SPECIAL VALUES OF HECKE L-FUNCTIONS AND ABELIAN INTEGRALS G ., Harder and N. Schappacher Max-Planck-Institut f~r Mathematik Gottfried-Claren-Str. 26 5300 Bonn 3 In this article we attempt to explain the formalism of Deligne's ratio- nality conjecture for special values of motivic L-functions (see [DI]) in the particular case of L-functions attached to algebraic Hecke charac- ters ("Gr~Bencharaktere of type A0"). In this case the conjecture is now a theorem by virtue of two complementary results, due to D. Blasius and G. Harder, respectively: see §5 below. For any "motive" over an algebraic number field, Deligne's conjecture relates certain special values of its L-function to certain periods of the motive. Most of the time when motives come up in a geometric situa- tion, we tend to know very little about their L-functions. In the special case envisaged here, however, the situation is quite different: The L- functions of algebraic Hecke characters are among those for which Hecke proved analytic continuation to the whole complex plane and functional equation. But the "geometry" of the corresponding motives has emerged only fairly recently - see §3 below. The relatively good command we now have of the motives attached to alge- braic Hecke characters reveals that many non-trivial period relations are in fact but reflections of character-identities. This point of view is systematically perused in [Seh], and we shall illustrate it here by the so-called formula of Chowla and Selberg: see § 6. This formula, in fact, goes back to the year 1897, as does the instance of Deligne's conjecture with which we start in § .I Tying up these two relations in the motivic formalism, we hope to make it apparent that both results really should be viewed "comme les deux volets d'un m~me diptyque ,~' as A. Weil has pointed but in [WIII], p. 463. 81 Contents § I A formula of Hurwitz § 2 Algebraic Hecke characters § 3 Motives § 4 Periods § 5 The rationality conjecture for Hecke L-functions § 6 A formula of Lerch § 1o A formula of Hurwitz In 1897, Hurwitz [Hu] proved that 1 4v (I) 'E 4v ~ × (rational number), a,b6~ (a+bi) for all ~ = 1,2,3,... , where rl dx £( (2) ~ = 21_ - 2.62205755,.. = u Notice the analogy of these identities with the well-known formula for the Riemann zeta-function at positive even integers: 1 2v (3) E' 2~ - (2~i) x (rational number). a£ ~ a Both formulas are special cases of Deligne's conjecture. To understand this in Hurwitz' case, we look at the elliptic curve A given by the equation A : y2 = 4x 3 _ 4x . A is defined over ~ , but we often prefer to look at it as defined over the field k = ~(i)c ~ . Over this field of definition, we can see that A admits complex multiplication by the same field k : 19 k > End )A( ® --X i I > I---> iy Deligne's account of Hurwitz' formula would start from the observation that both sides of )I( express information about the homology I H (A) ®49 c H4~ (A 4~) The left hand side of )I( carries data collected at the finite places of k , as does the right hand side for the infinite places. In fact, look at the different cohomology theories: - Etale cohomology: Fix a rational prime number ~ , and denote, for n~ 1 , by A[£ ]n the group of ~ n -torslon points in A(~), ~ being the algebraic closure of ~ in ~ . Then V£(A) = I<lim A[£n])®ZZ£ ~£ n is the dual of the first £-adic cohomology of A×k~ with coefficients in ~£ By functoriality, the isomorphism ~ ~ k ®~ End A makes Vi(A) into a k ® ~£-module, free of rank .I The natural continuous action of Gal (W/k) on V£(A) is k ® ~ -linear, and therefore given by a continuous character ~£: Gal (~/k)ab - - > GL k®~£(V Z(A)) = (k®~z)* . This character was essentially determined - if from a rather different point of view - on July 6, 1814 by Gauss, [Ga] . The explicit analysis of the Galois-action on torsion-points of A was carried out (in a stunning- ly "modern" fashion) in 1850 by Eisenstein, [El]. - In any case, if is a prime element of ~ [i] not dividing 2Z , normalized so that ~ I (mod (I+i) )3 , and if F 6 Gal (~/k)ab is a geometric Frobenius 20 element at )~( (i.e., F-1(x)~ ~ ~ x ~ (mod P ) for any prime P of kab dividing (~), any algebraic integer x 6 kab), then one finds ~ (F) = -I 6 k* c > (k® Q£)* The characters ~£ all fit together to give an "algebraic Hecke charac- ter" ~ defined on the group 12 of ideals of k that are prime to 2: I ~ > k* 2 F ~ I2~ > Gal (~/k)ab > (k®~g)* Then for all Z ~ I, the character ~4~ can be defined on all ideals of k by (~) ~__> -4v . Remember that k is embedded into ~ , so that it makes sense to consider the L-functions L(~ 4~ ,s) = ~ I (Re(s) > I-2~) , ~ (~) 4~ p I s where p ranges over all prime ideals of ~ [i] . Then the left hand side of Hurwitz ' formula )I( is simply 4 . L(~ 4~ ,0). We have shown how this is a special value, of the L-function afforded by the l-adic cohomologies V~(A)®k®~ 4~ - Betti and de Rham cohomology. Here we shall use the fact that the curve A (if not its complex multiplication) is already defined over ~. B Denote by HI(A) = HI(A(~),~) the first rational singular homology of the Riemann surface A(~) , with the Hodge decomposition B H -I'0 @ H 0'-I (A) I H ®~ • = Complex conjugation on A×~ induces an endomorphism F ~ HI(A) : (the "Frobenius at ~"). 12 Call HB the fixed part of HI(A) B under F , and let be a basis of this onedimensional G-vector space. Let H R(A) = HDR 1 (A)V be the dual of the first algebraic de Rham cohomo- logy of A over ~ , given with the Hodge filtration DR + I H )A( ~F D {0} where + F ®~ H ~ 0'-1 under the GAGA isomorphism over ¢: I : H~(A) ®~ 1( > HIR )A( E(~(® I induces an isomorphism of onedimensional ~-vector spaces + i + : HB(A) ®~ .... > (HIR(A)/F )+ ®~ Then, I~ . i +(n) 6 HIR(A)/F + , for ~ defined by )2( . In fact, = f 1 ~ dx is a real fundamental period of our curve, and so, up to ~* , Q is the determinant of the integration-pairing (HB(A) )~(~(® x (H0(A,~ )I )Z(~(® S > ~( calculated in terms of R-rational bases of both spaces. This determinant equals that of the map + I since H0(A,~ )I cH I )A( is the dual of DR H~R )A( /F + Passing to tenser powers of the onedimensional vector spaces above we find the periods ~4~ occuring in (I). In a sense, we have cheated a little in deriving the period ~ from the cohomological setup: In the ~tale case we have used the action of k via complex multiplication to obtain a onedimensional situation (i.e., the k-valued character ). ~ In the calculation of the period, too, we should have considered H~R(A/k). = H~R(A)j ®~ k , endowed with the further action B of k via complex multiplication, and two copies of HI(A), indexed by the two possible embeddings of the base field k into • .... But in the 22 presence of an elliptic curve over ~ , this would have seemed too arti- ficial, and the general procedure will be treated in § 4. As a final remark about formula (I), it should be noted that it is proved fairly easily. Any lattice F = I . ZZ( +~i) gives a WeierstraB ~func- tion such that 3 ~' (z,F) = 4~(z,r) - g2lF) ~(z,F), and for = ~ I we get g2(F) = 4. The rational numbers left unspecified in )1( are then essentially the coefficients of the z-expansion of ~(z,F) . It is these numbers that Hurwitz studied in his papers. § 2. Algebraic Hecke Characters Let k and E be totally imaginary number fields (of finite degree over ~) , and write Z = Hom (k,~) and T = Hom (E,~) the sets of complex embeddings of k and E. The group Gal(~/~) acts on Z×T , transitively on each individual factor. An algebraic homomor- phism 8 : k* > E* is a homomorphism induced by a rational character : Rk/~ (~m) > RE/~ (~m)- This means that, for all • 6T, the composite • oS : k* > ~* is given by = ~E-~ (4) ~oB (x) O(x) n(~,~) , 23 for certain integers n(o,T) , such that n(~o,~) = n(a,T) for all p6 Gal (~/~) . Let k~ ,f ,,~ > k~ be the topological group of finite id~les of k - i.e., those id~les whose components at the infinite places are I. For x 6k*, let x also denote the corresponding principal idele in k~ ,and xf the finite id~le obtained by changing the infinite components of x to 1. An algebraic Hecke character ~ of k with values in E , of (infinity-) type ~ , is a continuous homomorphism : k* > E* ~,f such that, for all x £ k*, (xf) = B )x( If ~ is the infinity-type of an algebraic Hecke character Y , then, by continuity, 8 has to kill a subgroup of finite index of the units of k. It follows that the integer (5) w = n(o,T) + n(co,T) = n(o,T) + n(o,cT) (where c : complex conjugation on ~) is independent of ~,T. It is called the weight of For any T6 T , we get a complex valued Gr~Sencharakter Toy which extends to a quasicharacter of the id~le-class-group: * ToY ~. k]A,f .......... > Alk * ToY ~. klA/k. > Consider the array of L-functions, indexed by T: 24 L (~,s) = (L(To~,S))~6 T , where, for Re(s)> ~ W + I , I # the product being over all prime ideals p of k for