CONGRUENCES FOR FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS AND SPECIAL VALUES OF L FUNCTIONS Antal Balog, Henri Darmon and Ken Ono Abstract. Congruences for Fourier coefficients of integer weight modular forms have been the focal oint of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different tye. Let f(z = P n=0 a(nqn be a holomorhic half integer weight modular form with integer coefficients. If l is rime, then we shall be interested in congruences of the form a(ln 0 mod l where N is any quadratic residue (res. non-residue modulo l. For every rime l > 3 we exhibit a natural holomorhic weight l + 1 modular form whose coefficients satisfy the congruence a(ln 0 mod l for every N satisfying ` N l = 1. This is roved by using the fact that the Fourier coefficients of these forms are essentially the secial values of real Dirichlet L series evaluated at s = 1 l which are exressed as generalized Bernoulli numbers whose numerators we show are multiles of l. From the works of Carlitz and Leooldt, one can deduce that the Fourier coefficients of these forms are almost always a multile of the denominator of a suitable Bernoulli number. Using these examles as a temlate, we establish sufficient conditions for which the Fourier coefficients of a half integer weight modular form are almost always divisible by a given ositive integer M. We also resent two examles of half-integer weight forms, whose coefficients are determined by the secial values at the center of the critical stri for the quadratic twists of two modular L functions, ossess such congruence roerties. These congruences are related to the non-triviality of the l rimary arts of Shafarevich-Tate grous of certain infinite families of quadratic twists of modular ellitic curves with conductors 11 and 14. 1. Congruences for Fourier coefficients First we shall fix the following notation. If D 0, 1 mod 4 is the fundamental discriminant of the quadratic field Q( D, then let χ D denote the Kronecker character Key words and hrases. congruences, modular forms, secial values of L functions. The third author is suorted by grant DMS-9508976 from the National Science Foundation. 1 Tyeset by AMS-TEX
ANTAL BALOG, HENRI DARMON AND KEN ONO with conductor. It turns out that χ D, in terms of Jacobi symbols, is given by: ( n if D 1 mod 4 ( 1 ( n n if D = 4D 1, D 1 3 mod 4 (0 χ D (n := ( n χd1 (n if D = D 1, D 1 1 mod 4 ( n χ4d1 (n if D = D 1, D 1 3 mod 4. ( ( 1 Here := ( 1 n 1 if n is odd and is zero otherwise, and := ( 1 n 1 8 if n is n n odd and is zero otherwise. By χ 0 we shall mean the identity character. If χ is a Dirichlet character modulo N and k is a ositive integer or k Z 0 + 1, then let M k (N, χ (res. S k (N, χ denote the finite dimensional comlex vector sace of holomorhic modular forms (res. cus forms with resect to Γ 0 (N. Similarly let M k (N (res. S k (N denote the sace of holomorhic modular forms (res. cus forms with resect to Γ 1 (N. If f(z is such a modular form, then it has a Fourier exansion in q := e πiz of the form f(z = a(nq n. n=0 For more on the theory of modular forms see [ ]. The congruence roerties of Fourier coefficients have been investigated by a number of authors (see [ ]. For examle, if (z = τ(nqn is the unique normalized cus form of weight 1 with resect to SL (Z, then it is well known that and τ(n σ 11 (n mod 691 (1 τ(n 0 mod 3 if ( n = 1 3 where σ 11 (n := 0<d n d11. These congruences are exlained by the theory of modular l adic Galois reresentations as develoed by Deligne, Serre, and Swinnerton Dyer. Still there are other examles of congruences for Fourier coefficients which have been the focus of some attention. For examle, if (n denotes the number of artitions of n, then it is well known that (5n + 4 0 mod 5, ( (7n + 5 0 mod 7, and (11n + 6 0 mod 11 for every non-negative integer n. These congruences may be viewed as consequences of the action of certain Hecke oerators. Here we illustrate this fact for (. To see this we construct a holomorhic integer weight modular form whose coefficients inherit these
