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 point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let f(z) = ∑_{n=0}^{∞} a(n)q^{n} be a holomorphic half integer weight modular form with integer coefficients. If ℓ is prime, then we shall be interested in congruences of the form

a(ℓN) ≡ 0 mod ℓ

where N is any quadratic residue (resp. non-residue) modulo ℓ. For every prime ℓ > 3 we exhibit a natural holomorphic weight \frac{ℓ}{2} + 1 modular form whose coefficients satisfy the congruence a(ℓN) ≡ 0 mod ℓ for every N satisfying (\frac{−N}{ℓ}) = 1. This is proved by using the fact that the Fourier coefficients of these forms are essentially the special values of real Dirichlet L−series evaluated at s = \frac{1−ℓ}{2} which are expressed as generalized Bernoulli numbers whose numerators we show are multiples of ℓ. ¿From the works of Carlitz and Leopoldt, one can deduce that the Fourier coefficients of these forms are almost always a multiple of the denominator of a suitable Bernoulli number. Using these examples as a template, we establish sufficient conditions for which the Fourier coefficients of a half integer weight modular form are almost always divisible by a given positive integer M. We also present two examples of half-integer weight forms, whose coefficients are determined by the special values at the center of the critical strip for the quadratic twists of two modular L−functions, possess such congruence properties. These congruences are related to the non-triviality of the ℓ−primary parts of Shafarevich-Tate groups of certain infinite families of quadratic twists of modular elliptic 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(\sqrt{D}), then let χ_{D} denote the Kronecker character

Key words and phrases. congruences, modular forms, special values of L−functions. The third author is supported by grant DMS-9508976 from the National Science Foundation.

Typeset by AMS-TEX

with conductor |D|. It turns out that χ_{D}, in terms of Jacobi symbols, is given by:

(0)  χ_{D}(n) := \{ \begin{matrix}(\frac{n}{|D|}) if D ≡ 1 mod 4 \\ (\frac{−1}{n})(\frac{n}{|D|}) if D = 4D_{1}, D_{1} ≡ 3 mod 4 \\ (\frac{2}{n})χ_{D_{1}}(n) if D = 2D_{1}, D_{1} ≡ 1 mod 4 \\ (\frac{2}{n})χ_{4D_{1}}(n) if D = 2D_{1}, D_{1} ≡ 3 mod 4. \end{matrix}

Here (\frac{−1}{n}) := (−1)^{\frac{n−1}{2}} if n is odd and is zero otherwise, and (\frac{2}{n}) := (−1)^{\frac{n^{2}−1}{8}} if n is

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 positive integer or k ∈ ℤ_{≥0} + \frac{1}{2}, then let M_{k}(N, χ) (resp. S_{k}(N, χ)) denote the finite dimensional complex vector space of holomorphic modular forms (resp. cusp forms) with respect to Γ_{0}(N). Similarly let M_{k}(N) (resp. S_{k}(N)) denote the space of holomorphic modular forms (resp. cusp forms) with respect to Γ_{1}(N). If f(z) is such a modular form, then it has a Fourier expansion in q := e^{2πiz} of the form

f(z) = ∑_{n=0}^{∞} a(n)q^{n}.

For more on the theory of modular forms see [ ].

The congruence properties of Fourier coefficients have been investigated by a number of authors (see [ ]). For example, if Δ(z) = ∑_{n=1}^{∞} τ(n)q^{n} is the unique normalized cusp form of weight 12 with respect to SL_{2}(ℤ), then it is well known that

τ(n) ≡ σ_{11}(n) mod 691

and

(1)  τ(n) ≡ 0 mod 23 if (\frac{n}{23}) = −1

where σ_{11}(n) := ∑_{0<d|n} d^{11}. These congruences are explained by the theory of modular ℓ−adic Galois representations as developed by Deligne, Serre, and Swinnerton Dyer.

Still there are other examples of congruences for Fourier coefficients which have been the focus of some attention. For example, if p(n) denotes the number of partitions of n, then it is well known that

p(5n + 4) ≡ 0 mod 5,
(2)  p(7n + 5) ≡ 0 mod 7,

and

p(11n + 6) ≡ 0 mod 11

for every non-negative integer n. These congruences may be viewed as consequences of the action of certain Hecke operators. Here we illustrate this fact for (2). To see this we construct a holomorphic integer weight modular form whose coefficients inherit these congruence properties. Recall that Euler’s generating function for p(n) is given by the infinite product

∏_{n=1}^{∞} \frac{1}{1 − q^{n}} .

Also recall that η(z) := q^{\frac{1}{24}} ∏_{n=1}^{∞}(1 − q^{n}), the Dedekind eta-function, is a weight \frac{1}{2} cusp form. If we define F_{7}(z) and a(n) by

F_{7}(z) = \frac{η^{7}(7z)}{η(z)} = q^{2} ∏_{n=1}^{∞} \frac{(1 − q^{7n})^{7}}{(1 − q^{n})} = ∑_{n=2}^{∞} a(n)q^{n},

then (1) holds if and only if

a(7n) ≡ 0 mod 7

for every positive integer n. However F_{7}(z) is a holomorphic modular form in M_{3}(7, χ_{−7}) and may be rewritten as

F_{7}(z) = \frac{1}{8}E_{3}(z) − \frac{1}{8}η^{3}(z)η^{3}(7z)

where E_{3}(z) = ∑_{n=1}^{∞} σ_{3,χ_{−7}}(n)q^{n} and σ_{3,χ_{−7}}(n) = ∑_{0<d|n} χ_{−7}(d)\frac{n^{2}}{d^{2}} .