which the value ~(~p) does not depend on the choice of uniformizing parameter wp of kp The point s = 0 is called critical for ~ , if for any T , no F- factor on either side of the functional equation of L(To~,s) has a pole at s = 0 . This is really a property of the infinity-type B of , for it turns out that s = 0 is critical for ~ if and only if there is a disjoint decomposition Z ×T : { (o,T) n(o,T) <0} 0 { (d,T) I n(co,7) < 0} In other words, for every TET , there is a "CM-type" ~(TOB)C Z such that • (~ToB) = ~(ToB) ~ , for ~6 Gal(~/~) (6) • O6~(~8) ¢¢ n(d,T) <0 ~ n(co,<) ->0 For ~ such that s =0 is critical Deligne defined an array of periods ~(~) : (Q(~}''T))T6T 6 ({*)T = (E®~{)* , and conjectured that (7) L(~,0) 6 E c > E®{ ~(~) In other words, he conjectured that there is x 6 E such that for all Y : E ~r >{ , L(To~,0) = T(X) . ~(~,T) 25 The definition of ~(~) is discussed in § 4. It requires attaching a motive to an algebraic Hecke character. § 3. Motives 3.1 In the example of § 1, we constructed a "motive" for our Hecke characters ~4m by taking tensor powers of HI )A( , i.e., a certain di- rect factor of H4~(A4m), in the various cohomology theories. This illustrates fairly well the general idea of what a motive should be: Starting from an algebraic variety over a number field, we have the right to consistently choose certain parts of its cohomology. Just what "consistenly" means constitutes the difference between various notions of motive. Here we shall be concerned with a fairly weak and therefore half way manageable version: motives defined using "absolute Hodge cycles" - see [DMOS], I and II. In this theory motives can often be shown to be isomorphic when their L - func- tions and periods coincide. A little more precisely, giving a homo- morphism between two such motives M and N amounts to giving a fa- mily of homomorphisms O H )M( --> ° H (N) (Betti cohomology depends on the choice of o : k--> ~ yielding M}--> Mxo~ ) HDR(M) --> HDR(N) H£ (M)--> H£ )N( (for all £ ) compatible with all the natural structures on these cohomology groups: Hodge decomposition, Hodge filtration, Gal(k/k)-action, as well as with the comparison isomorphisms between B H and HDR B , H and the His • 3.2 Let us state more precisely what a motive attached to an alge- braic Hecke character ~ should be[ - In the example of § I, the curve A/~ defines the motive HI )A( over ~ whose L-function is L(~,s). (This is really what Gauss observed in 1814; nowadays this follows from a result of Deuring, which has been further generalized by Shimura [Sh I]...) But this is not what we are looking for. The complex multi- plication of A and therefore the Hecke character ~ are not visible 28 over ~ . That is why we considered A over k in our treatment of the @tale cohomology, and used the field of values of ~ (which again happened to be k ) to obtain onedimensional Galois-representations, and thus Given a general algebraic Hecke character ~ like in § 2, a motive M for ~ has to be a motive defined over the base field k such that the field E acts on all the realizations of M in the various eoho- mology theories, and such that for all Z, Hz(M) is an E®~i-module of rank I with Gal (T/k) acting via ~ . The action of E on the various realizations of M should of course be compatible with their extra structures and with the various comparison isomorphisms. In other words (see 3.1), E should embed into End M . Thus the rank-condition on HI(M ) can also be stated by saying that Betti cohomology H (M) should form a onedimensional E-vector space. 3.3 The typical example is HI )A( , for an abelian variety A/k with E ~ ~®~ End/kA and 2 dim A = [E : Q]. The fact that these motives always give rise to an algebraic Hecke character was one of the main results of the theory of complex multiplication by Shimura and Taniyama. The Hecke characters occuring with abelian varieties of CM-type are precisely those of weight -I such that n(o,7) 6 {-1,0} , for all (o,T) 6 Z×T . In fact, given such an algebraic Hecke character ~ of k with values in E , we can assume without loss of generality that E is the field generated by the values of ~ on the finite id~les of k . Then E is a CM-field (i.e., quadratic over a totally real subfield), and a theorem of Casselman, [Sh 1], can be applied to get an abelian variety A defined over k such that: . 2 dim A = [E : ~] • there is an isomorphism E N > ~®~ End/kA • HI )A( is a motive for ~ .