CONGRUENCES FOR SPECIAL VALUES 3 congruence roerties. Recall that Euler s generating function for (n is given by the infinite roduct 1 1 q n. Also recall that η(z := q 1 4 (1 qn, the Dedekind eta-function, is a weight 1 cus form. If we define F 7 (z and a(n by F 7 (z = η7 (7z η(z = q (1 q 7n 7 (1 q n = a(nq n, n= then (1 holds if and only if a(7n 0 mod 7 for every ositive integer n. However F 7 (z is a holomorhic modular form in M 3 (7, χ 7 and may be rewritten as F 7 (z = 1 8 E 3(z 1 8 η3 (zη 3 (7z where E 3 (z = σ 3,χ 7 (nq n and σ 3,χ 7 (n = 0<d n χ 7(d n d. It is easy to verify that F 7 (z T 7 = a(7nq n = 49 8 E 3(z + 7 8 η3 (zη 3 (7z; this imlies that a(7n 0 mod 7 for all n which imlies (. However there are other congruences that the artition function satisfies. For examle it is known that (49n + 19 (49n + 33 (49n + 40 0 mod 49 for every non-negative integer n (see [ ]. In a more convenient form, (49n + 7δ 0 mod 49 for all n if δ is a quadratic non-residue modulo 7. Following an argument similar to the one above, this means that the Fourier coefficients of the holomorhic integer weight modular form satisfies the congruence η 49 (49z η(z = q 100 (1 q 49n 49 1 q n = a(49n + 7δ 0 mod 49 a(nq n for all n where δ is a quadratic non-residue modulo 7. We shall be interested in the arithmetic imlications of other congruences of this tye. n=0
4 ANTAL BALOG, HENRI DARMON AND KEN ONO Definition. Let F (n be an integer valued arithmetic function, M a ositive integer, and l a rime. If F (ln 0 mod M for every ositive integer N that is a quadratic residue (res. non-residue modulo l, then we say that F has a quadratic congruence modulo M of tye (l, +1 (res. (l, 1. Therefore if F (n = (n, then F has a quadratic congruence modulo 49 of tye (7, 1. Moreover by (1 we see that the Fourier coefficients of (3z satisfy the quadratic congruence modulo 3 of tye (3, 1. In the following roosition we will give an exlicit criterion for determining the truth of certain congruences of this tye that will be used in the sequel. Proosition 1. Let f(z = n=0 a(nqn M k (N, χ 0 be a holomorhic integer weight modular form with rational integer coefficients. If l is a rime dividing N, then the coefficients ossess a quadratic congruence of tye (l, +1 (res. (l, 1 if and only if a(ln 0 N mod l for every ositive integer n C satisfying ( ( n l = 1 (res. n l = 1 where (3 C := knl3 ( 1 + 1. 1 Proof. In [ ], Sturm roves that if f(z = n=0 a(nqn, and g(z = n=0 b(nqn M k (N, χ 0 have algebraic integer Fourier coefficients, then f(z g(z mod M if a(n b(n for every n kn 1 N (1 + 1. Define f 1 (z and f (z by f 1 (z := f(z T l a(lnq n mod l and n=0 f (z := f(z T l a(l nq n mod l. n=0 Since the Hecke oerators reserve saces of modular forms, we find that f 1 (z, f (z M k (N, χ 0. If f 3 (z = f (lz M k (Nl, χ 0, then f 4 (z := f 1 (z f 3 (z M k (Nl, χ 0 and f 4 (z a(lnq n mod l. n = 0 (n, l = 1 Now let f 5 (z denote the modular form that is the quadratic twist of f 4 (z by ( n l ; therefore we find that f 5 (z n = 0 (n, l = 1 ( n a(lnq n mod l. l
CONGRUENCES FOR SPECIAL VALUES 5 By [ ] it turns out that f 5 (z M k ( Nl 3, χ 0. Therefore if we define the modular forms f + (z and f (z by f + (z := 1 (f 4(z + f 5 (z and f (z := 1 (f 4(z f 5 (z, then we will find that f + (z 0 mod l (res. f (z 0 mod l if and only if the Fourier coefficients of f(z satisfy a quadratic congruence modulo l of tye (l, +1 (res. (l, 1. The claim then follows from Sturm s theorem if we let g(z = 0.. Secial values of L(s, χ D In this section we resent a number of examles of such congruences which involve secial values of real Dirichlet L functions at negative integers. If χ is a Dirichlet character with conductor f and n is any ositive integer, then it is well known that L(1 n, χ = B n,χ where B n,χ is the n th generalized Bernoulli n number with character χ defined by f χ(ate at e ft 1 = t n B n,χ n!. The roerties of these numbers are imortant in the construction of adic L-functions (see [ ]. These numbers also occur in congruences involving the secial values of L functions of ellitic curves with comlex multilication and also the class numbers of real quadratic fields (see [ ]. In [ ] H. Cohen exlicitly constructed holomorhic modular forms of half integer weight whose Fourier coefficients are exlicit exressions involving the secial values at negative integers of Dirichlet