than expected. In many cases, Theorem 1.3 reduces the conjecture of Birch and SwinnertonDyer to a finite amount of computation. For example, let E = X0(37)/w37 be the curve y2 + y = x3 — x, and let ip be the modular parametrization of degree 2. Then c = 1 and m37 = 1 in part (2) of Conjecture 1.2, so we expect that #Ul(E/K) = (IK)2 when yK has infinite order. Kolyvagin shows that tE/K is a power of 2 in this case, and that tEjK = 1 when IK is odd. To prove the full conjecture of Birch and Swinnerton-Dyer for E over K, one must construct non-trivial elements in Ul(E/K) when IK > 1. (Kolyvagin's method suggests such a construction - see §11). We remark that in this case the point yK lies in i?(Q), which is infinite cyclic and generated by P = (0,0).

Kolyvagin's work on modular elliptic curves

237

Writing yK = rnK • P we find IK = ±mK; the integers mK appear as Fourier coefficients of a modular form of weight 3/2 for F0(4 • 37) [Z; §5]. 2. We will not prove all of Theorem 1.3, but will sketch the proof of a slightly weaker result to illustrate Kolyvagin's main argument. In all that follows, we assume that the curve E does not have complex multiplication over C. (This excludes only thirteen j-invariants.) Then Serre has shown that the extension Q(EP) generated by the p-division points of E has Galois group isomorphic to GL2(l/pl) over Q for all sufficiently large primes p [S; Thm. 2]. In fact, if E is semi-stable (i.e., if N is square-free), the Galois group of Q(J5P)/Q is isomorphic to GL2(l/pl) for all p > 11 [Ma; Thm. 4]. The first (crucial) observation is the following. If yK has infinite order in E(K), one does not know a priori that the index [E(K) : lyx] is finite. However, since the group E(K) is finitely generated, the point yK is not infinitely divisible in E(K). In other words, there are only finitely many integers n such that yK = nP with P G E(K). Proposition 2.1 Let p be an odd prime such that the extension Q(Ep) has Galois group GL2(l/pl), and assume that p does not divide yK in E(K). Then (1) the group E(K) has rank 1, (2) the p-torsion subgroup Ul(E/K)p is trivial. When yK has infinite order in E(K), Proposition 2.1 applies for almost all primes p. Our hypotheses imply that p does not divide the index IK = [E(K) : Zy/r], so the conclusion is consistent with part (2) of Conjecture 1.2. Kolyvagin obtains Theorem 1.3 by refining the argument for primes p which divide yK, using the fact that pn does not divide yK for large n. The p-primary component of Ul(E/K) is bounded using his techniques on ideal class groups (see [R2]). When the Galois group of Q(EP) is strictly contained in GfX2(Z/pZ), he uses Serre's result that the Galois group of Q(Epn) has bounded index in GL2(2/pnl) for n -> oo. In fact, what we will prove involves the Selmer group Sel(E/K)p at p, which sits in an exact sequence of Z/pZ-vector spaces (2.2)

0 —+ E(K)/pE(K)

- ^ Sel(E/K)p

—> W(E/K)P

—> 0.

By our hypothesis on Q(.EP), the group E(K) contains no p-torsion and the dimension of E(K)/pE(K) over Z/pZ is equal to the rank of E(K).

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Gross - Kolyvagin's work on modular elliptic curves

Proposition 2.3 Let p be an odd prime such that the extension Q(EP) has Galois group GL2(Z/pZ), and assume that p does not divide yK in E(K). Then the group Se\(E/K)p is cyclic, generated by 8yK. The proof of Proposition 2.3 (following Kolyvagin) has three steps. The first is the construction of certain cohomology classes c(n) E HX{K^EP) from Heegner points of conductor n for K, and the study of their amazing properties. The second is the use of Tate duality to obtain information on the local components of elements in the Selmer group Se\p(E/K) from the classes c(n). The third is the use of the Cebotarev density theorem to convert information on the local components of the Selmer group to an upper bound on its order. Proposition 2.1 is an immediate corollary of Proposition 2.3, using (2.2). 3. We begin with a construction of the cohomology classes c(n), or rather, with a description of the properties of Heegner points on which the construction depends. Let n > 1 be an integer which is prime to TV, and let On = Z + nOK be the order of index n in OK- The ideal Afn = M fl On is an invertible 0n-module with On/Afn ~ l/Nl. Consequently the elliptic curve C/On (with its cyclic AMsogeny to C/M~l) defines a complex point xn on XQ(N). The theory of complex multiplication shows that the point xn is rational over Kn, the ring class field of conductor n over K. We have a field diagram with Galois groups marked:

Here r is complex conjugation, which lifts to an involution of Kn and acts on

Gal(Kn/K) by:

We will only consider the points xn on X0(N), and their images yn =
£ does not divide N • D -p.

Kolyvagin's work on modular elliptic curves

239

This hypothesis implies that the prime £ is unramified in the extension K(EP). We let Frob(^) be the conjugacy class in Gal(K(Ep)/Q) containing the Frobenius substitutions of the prime factors of £, and further insist that (3.2)

Frob(£) = Frob(oo)

as conjugacy classes in Gal(K(Ep)/Q,). Here Frob(oo) is the conjugacy class of complex conjugation r. There are an infinite number of primes £ satisfying (3.2), by Cebotarev's density theorem. A simple implication of (3.2) is that Frob(^) = r in Gal(/i7Q). Hence the prime {£) remains inert in K] we let A denote its unique prime factor. The implication Frob(^) = Frob(oo) in Gal(Q(£>p)/Q) is equivalent to the congruences: (3.3)

at=£+l=0

(mod p),

where £ + 1 — at is the number of points on the reduction E over the finite field Ft = l/£l. Indeed, the characteristic polynomial of Frob(£) acting on Ep is known to be x2 — atx + £, whereas the characteristic polynomial of Frob(oo) = r is known to be x2 — 1 = (x — l)(x + 1). Let F\ denote the residue field of K at A, which has £2 elements. By (3.2) the prime A splits completely in the extension K(EP). Hence E(FX)P — (Z/pZ) 2; in fact we have: (3.4)

E(Fx)f

~ Z/pZ

where ± denote the eigenspaces for the automorphism group ( 1 , T ) . Indeed E(FX)+ has order £+l—a t^ and E(FX)~ has order £4-1 + at\ both are divisible by p by (3.3). We recall that n is square-free. Write n = JJ£ and let Gn be the Galois group of the extension KnjKx. Then Gn ~ Y\G£ where, for each £\n, Gt is the subgroup fixing the subfield Kn/i. The subgroups Gt ~ F A x /i^ x are cyclic of order £ + 1. Let at be a fixed generator of Gi\ the augmentation ideal of the group ring 2.[Gt] is principal and generated by (at — 1). Let Tr^ be the element ^TV in Z[G^], and let Dt be a solution of (3.5)

(at-l)-Dl=£+l-Trl

in l[Gf\. Then Dt is well-defined up to addition of elements in the subgroup i

t+i

Z • Tr* (Kolyvagin uses the solution D°t = ]Ti • a\ = - ] £ {a\ - l)/(at

- 1)

240

Gross - Kolyvagin's work on modular elliptic curves

but this has little advantage over the others). Finally, define Dn =Y[Dt in Proposition 3.6 The point Dnyn in E{Kn) gives a class [Dnyn] in (E(Kn)/pE(K