It is easy to verify that

F_{7}(z) | T_{7} = ∑_{n=1}^{∞} a(7n)q^{n} = \frac{49}{8}E_{3}(z) + \frac{7}{8}η^{3}(z)η^{3}(7z);

this implies that a(7n) ≡ 0 mod 7 for all n which implies (2).

However there are other congruences that the partition function satisfies. For example it is known that

p(49n + 19) ≡ p(49n + 33) ≡ p(49n + 40) ≡ 0 mod 49

for every non-negative integer n (see [ ]). In a more convenient form, p(49n + 7δ − 2) ≡ 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 holomorphic integer weight modular form

\frac{η^{49}(49z)}{η(z)} = q^{100} ∏_{n=1}^{∞} \frac{(1 − q^{49n})^{49}}{1 − q^{n}} = ∑_{n=0}^{∞} a(n)q^{n}

satisfies the congruence

a(49n + 7δ) ≡ 0 mod 49

for all n where δ is a quadratic non-residue modulo 7.

We shall be interested in the arithmetic implications of other congruences of this type.

Definition. Let F (n) be an integer valued arithmetic function, M a positive integer, and ℓ a prime. If

F (ℓN) ≡ 0 mod M

for every positive integer N that is a quadratic residue (resp. non-residue) modulo ℓ, then we say that F has a quadratic congruence modulo M of type (ℓ, +1) (resp. (ℓ, −1)).

Therefore if F (n) = p(n − 2), then F has a quadratic congruence modulo 49 of type (7, −1). Moreover by (1) we see that the Fourier coefficients of Δ(23z) satisfy the quadratic congruence modulo 23 of type (23, −1).

In the following proposition we will give an explicit criterion for determining the truth of certain congruences of this type that will be used in the sequel.

Proposition 1. Let f(z) = ∑_{n=0}^{∞} a(n)q^{n} ∈ M_{k}(N, χ_{0}) be a holomorphic integer weight modular form with rational integer coefficients. If ℓ is a prime dividing N, then the coefficients possess a quadratic congruence of type (ℓ, +1) (resp. (ℓ, −1)) if and only if

a(ℓn) ≡ 0 mod ℓ

for every positive integer n ≤ C satisfying (\frac{n}{ℓ}) = 1 (resp. (\frac{n}{ℓ}) = −1) where

(3)  C := \frac{kNℓ^{3}}{12} ∏_{p|N} (1 + \frac{1}{p}) .

Proof. In [ ], Sturm proves that if f(z) = ∑_{n=0}^{∞} a(n)q^{n}, and g(z) = ∑_{n=0}^{∞} b(n)q^{n} ∈ 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 ≤ \frac{kN}{12} ∏_{p|N}(1 + \frac{1}{p}).

Define f_{1}(z) and f_{2}(z) by

f_{1}(z) := f(z)|T_{ℓ} ≡ ∑_{n=0}^{∞} a(ℓn)q^{n} mod ℓ

and

f_{2}(z) := f(z)|T_{ℓ^{2}} ≡ ∑_{n=0}^{∞} a(ℓ^{2}n)q^{n} mod ℓ.

Since the Hecke operators preserve spaces of modular forms, we find that f_{1}(z), f_{2}(z) ∈ M_{k} (N, χ_{0}) . If f_{3}(z) = f_{2}(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) ≡ ∞ ; ∑ ; \begin{matrix}n = 0 \\ (n, ℓ) = 1 \end{matrix} a(ℓn)q^{n} mod ℓ.

Now let f_{5}(z) denote the modular form that is the quadratic twist of f_{4}(z) by (\frac{n}{ℓ}); therefore we find that

f_{5}(z) ≡ ∞ ; ∑ ; \begin{matrix}n = 0 \\ (n, ℓ) = 1 \end{matrix} (\frac{n}{ℓ})a(ℓn)q^{n} mod ℓ.

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) := \frac{1}{2} (f_{4}(z) + f_{5}(z))

and

f_{−}(z) := \frac{1}{2} (f_{4}(z) − f_{5}(z)) ,

then we will find that f_{+}(z) ≡ 0 mod ℓ (resp. f_{−}(z) ≡ 0 mod ℓ) if and only if the Fourier coefficients of f(z) satisfy a quadratic congruence modulo ℓ of type (ℓ, +1) (resp. (ℓ, −1)).

The claim then follows from Sturm’s theorem if we let g(z) = 0. □

2. Special values of L(s, χ_{D})

In this section we present a number of examples of such congruences which involve special values of real Dirichlet L−functions at negative integers.

If χ is a Dirichlet character with conductor f and n is any positive integer, then it

is well known that L(1 − n, χ) = −\frac{B_{n,χ}}{n} where B_{n,χ} is the n^{th} generalized Bernoulli number with character χ defined by

∑_{a=1}^{f} \frac{χ(a)te^{at}}{e^{ft} − 1} = ∑_{n=0}^{∞} B_{n,χ} \frac{t^{n}}{n!} .

The properties of these numbers are important in the construction of p−adic L-functions (see [ ]). These numbers also occur in congruences involving the special values of L−functions of elliptic curves with complex multiplication and also the class numbers of real quadratic fields (see [ ]).