L functions of quadratic characters. Fix a ositive integer r. If N is a ositive integer and Dn = ( 1 r N where D is the fundamental discriminant of a quadratic number field, then define H(r, N by (4 H(r, N := L(1 r, χ D d n n=0 µ(dχ D (dd r 1 σ r 1 ( n d. If N = 0, then let H(r, 0 := ζ(1 r; otherwise let H(r, N := 0. In articular, if D = ( 1 r N is the discriminant of a quadratic field, then (5 H(r, N = L(1 r, χ D = B r,χ D. r If r and F r (z := n=0 H(r, Nqn, then F r (z M r+ 1 (4, χ 0 (see [Th. 3.1, Cohen]. We shall show that lots of these modular forms have the desired congruence roerties. First we need the following little lemma:
6 ANTAL BALOG, HENRI DARMON AND KEN ONO Lemma 1. Let > 3 be a rime and suose that D is a fundamental discriminant of Q( D of the form D = ( 1 +1 N where N is a ositive integer satisfying ( N = 1. Then χ D (aa +1 0 mod. Proof of Lemma 1. From (0 and the law of quadratic recirocity, we may factor χ D (n as ( n χ D (n = χ(n where χ is a non-trivial real character satisfying χ( = ( N. Therefore the sum, which we denote T, may be rewritten as T := χ D (aa +1 = N ( a χ(aa +1. We first show that T 0 mod N, a fact which does not deend on the condition that ( N = 1. We slit T into residue classes modulo N, and by the Binomial Theorem deduce that T = 1 b=0 r=1 N ( bn + r N r=1 χ(rr +1 χ(r(bn + r +1 1 ( bn + r b=0 mod N. Since ( n is a non-trivial Dirichlet character with conductor and since gcd(n, = 1, we find that 1 ( bn + r 1 ( d = = 0. b=0 Therefore it is easy to see that T 0 mod N. Using the same argument, it is also easy to deduce that T 0 mod. However to comlete the roof we need to establish that T 0 mod since. To establish this claim, we slit the sum T into residue classes modulo and from the Binomial Theorem we find that T = c=0 s=1 c=0 s=1 d=0 ( c + s χ(c + s(c + s +1 ( ( s χ(c + s s +1 + + 1 cs 1 mod.
CONGRUENCES FOR SPECIAL VALUES 7 Since χ(n is a non-trivial Dirichlet character with conductor N, and gcd(n, = 1, we find that for every integer s (6 Therefore we find that ( s T s +1 Since s=1 c=0 ( + 1 c=0 χ(c + s = χ(c + s + c=0 s=1 c=0 d=0 s=1 χ(d = 0. ( s χ(c + scs 1 mod. ( s s 1 1 mod, for every integer s 0 mod then Finally we show that T ( + 1 S := c=0 s=1 c=0 s=1 ( s χ(c + s( + 1cs 1 mod 1 χ(c + sc mod. 1 N χ(c + sc = (1 χ( χ(bb. This imlies that T S 0 mod since χ( = ( N roof. Note that this identity holds for arbitrary Dirichlet characters. By (6 we have S = 1 1 χ(c + sc + 1 1 c=0 s=1 = 1 = 1 = 1 c=0 s=1 s=1 s b=1 = 1 which would comlete the χ(c + s = c=0 1 χ(c + s(c + s = c=0 s=1 N χ(c + s(c + s 1 χ(aa χ( N χ(rr. r=1 c=0 χ(c + (c + We slit the first sum again into residue classes modulo N and obtain 1 N χ(aa = 1 1 = 1 b=0 r=1 1 b=0 r=1 N χ(r (bn + r = N N χ(rr = χ(rr. r=1
8 ANTAL BALOG, HENRI DARMON AND KEN ONO Therefore we find that T S = 0 mod ; this comletes the roof. Although the congruence roerties of the denominator of ordinary and generalized Bernoulli numbers are well known by the Von Staudt-Clausen tye theorems (see [ and Carlitz], the nature of the rime divisors of the numerators seems to be quite elusive although some results in this direction are known. In the next Lemma we resent circumstances for which a given rime > 3 must divide the numerator of the generalized Bernoulli number B +1,χ D. Theorem 1. If > 3 is rime and D is a fundamental discriminant of Q( D of the form D = ( 1 +1 N where N is a ositive integer satisfying ( N = 1, then B +1,χ 0 mod. D Moreover the secial value L( 1, χ D 0 mod. Proof Theorem 1. It is well known that where B +1 1 B +1,χ = D χ D (ab +1 (x is the +1 st Bernoulli olynomial defined by Therefore one finds that B +1 (x = +1 i=0 B +1,χ = χ D D (a +1 ( +1 i B i x +1 i. i=0 ( a ( +1 i B i i 1 a +1 i. Since > 3 and i +1, by Von Staudt-Clausen it follows that all of the B i in the above sum are integral (i.e. denominators are rime to. Therefore since 0 mod we find that B +1,χ D ( χ D (a B 0 1 a +1 + + 1 a 1 B1 This reduces to B +1,χ B 0 1 χ D D (aa +1 + + 1 ( a B 1 χ D (a mod. mod. Since χ D (n ( n is a character modulo, the second sum is identically zero and hence we find that B +1,χ D 1 B 0 χ D (aa +1 mod. The result now follows as a consequence of (5 and Lemma 1.