In [ ] H. Cohen explicitly constructed holomorphic modular forms of half integer weight whose Fourier coefficients are explicit expressions involving the special values at negative integers of Dirichlet L−functions of quadratic characters.

Fix a positive integer r. If N is a positive integer and Dn^{2} = (−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} μ(d)χ_{D}(d)d^{r−1}σ_{2r−1} (\frac{n}{d}) .

If N = 0, then let H(r, 0) := ζ(1 − 2r); otherwise let H(r, N) := 0. In particular, if D = (−1)^{r}N is the discriminant of a quadratic field, then

(5)  H(r, N) = L(1 − r, χ_{D}) = −\frac{B_{r,χ_{D}}}{r} .

If r ≥ 2 and F_{r}(z) := ∑_{n=0}^{∞} H(r, N)q^{n}, then F_{r}(z) ∈ M_{r+ \frac{1}{2}} (4, χ_{0}) (see [Th. 3.1, Cohen]). We shall show that lots of these modular forms have the desired congruence properties. First we need the following little lemma:

Lemma 1. Let p > 3 be a prime and suppose that D is a fundamental discriminant of ℚ(\sqrt{D}) of the form D = (−1)^{\frac{p+1}{2}} pN where N is a positive integer satisfying (\frac{−N}{p}) = 1. Then

∑_{a=1}^{|D|} χ_{D}(a)a^{\frac{p+1}{2}} ≡ 0 mod p|D|.

Proof of Lemma 1. ¿From (0) and the law of quadratic reciprocity, we may factor χ_{D}(n) as

χ_{D}(n) = (\frac{n}{p})χ(n)

where χ is a non-trivial real character satisfying χ(p) = (\frac{−N}{p}). Therefore the sum, which we denote T, may be rewritten as

T := ∑_{a=1}^{|D|} χ_{D}(a)a^{\frac{p+1}{2}} = ∑_{a=1}^{pN} (\frac{a}{p})χ(a)a^{\frac{p+1}{2}} .

We first show that T ≡ 0 mod N, a fact which does not depend on the condition that (\frac{−N}{p}) = 1. We split T into residue classes modulo N, and by the Binomial Theorem deduce that

T = ∑_{b=0}^{p−1} ∑_{r=1}^{N} (\frac{bN + r}{p})χ(r)(bN + r)^{\frac{p+1}{2}} ≡

≡ ∑_{r=1}^{N} χ(r)r^{\frac{p+1}{2}} ∑_{b=0}^{p−1} (\frac{bN + r}{p}) mod N.

Since (\frac{n}{p}) is a non-trivial Dirichlet character with conductor p and since gcd(N, p) = 1, we find that

∑_{b=0}^{p−1} (\frac{bN + r}{p}) = ∑_{d=0}^{p−1} (\frac{d}{p}) = 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 p. However to complete the proof we need to establish that T ≡ 0 mod p^{2} since p||D|.

To establish this claim, we split the sum T into residue classes modulo p and from the Binomial Theorem we find that

T = ∑_{c=0}^{N−1} ∑_{s=1}^{p} (\frac{cp + s}{p})χ(cp + s)(cp + s)^{\frac{p+1}{2}} ≡

≡ ∑_{c=0}^{N−1} ∑_{s=1}^{p} (\frac{s}{p})χ(cp + s) (s^{\frac{p+1}{2}} + \frac{p + 1}{2} · cps^{\frac{p−1}{2}} ) mod p^{2}.

Since χ(n) is a non-trivial Dirichlet character with conductor N, and gcd(N, p) = 1, we find that for every integer s

(6)  ∑_{c=0}^{N−1} χ(cp + s) = ∑_{d=0}^{N−1} χ(D) = 0.

Therefore we find that

2T ≡ 2 ∑_{s=1}^{p}(\frac{s}{p})s^{\frac{p+1}{2}} ∑_{c=0}^{N−1} χ(cp + s) + ∑_{c=0}^{N−1} ∑_{s=1}^{N−1} (\frac{s}{p})χ(cp + s)(p + 1)cps^{\frac{p−1}{2}} mod p^{2}

≡ (p + 1) ∑_{c=0}^{N−1} ∑_{s=1}^{p} (\frac{s}{p})χ(cp + s)cps^{\frac{p−1}{2}} mod p^{2}.

Since (\frac{s}{p})s^{\frac{p−1}{2}} ≡ 1 mod p, for every integer s ≢ 0 mod p then

2T ≡ (p + 1) ∑_{c=0}^{N−1} ∑_{s=1}^{p−1} χ(cp + s)cp mod p^{2}.

Finally we show that

S := ∑_{c=0}^{N−1} ∑_{s=1}^{p−1} χ(cp + s)c = (1 − χ(p)) ∑_{b=1}^{N} χ(b)b.

This implies that 2T ≡ S ≡ 0 mod p^{2} since χ(p) = (\frac{−N}{p}) = 1 which would complete the proof.

Note that this identity holds for arbitrary Dirichlet characters. By (6) we have

S = \frac{1}{p} ∑_{c=0}^{N−1} ∑_{s=1}^{p−1} χ(cp + s)cp + \frac{1}{p} ∑_{s=1}^{p−1} s ∑_{c=0}^{N−1} χ(cp + s) =

=\frac{1}{p} ∑_{c=0}^{N−1} ∑_{s=1}^{p−1} χ(cp + s)(cp + s) =

=\frac{1}{p} ∑_{c=0}^{N−1} ∑_{s=1}^{p} χ(cp + s)(cp + s) − \frac{1}{p} ∑_{c=0}^{N−1} χ(cp + p)(cp + p)

=\frac{1}{p} ∑_{a=1}^{pN} χ(a)a − χ(p) ∑_{r=1}^{N} χ(r)r.