CONGRUENCES FOR SPECIAL VALUES 9 Corollary 1. Let l > 3 be rime. Then the Fourier coefficients H( l+1, n of the weight l + 1 modular form F l+1 (z satisfy a quadratic congruence modulo l of tye (l, ( 1 l. Proof Corollary 1. By (5 and Lemma we see that if D if a fundamental discriminant of the form D = ( 1 l+1 ln where N is a ositive integer satisfying ( N = 1, then ( l + 1 H, ln B l+1 =,χ D l + 1 0 mod l. Then by (4 it is easy to deduce that for every integer n that ( l + 1 H, lnn 0 mod l. Therefore it follows that ( l + 1 H, M 0 mod l ( m for every ositive integer M = lm where = 1. However this is recisely the ( ( l m 1 condition that =. Therefore the modular form F l+1 (z satisfies a quadratic l l congruence modulo l of tye (l, ( 1 l. There are various other congruences for H(r, N which are not of this tye which are also of interest. To illustrate this we now rove the following theorem. Theorem. For every ositive integer N 1 mod 5 the function H(5, N satisfies the congruence H(5, N 0 mod 5 for every ositive integer N 1 mod 5. Proof. Let f 1 (z := F 5 (zθ(5z M 6 (0, χ 5, its Fourier exansion is given by f 1 (z = n=0 b(nq n = 1 ( Θ 11 (z Θ7 (zη 8 (4z 13 η 4 + 88Θ3 (zη 16 (4z (z η 8 Θ(5z. (z Since Θ(5z = 1 + q5n, it is clear that it suffices to check that b(n 0 mod 5 for every ositive integer n 1 mod 5. Now it is known that f 3 (z = n 1 mod 5 b(nq n l
10 ANTAL BALOG, HENRI DARMON AND KEN ONO is a weight 6 modular form with resect to Γ 1 (500. Therefore by Sturm s theorem (see [ ] it suffices to check that b(n 0 mod 5 for every n 1 mod 5 u to 90000. The congruence has been verified with machine comutation. In [Th 4 ] Carlitz roved that if χ is a rimitive Dirichlet character with conductor f and is a rime for which f and n is a ositive integer for which e n, then e divides the numerator of B n,χ. For examle if D = N is the fundamental discriminant of the quadratic number field Q( D where N 1 mod 5, then this result imlies that the numerator of B 5,χD is a multile of 5. However by Theorem we find that even more is true. We obtain: Corollary. Let D = N be the fundamental discriminant of Q( D where N 1 mod 5 is a ositive integer. Then the numerator of B 5,χD is a multile of 5. Proof Corollary. By (5 and Theorem it follows that H(5, N = L( 4, χ D = B 5,χ D 5 0 mod 5. This immediately imlies that the numerator of B 5,χD is a multile of 5. Now we make some observations regarding the divisibility of the Fourier coefficients of half-integer weight modular forms. In [ ] Serre roved a remarkable theorem regarding the divisibility of the Fourier coefficients of holomorhic integer weight modular forms. Let f(z = n=0 a(nqn be the Fourier exansion of a holomorhic integer weight holomorhic modular form with resect to some congruence subgrou of SL (Z whose coefficients a(n are algebraic integers in a fixed number field. Then he roved that given a ositive integer M, the set of non-negative integers n for which a(n 0 mod M has arithmetic density one. Unfortunately much less is known regarding the divisibility roerties of the coefficients of holomorhic half-integer weight forms. Let r be a non-negative integer and let f(z = n=0 a(nqn M r+ 1 (N, χ with rational integer coefficients. If r = 0, then by the Serre-Stark basis theorem (see [ ], it is known that f(z is a finite linear combination of theta functions. Moreover these functions are of the form Θ a,m (dz where d is a ositive integer and Θ a,m (z := q n. n a mod M In articular for all but a finite number of square-free ositive integers t, it is the case that a(tn = 0 for every integer n. Therefore the number of integers n x for which a(n 0 is O( x. However if r 1, then the situation is very different and is of significant interest. A thorough understanding of the divisilibity roerties of Fourier coefficients when r = 1 will shed some light on the divisors of class numbers of imaginary quadratic fields and the Shafarevich-Tate grous of twists of certain modular ellitic curves. Therefore it is worth examining any analogs of Serre s divisibility result which may hold for half-integer weight forms.
CONGRUENCES FOR SPECIAL VALUES 11 From the works of Carlitz and Leooldt we find that the modular forms F r (z rovide us with an infinite family of interesting modular forms, which are not trivially zero modulo an integer M, for which we find obvious analogs of Serre s divisibility theorem. Both roved that if χ is a Dirichlet character with conductor f that is not a ower of a rime, then L(1 r, χ is an algebraic integer. However if χ is a character with a conductor that is a ower of a rime, then the rime ideal divisors of the denominator of L(1 r, χ are rime ideal divisors of. However if r fixed, it known that the denominators of L(1 r, χ D are bounded which imlies that there are at most finitely many D for which L(1 r, χ D is not an integer. Therefore since the modular form F r (z may be written as F r (z := ζ(1 r n=0 a r (nq n = B r r a r (nq n where the coefficients a r (n are rational with denominators bounded by D r, the least common multile of all the denominators occuring in the a r (n, it follows that the numerator of almost every a r (n is a multile of the denominator of ζ(1 r = B r r. By Von Staudt-Clausen and the Voronoi congruences (see [15..4, Ireland and Rosen], the denominator of B r is ( 1 r. We have roved: Proosition. Let r be a ositive integer. Then for all but finitely many square-free integers t we find that a r (tn 0 mod M for every integer n where M = ( 1 r. In articular, the number of non-negative integers n x for which a r (n 0 mod M is O( x. Using this roosition as a temlate we establish circumstances for which the Fourier coefficients of a half-integer weight modular form are almost always a multile of a ower of a rime. From the discussion above it is clear that we only need to consider those half-integer weight forms with weight 3. First we note that the classical theta function Θ(z = 1 + 1 mod. Therefore if f(z = qn n=0 a(nqn is a holomorhic half-integer weight modular form with integer coefficients, then f(z Θ(z f(z mod is an integer weight holomorhic modular form for which the Fourier coefficients are almost always even by Serre s theorem. Therefore the coefficients a(n are almost always even. Therefore we may assume that is an odd rime. Let f (z be the weight 1 modular form defined by f (z = η (z η(z. It is easy to verify that f (z M 1 (, χ D where D := ( 1 1. More imortantly if s is a ositive integer, then since 1 X (1 X mod, we find that (7 f s (z 1 mod s+1. Using this notation we observe: n=0
1 ANTAL BALOG, HENRI DARMON AND KEN ONO Proosition 3. Let f(z = n=0 a(nqn M r+ 1 (N, χ with rational integer coefficients. Let be an odd rime and let s and k be ositive integers for which If for every cus c d of Γ 0(N r = k s ( 1. Ord(f, c d Nk s+1 ( 4 gcd(d, N/dd gcd(d, where Ord(f, c d is the analytic order of f(z at the cus c d, then for all but finitely many square-free ositive integers t a(tn 0 mod s+1 for every integer n. In articular, the set of non-negative integers n x for which a(n 0 mod s+1 is O( x. k Proof. If r = k s ( 1, then by (7 the weight of (f s (z is exactly 1 more than r + 1, the weight of f(z. The system of inequalities imlies that the modular form f(z ( k η s+1 (z f(z mod s+1 η s (z is holomorhic at all the cuss of Γ 0 (N since the order of f s by (see [ ] N s+1 ( 4 gcd(d, N/ gcd(d,. (z at a cus c d is given Therefore this form is a holomorhic weight 1 form, hence is a finite linear combination of theta functions by the Serre-Stark basis theorem. This comletes the roof. 3. Secial values of modular L functions In this section we investigate the the congruence roerties of secial values of quadratic twists of a modular L function at the center of the critical stri on the real line. By the work of Kohnen, Shimura, Waldsurger, and Zagier, the Fourier coefficients of certain secial half-integer weight forms are essentially (u to a transcendental factor the square-root of these secial values. We now resent two simle examles for which congruences exist. Let L(, s denote the modular L function defined by L(, s := τ(n n s