We split the first sum again into residue classes modulo N and obtain

\frac{1}{p} ∑_{a=1}^{pN} χ(a)a = \frac{1}{p} ∑_{b=0}^{p−1} ∑_{r=1}^{N} χ(r) (bN + r) =

= \frac{1}{p} ∑_{b=0}^{p−1} ∑_{r=1}^{N} χ(r)r = ∑_{r=1}^{N} χ(r)r.

Therefore we find that 2T ≡ S = 0 mod p^{2}; this completes the proof.

□

Although the congruence properties of the denominator of ordinary and generalized Bernoulli numbers are well known by the Von Staudt-Clausen type theorems (see [ and Carlitz]), the nature of the prime divisors of the numerators seems to be quite elusive although some results in this direction are known. In the next Lemma we present circumstances for which a given prime p > 3 must divide the numerator of the generalized Bernoulli number B_{\frac{p+1}{2} ,χ_{D}}.

Theorem 1. If p > 3 is prime and D is a fundamental discriminant of ℚ(\sqrt{D}) of the form D = (−1)^{\frac{p+1}{2}} pN where N is a positive integer satisfying (\frac{−N}{p}) = 1, then

B_{\frac{p+1}{2} ,χ_{D}} ≡ 0 mod p.

Moreover the special value L(\frac{1−p}{2}, χ_{D}) ≡ 0 mod p.

Proof Theorem 1. It is well known that

B_{\frac{p+1}{2} ,χ_{D}} = |D|^{\frac{p−1}{2}} ∑_{a=1}^{|D|} χ_{D}(a)B_{\frac{p+1}{2}} (\frac{a}{|D|})

where B_{\frac{p+1}{2}} (x) is the \frac{p+1}{2}^{st} Bernoulli polynomial defined by

B_{\frac{p+1}{2}} (x) = ∑_{i=0}^{\frac{p+1}{2}} ( i^{\frac{p+1}{2}} ) B_{i}x^{\frac{p+1}{2} −i}.

Therefore one finds that

B_{\frac{p+1}{2} ,χ_{D}} = ∑_{a=1}^{|D|} χ_{D}(a) ∑_{i=0}^{\frac{p+1}{2}} ( i^{\frac{p+1}{2}} ) B_{i}|D|^{i−1}a^{\frac{p+1}{2} −i}.

Since p > 3 and i ≤ \frac{p+1}{2}, by Von Staudt-Clausen it follows that all of the B_{i} in the above sum are p−integral (i.e. denominators are prime to p). Therefore since |D| ≡ 0 mod p we find that

B_{\frac{p+1}{2} ,χ_{D}} ≡ ∑_{a=1}^{|D|} χ_{D}(a) (B_{0}|D|^{−1}a^{\frac{p+1}{2}} + \frac{p + 1}{2} · a^{\frac{p−1}{2}} B_{1}) mod p.

This reduces to

B_{\frac{p+1}{2} ,χ_{D}} ≡ B_{0}|D|^{−1} ∑_{a=1}^{|D|} χ_{D}(a)a^{\frac{p+1}{2}} + \frac{p + 1}{2} · B_{1} ∑_{a=1}^{|D|} χ_{D}(a)(\frac{a}{p}) mod p.

Since χ_{D}(n)(\frac{n}{p}) is a character modulo |D|, the second sum is identically zero and hence we find that

B_{\frac{p+1}{2} ,χ_{D}} ≡ |D|^{−1}B_{0} ∑_{a=1}^{|D|} χ_{D}(a)a^{\frac{p+1}{2}} mod p.

The result now follows as a consequence of (5) and Lemma 1.

□

Corollary 1. Let ℓ > 3 be prime. Then the Fourier coefficients H(\frac{ℓ+1}{2}, n) of the weight \frac{ℓ}{2} + 1 modular form F_{\frac{ℓ+1}{2}} (z) satisfy a quadratic congruence modulo ℓ of type (ℓ, (\frac{−1}{ℓ})).

Proof Corollary 1. By (5) and Lemma 2 we see that if D if a fundamental discriminant of the form D = (−1)^{\frac{ℓ+1}{2}} ℓN where N is a positive integer satisfying (\frac{−N}{ℓ}) = 1, then

H (\frac{ℓ + 1}{2}, ℓN) = −\frac{2B_{\frac{ℓ+1}{2} ,χ_{D}}}{ℓ + 1} ≡ 0 mod ℓ.

Then by (4) it is easy to deduce that for every integer n that

H (\frac{ℓ + 1}{2}, ℓNn^{2}) ≡ 0 mod ℓ.

Therefore it follows that

H (\frac{ℓ + 1}{2}, M) ≡ 0 mod ℓ

for every positive integer M = ℓm where (\frac{−m}{ℓ}) = 1. However this is precisely the

condition that (\frac{m}{ℓ}) = (\frac{−1}{ℓ}). Therefore the modular form F_{\frac{ℓ+1}{2}} (z) satisfies a quadratic

congruence modulo ℓ of type (ℓ, (\frac{−1}{ℓ})).

□

There are various other congruences for H(r, N) which are not of this type which are also of interest. To illustrate this we now prove the following theorem.

Theorem 2. For every positive integer N ≡ 1 mod 5 the function H(5, N) satisfies the congruence

H(5, N) ≡ 0 mod 5

for every positive integer N ≡ 1 mod 5.

Proof. Let f_{1}(z) := F_{5}(z)Θ(5z) ∈ M_{6}(20, χ_{5}), its Fourier expansion is given by

f_{1}(z) = ∑_{n=0}^{∞} b(n)q^{n} = −\frac{1}{132} (Θ^{11}(z) − \frac{22Θ^{7}(z)η^{8}(4z)}{η^{4}(2z)} + \frac{88Θ^{3}(z)η^{16}(4z)}{η^{8}(2z)}) Θ(5z).

Since Θ(5z) = 1 + 2 ∑_{n=1}^{∞} q^{5n^{2}}, it is clear that it suffices to check that

b(n) ≡ 0 mod 5

for every positive integer n ≡ 1 mod 5. Now it is known that

f_{3}(z) = ∑_{n≡1 mod 5}^{∞} b(n)q^{n}

is a weight 6 modular form with respect 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 up to 90000. The congruence has been verified with machine computation.

□

In [Th 4 ] Carlitz proved that if χ is a primitive Dirichlet character with conductor f and p is a prime for which p ∤ f and n is a positive integer for which p^{e} | n, then p^{e} divides the numerator of B_{n,χ}. For example if D = −N is the fundamental discriminant of the quadratic number field ℚ(\sqrt{D}) where N ≡ 1 mod 5, then this result implies that the numerator of B_{5,χ_{D}} is a multiple of 5. However by Theorem 2 we find that even more is true. We obtain:

Corollary 2. Let D = −N be the fundamental discriminant of ℚ(\sqrt{D}) where N ≡ 1 mod 5 is a positive integer. Then the numerator of B_{5,χ_{D}} is a multiple of 25.

Proof Corollary 2. By (5) and Theorem 2 it follows that

H(5, N) = L(−4, χ_{D}) = −\frac{B_{5,χ_{D}}}{5} ≡ 0 mod 5.

This immediately implies that the numerator of B_{5,χ_{D}} is a multiple of 25.

□

Now we make some observations regarding the divisibility of the Fourier coefficients of half-integer weight modular forms. In [ ] Serre proved a remarkable theorem regarding the divisibility of the Fourier coefficients of holomorphic integer weight modular forms. Let f(z) = ∑_{n=0}^{∞} a(n)q^{n} be the Fourier expansion of a holomorphic integer weight holomorphic modular form with respect to some congruence subgroup of SL_{2}(ℤ) whose coefficients a(n) are algebraic integers in a fixed number field. Then he proved that given a positive 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 properties of the coefficients of holomorphic half-integer weight forms. Let r be a non-negative integer and let f(z) = ∑_{n=0}^{∞} a(n)q^{n} ∈ M_{r+ \frac{1}{2}} (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 positive integer and

Θ_{a,M}(z) := ∑_{n≡a mod M} q^{n^{2}}.

In particular for all but a finite number of square-free positive integers t, it is the case that a(tn^{2}) = 0 for every integer n. Therefore the number of integers n ≤ x for which a(n) ≠ 0 is O(\sqrt{x}). However if r ≥ 1, then the situation is very different and is of significant interest. A thorough understanding of the divisilibity properties of Fourier coefficients when r = 1 will shed some light on the divisors of class numbers of imaginary quadratic fields and the Shafarevich-Tate groups of twists of certain modular elliptic curves. Therefore it is worth examining any analogs of Serre’s divisibility result which may hold for half-integer weight forms.

¿From the works of Carlitz and Leopoldt we find that the modular forms F_{r}(z) provide 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 proved that if χ is a Dirichlet character with conductor f that is not a power of a prime p, then L(1−r, χ) is an algebraic integer. However if χ is a character with a conductor that is a power of a prime p, then the prime ideal divisors of the denominator of L(1 − r, χ) are prime ideal divisors of p. However if r ≥ 2 fixed, it known that the denominators of L(1 − r, χ_{D}) are bounded which implies 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 − 2r) ∑_{n=0}^{∞} a_{r}(n)q^{n} = −\frac{B_{2r}}{2r} ∑_{n=0}^{∞} a_{r}(n)q^{n}

where the coefficients a_{r}(n) are rational with denominators bounded by D_{r}, the least common multiple of all the denominators occuring in the a_{r}(n), it follows that the nu-

merator of almost every a_{r}(n) is a multiple of the denominator of ζ(1 − 2r) = −\frac{B_{2r}}{2r} . By Von Staudt-Clausen and the Voronoi congruences (see [15.2.4, Ireland and Rosen]), the denominator of B_{2r} is ∏_{(p−1)|2r} p. We have proved:

Proposition 2. Let r ≥ 2 be a positive integer. Then for all but finitely many square-free integers t we find that

a_{r}(tn^{2}) ≡ 0 mod M

for every integer n where M = ∏_{(p−1)|2r} p. In particular, the number of non-negative integers n ≤ x for which a_{r}(n) ≢ 0 mod M is O(\sqrt{x}).

Using this proposition as a template we establish circumstances for which the Fourier coefficients of a half-integer weight modular form are almost always a multiple of a power of a prime p. From the discussion above it is clear that we only need to consider those half-integer weight forms with weight ≥ \frac{3}{2}. First we note that the classical theta function Θ(z) = 1 + 2 ∑_{n=1}^{∞} q^{n^{2}} ≡ 1 mod 2. Therefore if f(z) = ∑_{n=0}^{∞} a(n)q^{n} is a holomorphic half-integer weight modular form with integer coefficients, then f(z)·Θ(z) ≡ f(z) mod 2 is an integer weight holomorphic 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 p is an odd prime.