CONGRUENCES FOR SPECIAL VALUES 13 where (z = τ(nqn is the Fourier exansion of Ramanujan s weight 1 cus form. If χ D is the Kronecker character where D is a fundamental discriminant of Q( D, then let L(, D, s denote the twisted L function defined by L(, D, s := χ D (nτ(n n s. Let g(z = a(nqn be the weight 13 eigenform defined by g(z := Θ9 (zη 8 (4z η 4 (z 18Θ5 (zη 16 (4z η 8 (z + 3Θ(zη4 (8z η 1. (z Using the Shimura corresondence, Kohnen and Zagier [ ] roved a general theorem which in this case imlies that if D is the fundamental discriminant of a real quadratic field, then (8 L(, D, 6 = ( 6 π D < (z, (z > D 5! < g(z, g(z > (a(d where < (z, (z > and < g(z, g(z > are the relevant Peterson scalar roducts. Therefore we shall refer to (a(d as the rational factor of L(, D, 6. With this notation we rove the following congruences for the rational factors of L(, D, 6 : Theorem 3. If N is a ositive integer satisfying ( N 11 = 1, then the Fourier coefficient a(11n satisfies a(11n 0 mod 11. Proof Theorem 3. It suffices to check that the Fourier coefficients a(n satisfy a quadratic congruence modulo 11 of tye (11, 1. Let T (z M 1 (44, χ 0 be defined by T (z := c(nq n = g(z Θ(11z η11 (z η(11z. Since the right hand factor is a modular form whose Fourier exansion is 1 mod 11, it suffices to check that the Fourier exansion of T (z has a quadratic congruence modulo 11 of tye (11, 1. However by Proosition 1 it suffices to check that c(11n 0 mod 11 for all n 9583 that satisfy ( n 11 = 1. This congruence has been verified by machine comutation. Therefore by (8 we obtain:
14 ANTAL BALOG, HENRI DARMON AND KEN ONO Corollary 3. Using the notation above, if D = 11N is the fundamental discriminant of Q( D where N is a ositive integer satisfying ( N 11 = 1, then the rational factor of L(, D, 6 is a multile of 11. Now we resent a second examle of such congruences. In this second examle let f(z := c(nqn = η 8 (zη 8 (z; hence f(z is the unique normalized weight 8 eigenform of level. Let L(f, s denote the modular L function c(n L(f, s := n s. As above, let L(f, D, s be the twisted modular L function defined by χ D (nc(n L(f, D, s := n s. Now let g(z denote the weight 9 eigenform h(z := d(nq n = Θ5 (zη 8 (4z η 4 (z 16Θ(zη16 (4z η 8. (z The Shimura lift of h(z is f(z, hence by [ro??? Wa] it turns out that if N 1 N 5 mod 8 are two ositive square-free integers with corresonding quadratic characters χ D1 and χ D where d(n 1 0, then (9 L(g, D, 4 = d (N L(g, D 1, 4N 7 1. d (N 1 N 7 With this notation we rove: Theorem 4. If N is a ositive integer satisfying ( N 7 = 1, then the Fourier coefficient d(7n satisfies the congruence d(7n 0 mod 7. Proof Theorem 4. It suffices to check that the Fourier coefficients d(n satisfy a quadratic congruence modulo 7 of tye (7, 1. Let S(z M 8 (8, χ 0 be the modular form defined by S(z := b(nq n := h(z Θ(7z η7 (z η(7z. Since the right hand factor is a modular form whose Fourier exansion is 1 mod 7, by Proosition 1 it suffices to check that b(7n 0 mod 7 for all n 10976 where ( n 7 = 1. This has been verified by machine comutation. By (9 we obtain: Corollary ( 4. Let N be a ositive square-free integer for which 7N 5 mod 8 and N 7 = 1. If the character of Q( 7N is χd, then the rational factor of L(f, D, 4 is a multile of 49.
CONGRUENCES FOR SPECIAL VALUES 15 4. An ellitic curve exlanation We interret the congruences of corollaries 3 and 4 in terms of the Bloch-Kato conjecture on secial values of L-functions associated to arithmetic objects of a very general tye (the motives in the sense of Grothendieck and Deligne, as defined for examle in [Deligne]. Motives: It is beyond our scoe to give a detailed account of motives. A good reference for this is [Deligne]. For the urose of our discussion, a motive M (over Q, with rational coefficients, of rank r can be thought of as a iece of the cohomology of an algebraic variety over Q, giving rise to: For each rime l, an l-adic reresentation M l of G Q = Gal( Q/Q, arising from l- adic étale cohomology. The object M l is an r-dimensional Q l -vector sace equied with an action of G Q. Since G Q acts through a comact quotient, it leaves stable a Z l - sublattice T l of M l, and one can define the mod l reresentation associated to M l to be the F l -vector sace M l = T l /lt l. This sace deends on the choice of T l in general, but its semi-simlification does not, by the Brauer-Nesbitt theorem. To obtain a canonical object we simly define M l to be the semi-simlification of T l /lt l. The system {M l } should form a comatible system of rational l-adic reresentations in the sense of [Serre.McGill], I.11. More recisely, the action of G Q is unramified outside S {l}, where S is a fixed finite set of rimes not deending on l. Let D be a decomosition grou at in G Q and let I denote an inertia subgrou of D. Let Frob be the canonical ( frobenius generator of D /I which gives the ma x x on residue fields. If l is a rime, then the characteristic olynomial Z (M, T of Frob acting on M I l Z (M, T = det((1 Frob T M I l has integer coefficients and should not deend on the choice of l. A rational vector sace M