Let f_{p}(z) be the weight \frac{p−1}{2} modular form defined by

f_{p}(z) = \frac{η^{p}(z)}{η(pz)}.

It is easy to verify that f_{p}(z) ∈ M_{\frac{p−1}{2}} (p, χ_{D}) where D := (−1)^{\frac{p−1}{2}} p. More importantly if s is a positive integer, then since 1 − X^{p} ≡ (1 − X^{p}) mod p, we find that

(7)  f_{p}^{p^{s}}(z) ≡ 1 mod p^{s+1}.

Using this notation we observe:

Proposition 3. Let f(z) = ∑_{n=0}^{∞} a(n)q^{n} ∈ M_{r+ \frac{1}{2}} (N, χ) with rational integer coefficients. Let p be an odd prime and let s and k be positive integers for which

2r = kp^{s}(p − 1).

If for every cusp \frac{c}{d} of Γ_{0}(Np)

Ord(f, \frac{c}{d}) ≥ \frac{Nkp^{s+1}}{24 · gcd(d, Np/d)d} (p − \frac{gcd(d, p)^{2}}{p} )

where Ord(f, \frac{c}{d}) is the analytic order of f(z) at the cusp \frac{c}{d}, then for all but finitely many square-free positive integers t

a(tn^{2}) ≡ 0 mod p^{s+1}

for every integer n. In particular, the set of non-negative integers n ≤ x for which a(n) ≢ 0 mod p^{s+1} is O(\sqrt{x}).

Proof. If 2r = kp^{s}(p − 1), then by (7) the weight of (f_{p}^{p^{s}}(z))^{k} is exactly \frac{1}{2} more than

r + \frac{1}{2}, the weight of f(z). The system of inequalities implies that the modular form

f(z) · (\frac{η^{p^{s+1}}(z)}{η^{p^{s}}(pz)})^{−k} ≡ f(z) mod p^{s+1}

is holomorphic at all the cusps of Γ_{0}(Np) since the order of f_{p}^{p^{s}}(z) at a cusp \frac{c}{d} is given by (see [ ])

\frac{Np^{s+1}}{24 · gcd(d, Np/p)} (p − \frac{gcd(d, p)^{2}}{p} ) .

Therefore this form is a holomorphic weight \frac{1}{2} form, hence is a finite linear combination of theta functions by the Serre-Stark basis theorem. This completes the proof.

□

3. Special values of modular L−functions

In this section we investigate the the congruence properties of special values of quadratic twists of a modular L−function at the center of the critical strip on the real line. By the work of Kohnen, Shimura, Waldspurger, and Zagier, the Fourier coefficients of certain special half-integer weight forms are essentially (up to a transcendental factor) the square-root of these special values. We now present two simple examples for which congruences exist.

Let L(Δ, s) denote the modular L−function defined by

L(Δ, s) := ∑_{n=1}^{∞} \frac{τ(n)}{n^{s}}

where Δ(z) = ∑_{n=1}^{∞} τ(n)q^{n} is the Fourier expansion of Ramanujan’s weight 12 cusp form. If χ_{D} is the Kronecker character where D is a fundamental discriminant of ℚ(\sqrt{D}), then let L(Δ, D, s) denote the twisted L−function defined by

L(Δ, D, s) := ∑_{n=1}^{∞} \frac{χ_{D}(n)τ(n)}{n^{s}}.

Let g(z) = ∑_{n=1}^{∞} a(n)q^{n} be the weight \frac{13}{2} eigenform defined by

g(z) := \frac{Θ^{9}(z)η^{8}(4z)}{η^{4}(2z)} − \frac{18Θ^{5}(z)η^{16}(4z)}{η^{8}(2z)} + \frac{32Θ(z)η^{24}(8z)}{η^{12}(2z)}.

Using the Shimura correspondence, Kohnen and Zagier [ ] proved a general theorem which in this case implies that if D is the fundamental discriminant of a real quadratic field, then

(8)  L(Δ, D, 6) = (\frac{π}{D})^{6} \frac{\sqrt{D}}{5!} \frac{< Δ(z), Δ(z) >}{< g(z), g(z) >} · (a(D))^{2}

where < Δ(z), Δ(z) > and < g(z), g(z) > are the relevant Peterson scalar products. Therefore we shall refer to (a(D))^{2} as the rational factor of L(Δ, D, 6). With this notation we prove the following congruences for the rational factors of L(Δ, D, 6) :

Theorem 3. If N is a positive integer satisfying (\frac{−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 type (11, −1). Let T (z) ∈ M_{12}(44, χ_{0}) be defined by

T (z) := ∑_{n=1}^{∞} c(n)q^{n} = g(z) · Θ(11z) · \frac{η^{11}(z)}{η(11z)}.

Since the right hand factor is a modular form whose Fourier expansion is ≡ 1 mod 11, it suffices to check that the Fourier expansion of T (z) has a quadratic congruence modulo 11 of type (11, −1). However by Proposition 1 it suffices to check that

c(11n) ≡ 0 mod 11

for all n ≤ 95832 that satisfy (\frac{n}{11}) = −1. This congruence has been verified by machine computation.

□

Therefore by (8) we obtain:

Corollary 3. Using the notation above, if D = 11N is the fundamental discriminant of ℚ(\sqrt{D}) where N is a positive integer satisfying (\frac{−N}{11}) = 1, then the rational factor of L(Δ, D, 6) is a multiple of 121.

Now we present a second example of such congruences. In this second example let f(z) := ∑_{n=1}^{∞} c(n)q^{n} = η^{8}(z)η^{8}(2z); hence f(z) is the unique normalized weight 8 eigenform of level 2. Let L(f, s) denote the modular L−function

L(f, s) := ∑_{n=1}^{∞} \frac{c(n)}{n^{s}}.

As above, let L(f, D, s) be the twisted modular L−function defined by

L(f, D, s) := ∑_{n=1}^{∞} \frac{χ_{D}(n)c(n)}{n^{s}}.

Now let g(z) denote the weight \frac{9}{2} eigenform

h(z) := ∑_{n=1}^{∞} d(n)q^{n} = \frac{Θ^{5}(z)η^{8}(4z)}{η^{4}(2z)} − \frac{16Θ(z)η^{16}(4z)}{η^{8}(2z)}.

The Shimura lift of h(z) is f(z), hence by [prop??? Wa] it turns out that if N_{1} ≡ N_{2} ≢ 5 mod 8 are two positive square-free integers with corresponding quadratic characters χ_{D_{1}} and χ_{D_{2}} where d(N_{1}) ≠ 0, then

(9)  L(g, D_{2}, 4) = \frac{d^{2}(N_{2})L(g, D_{1}, 4)N1^{\frac{7}{2}}}{d^{2}(N_{1})N2^{\frac{7}{2}}} .

With this notation we prove:

Theorem 4. If N is a positive integer satisfying (\frac{−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 type (7, −1). Let S(z) ∈ M_{8}(28, χ_{0}) be the modular form defined by

S(z) := ∑_{n=1}^{∞} b(n)q^{n} := h(z) · Θ(7z) · \frac{η^{7}(z)}{η(7z)}.

Since the right hand factor is a modular form whose Fourier expansion is ≡ 1 mod 7, by Proposition 1 it suffices to check that

b(7n) ≡ 0 mod 7

for all n ≤ 10976 where (\frac{n}{7}) = −1. This has been verified by machine computation.

□

By (9) we obtain:

Corollary 4. Let N be a positive square-free integer for which 7N ≢ 5 mod 8 and (\frac{−N}{7}) = 1. If the character of ℚ(\sqrt{7N}) is χ_{D}, then the rational factor of L(f, D, 4) is a multiple of 49.

4. An elliptic curve explanation

We interpret the congruences of corollaries 3 and 4 in terms of the Bloch-Kato conjecture on special values of L-functions associated to arithmetic objects of a very general type (the “motives” in the sense of Grothendieck and Deligne, as defined for example in [Deligne]).

Motives: It is beyond our scope to give a detailed account of motives. A good reference for this is [Deligne]. For the purpose of our discussion, a motive M (over ℚ, with rational coefficients, of rank r) can be thought of as a piece of the cohomology of an algebraic variety over ℚ, giving rise to:

• For each prime ℓ, an ℓ-adic representation M_{ℓ} of G_{ℚ} = Gal(ℚ̄/ℚ), arising from ℓadic étale cohomology. The object M_{ℓ} is an r-dimensional ℚ_{ℓ}-vector space equipped with an action of G_{ℚ}. Since G_{ℚ} acts through a compact quotient, it leaves stable a ℤ_{ℓ}- sublattice T_{ℓ} of M_{ℓ}, and one can define the mod ℓ representation associated to M_{ℓ} to be the F_{ℓ}-vector space M̄_{ℓ} = T_{ℓ}/ℓT_{ℓ}. This space depends on the choice of T_{ℓ} in general, but its semi-simplification does not, by the Brauer-Nesbitt theorem. To obtain a canonical object we simply define M̄_{ℓ} to be the semi-simplification of T_{ℓ}/ℓT_{ℓ}.

• The system {M_{ℓ}} should form a compatible system of rational ℓ-adic representations in the sense of [Serre.McGill], §I.11. More precisely, the action of G_{ℚ} is unramified outside S ∪ {ℓ}, where S is a fixed finite set of primes not depending on ℓ. Let D_{p} be a decomposition group at p in G_{ℚ} and let I_{p} denote an inertia subgroup of D_{p}. Let Frob_{p} be the canonical (“frobenius”) generator of D_{p}/I_{p} which gives the map x ↦ x^{p} on residue fields. If p ≠ ℓ is a prime, then the characteristic polynomial Z_{p}(M, T) of Frob_{p} acting on M_{ℓ}^{I_{p}}

Z_{p}(M, T) = det((1 − Frob_{p}T)|M_{ℓ}^{I_{p}})

has integer coefficients and should not depend on the choice of ℓ ≠ p.

• A rational vector space M_{B}: the so called “Betti realization”, arising from singular cohomology.

• A rational vector space M_{DR} coming from the algebraic DeRham cohomology, equipped with its natural Hodge structure. The structures M_{B} and M_{DR} will not play an explicit role in our discussion, but are used in defining certain (transcendental) periods associated to M as in [Deligne].