B : the so called Betti realization, arising from singular cohomology. A rational vector sace M DR coming from the algebraic DeRham cohomology, equied with its natural Hodge structure. The structures M B and M DR will not lay an exlicit role in our discussion, but are used in defining certain (transcendental eriods associated to M as in [Deligne]. The L-function: One defines the local L function at by L (M, s = Z (M, s 1. By the comatibility axiom, L (M, s does not deend in the choice of the rime l used to define it. One then defines the global L-function as a roduct over all rimes : L(M, s = L (M, s. This Euler roduct converges in a right half lane, by the Weil conjectures. It is conjectured that L(M, s has an analytic continuation and a functional equation (cf. [Deligne], 1.. We will assume this. (In the secial cases that we discuss, this conjecture is known to be true. The Bloch-Kato conjectures: Under the asumtion that the motive is critical (in the sense of [Deligne], Def. 1.3, Deligne has given a very general conjectural formula for
16 ANTAL BALOG, HENRI DARMON AND KEN ONO the secial value L(M, 0, modulo rational multiles, in terms of a certain eriod integral defined in terms of the structures M B and M DR. This conjecture has been refined by Bloch and Kato, and redicts that the rational art L rat (M, 0 of L(M, 0 (i.e., the secial value L(M, 0 divided by the Deligne eriod can be interreted as the order of a certain Selmer grou Sel(M = l Sel l (M, where Sel l (M := ker ( H 1 (Q, M l Q l /Z l v H 1 f (Q v, M l Q l /Z l. The grous Hf 1(Q v, M l Q l /Z l are certain subgrous of the local Galois cohomology grous H 1 (Q v, M l Q l /Z l. When v / S {l} then these are exactly the unramified cohomology classes. When v = l, the definition of Hf 1 is more subtle and relies on the crystalline cohomology theory develoed by Fontaine and Messing. See [Bloch.Kato] for details. Note that in general Sel(M need not be finite. If it is infinite, one conjectures that L rat (M, 0 = 0. Congruences between motives: We say that two motives M and N are congruent modulo l if the local L-factors Z (M, T and Z (N, T are congruent mod l for all rimes l. In articular, it follows from the Chebotarev density theorem that the mod l Galois reresentations M l and N l associated to M and N are isomorhic. For examle, modular forms whose Fourier coefficients are congruent modulo l give rise to congruent motives. (See, for examle, the discussion in the introduction to [Mazur]. Generally seaking, motivated by the hilosohy exressed in [Mazur], one exects a mod l congruence between two motives to translate into a congruence modulo l between the secial values of their associated L-functions. Conjecture C. If two motives M and N are congruent modulo l, then L rat (M, 0 L rat (N, 0 (mod l. The motives associated to L(, D, 6 and L(f, D, 4: Let M and M f denote the motives associated by Scholl [scholl] to the modular forms S 1 (SL (Z and f S 8 (Γ 0 ( resectively. By definition, we have: L(M, s = L(, s, L(M f, s = L(f, s. Set M,D,6 = M ( 6 M χd, M f,d,4 = M f ( 4 M χd. Here M(n is the n-th Tate twist of M, as defined in [Deligne], and M χd motive associated to the Dirichlet character χ D. From the definitions it follows immediately that we have is the Artin L(M,D,6, 0 = L(, D, 6, L(M f,d,4, 0 = L(f, D, 4.
CONGRUENCES FOR SPECIAL VALUES 17 A congruence for M,D,6 : Suose that D = 11N, with N rime to 11. Let E 11 be the ellitic curve X 0 (11. It is the (unique modular ellitic curve of conductor 11, and its associated eigenform f 11 S (Γ 0 (11 is f 11 = η(z η(11z = q (1 q n (1 q 11n = a n (E 11 q n. Let M E11,χ,1 be the motive satisfying L(M E11,χ,1, 0 = L(E 11, χ, 1, where χ is the Dirichlet character associated to the imaginary quadratic field Q( N. Then we have: Lemma. The motives M,D,6 and M E11,χ,1 are congruent modulo 11. Proof: We have = q (1 q n 4 q (1 q n (1 q 11n f 11 (mod 11, so that τ(n a n (E 11 (mod 11 for all n. Moreover, we have and the lemma follows directly from this. χ D ( 5 χ( (mod 11, lemma 3. If 11 is slit in the field Q( N, then L(M E11,χ,1, 0 = 0. Proof: This follows from the calculation of the sign in the functional equation for L(E 11, χ, s, which can be shown to be 1 when the conductor of E slits in the quadratic field Q( N. (See for examle [gross.zagier]. Since 1 is the symmetry oint for the functional equation, it follows that L(E 11, χ, 1 = L(M E11,χ,1, 0 = 0. Corollary. Assume conjecture C. Then L(, D, 6 0 (mod 11 whenever 11 is slit in Q( N. Proof: Since M,D,6 and M E11,χ,1 are congruent modulo 11 by lemma, conjecture C imlies the congruence L(, D, 6 = L(M,D,6, 0 L(M E11,χ,1, 0 = 0 (mod 11, where the last equality follows from lemma 3. A congruence for M f,d,4 : Suose that D = 7N with 7 not dividing N. Let E 14 be the ellitic curve X 0 (14. It is the (unique modular ellitic curve of conductor 14, and its associated eigenform f 14 S (Γ 0 (11 is f 14 = η(zη(zη(7zη(14z = q (1 q n (1 q n (1 q 7n (1 q 14n. Let M E14,χ,1 be the motive satisfying L(M E14,χ,1, 0 = L(E 14, χ, 1, where χ is the Dirichlet character corresonding to the quadratic imaginary field Q( N. Then we have:
18 ANTAL BALOG, HENRI DARMON AND KEN ONO Lemma 4. The motives M f,d,4 and M E11,χ,1 are congruent modulo 7. Proof: We have f = q (1 q n 8 (1 q n 8 q (1 q n (1 q n (1 q 7n (1 q 14n = f 14 (mod 14. The lemma follows directly from this, and from the fact that χ D ( 3 χ( (mod 7. Lemma 5. If and 7 are both slit or both inert in the field Q( N, then L(M E14,χ,1, 0 = 0. Proof: This follows from the calculation of the sign in the functional equation for L(E 14, χ, s, which can be shown to be 1 when χ(14 = 1. (See for examle [gross.zagier]. Since 1 is the symmetry oint for the functional equation, it follows that L(E 14, χ, 1 = L(M E14,χ,1, 0 = 0. Corollary. Assume conjecture C. Then L(f, D, 4 0 (mod 7 whenever and 7 are both slit or both inert in Q( N. Proof: By lemma 4, the motives M f,d,4 and M E14,χ,1 are congruent modulo 7. Conjecture C imlies the congruence L(f, D, 4 = L(M f,d,4, 0 L(M E14,χ,1, 0 = 0 (mod 7, where the last equality is by lemma 5. Acknowledgements The authors thank Will Galway (University of Illinois for his assistance regarding the machine comutations which were required for this aer. References?. G. Andrews, The theory of artitions, Addison-Wesley, 1976.?. A. Biagioli, The construction of modular forms as roducts of transforms of the Dedekind Eta function, Acta. Arith. 54 (1990, 73-300.?. L. Carlitz, Arithmetic roerties of generalized Bernoulli numbers, J. Reine und Angew. Math. 01-0 (1959, 173-18.?. H. Cohen, Sums involving the values at negative integers of L functions of quadratic characters, Math. Ann. 17 (1975, 71-85.?. D. Eichhorn and K. Ono, Partition function congruences, rerint.?. B. Gordon, rivate communication.?. K. Ireland and M. Rosen, A classical introduction to modern number theory, Sringer-Verlag, 198.?. K. Iwasawa, Lectures on adic L functions, Princeton Univ. Press, 197.?. M. Kno, Modular functions in analytic number theory, Markham, 1970.?. N. Koblitz, Introduction to ellitic curves and modular forms, Sringer-Verlag, 1984.?. W. Kohnen and D. Zagier, Values of L series of modular forms at the center of the critical stri, Invent. Math. 64 (1981, 173-198.
CONGRUENCES FOR SPECIAL VALUES 19?. H. Leooldt, Eine verallgemeinerung der Bernoullischen zahlen, Abh. Math. Sem. Univ. Hamburg (1958, 131-140.?. S. Ramanujan, Congruence roerties of artitions, Proc. London Math. Soc. ( 18 (??, 19-0.?. K. Rubin, Congruences for secial values of L functions of ellitic curves with comlex multilication, Invent. Math. 71 (1983, 339-364.?. J.-P. Serre and H. Stark, Modular forms of weight 1, Sringer Lect. Notes Math. 67 (1977, 7-68.?. J.-P. Serre, Congruences ét formes modulaires (d arés H.P.F. Swinnerton-Dyer, Seminaire Bourbaki 416 (1971.?., Divisibilité des coefficients des formes modulaires, C.R. Acad. Sci. Paris (A 79 (1974, 679-68.?. G. Shimura, Introduction to the arithmetic theory of automorhic functions, Princeton Univ. Press, 1971.?. On modular forms of half-integral weight, Ann. Math. 97 (1973, 440-481.?. J. Sturm, On the congruence of modular forms, Sringer Lect. Notes Math. 140 (1984.?. H.P.F. Swinnerton-Dyer, On l-adic reresentations and congruences for coefficients of modular forms, Sringer Lect. Notes Math. 350 (1973.?. J.L. Waldsurger, Sur les coefficients de Fourier des formes modulaires de oids demi-entier, J. Math. Pures et Al. 60 (1981, 375-484.?. L. Washington, Introduction to cyclotomic fields, Sringer-Verlag, 1980. Deartment of Mathematics, University of Illinois, Urbana, Illinois 61801 Current address: Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 17, Budaest 1364, Hungary E-mail address: h1165bal@ella.hu Deartment of Mathematics, McGill University, Montréal, Canada PQ H3A K6 Current address: Deartment of Mathematics, Princeton University, Princeton, New Jersey 08540 E-mail address: darmon@math.rinceton.edu Deartment of Mathematics, University of Illinois, Urbana, Illinois 61801 Current address: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 E-mail address: ono@math.ias.edu