The L-function: One defines the local L function at p by L_{p}(M, s) = Z_{p}(M, p^{−s})^{−1}. By the compatibility axiom, L_{p}(M, s) does not depend in the choice of the prime ℓ used to define it. One then defines the global L-function as a product over all primes p:

L(M, s) = ∏_{p} L_{p}(M, s).

This Euler product converges in a right half plane, by the Weil conjectures. It is conjectured that L(M, s) has an analytic continuation and a functional equation (cf. [Deligne], §1.2.) We will assume this. (In the special cases that we discuss, this conjecture is known to be true.)

The Bloch-Kato conjectures: Under the asumption that the motive is “‘critical” (in the sense of [Deligne], Def. 1.3), Deligne has given a very general conjectural formula for the special value L(M, 0), modulo rational multiples, in terms of a certain period integral defined in terms of the structures M_{B} and M_{DR}. This conjecture has been refined by Bloch and Kato , and predicts that the “rational part” L_{rat}(M, 0) of L(M, 0) (i.e., the special value L(M, 0) divided by the Deligne period) can be interpreted as the order of a certain Selmer group Sel(M) = ⊕_{ℓ}Sel_{ℓ}(M), where

Sel_{ℓ}(M) := ker (H^{1}(ℚ, M_{ℓ} ⊗ ℚ_{ℓ}/ℤ_{ℓ}) ⟶ ⊕_{v}H_{f}^{1}(ℚ_{v}, M_{ℓ} ⊗ ℚ_{ℓ}/ℤ_{ℓ})) .

The groups H_{f}^{1}(ℚ_{v}, M_{ℓ} ⊗ ℚ_{ℓ}/ℤ_{ℓ}) are certain subgroups of the local Galois cohomology groups 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 H_{f}^{1} is more subtle and relies on the crystalline cohomology theory developed 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_{p}(M, T) and Z_{p}(N, T) are congruent mod ℓ for all primes p ≠ ℓ. In particular, it follows from the Chebotarev density theorem that the mod ℓ Galois representations M̄_{ℓ} and N̄_{ℓ} associated to M and N are isomorphic. For example, modular forms whose Fourier coefficients are congruent modulo ℓ give rise to congruent motives. (See, for example, the discussion in the introduction to [Mazur].) Generally speaking, motivated by the philosophy expressed in [Mazur], one expects a mod ℓ congruence between two motives to translate into a congruence modulo ℓ between the special 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(Δ, D, 6) and L(f, D, 4): Let M_{Δ} and M_{f} denote the motives associated by Scholl [scholl] to the modular forms Δ ∈ S_{12}(SL_{2}(ℤ)) and f ∈ S_{8}(Γ_{0}(2)) respectively. By definition, we have:

L(M_{Δ}, s) = L(Δ, s), L(M_{f}, s) = L(f, s).

Set

M_{Δ,D,6} = M_{Δ}(−6) ⊗ M_{χ_{D}}, M_{f,D,4} = M_{f}(−4) ⊗ M_{χ_{D}}.

Here M(n) is the n-th Tate twist of M, as defined in [Deligne], and M_{χ_{D}} is the Artin motive associated to the Dirichlet character χ_{D}.

From the definitions it follows immediately that we have

L(M_{Δ,D,6}, 0) = L(Δ, D, 6), L(M_{f,D,4}, 0) = L(f, D, 4).

A congruence for M_{Δ,D,6}: Suppose that D = 11N, with N prime to 11. Let E_{11} be the elliptic curve X_{0}(11). It is the (unique) modular elliptic curve of conductor 11, and its associated eigenform f_{11} ∈ S_{2}(Γ_{0}(11)) is

f_{11} = η(z)^{2}η(11z)^{2} = q ∏(1 − q^{n})^{2}(1 − q^{11n})^{2} = ∑ a_{n}(E_{11})q^{n}.

Let M_{E_{11},χ,1} be the motive satisfying

L(M_{E_{11},χ,1}, 0) = L(E_{11}, χ, 1),

where χ is the Dirichlet character associated to the imaginary quadratic field ℚ(\sqrt{−N}). Then we have:

Lemma 2. The motives M_{Δ,D,6} and M_{E_{11},χ,1} are congruent modulo 11.

Proof: We have

Δ = q ∏(1 − q^{n})^{24} ≡ q ∏(1 − q^{n})^{2}(1 − q^{11n})^{2} ≡ f_{11} (mod 11),

so that τ(n) ≡ a_{n}(E_{11}) (mod 11) for all n. Moreover, we have

χ_{D}(p)p^{−5} ≡ χ(p) (mod 11),

and the lemma follows directly from this.

lemma 3. If 11 is split in the field ℚ(\sqrt{−N}), then

L(M_{E_{11},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 splits in the quadratic field ℚ(\sqrt{−N}). (See for example [gross.zagier].) Since 1 is the symmetry point for the functional equation, it follows that

L(E_{11}, χ, 1) = L(M_{E_{11},χ,1}, 0) = 0.

Corollary. Assume conjecture C. Then L(Δ, D, 6) ≡ 0 (mod 11) whenever 11 is split in ℚ(\sqrt{−N}).

Proof: Since M_{Δ,D,6} and M_{E_{11},χ,1} are congruent modulo 11 by lemma 2, conjecture C implies the congruence

L(Δ, D, 6) = L(M_{Δ,D,6}, 0) ≡ L(M_{E_{11},χ,1}, 0) = 0 (mod 11),

where the last equality follows from lemma 3.

A congruence for M_{f,D,4}: Suppose that D = 7N with 7 not dividing N. Let E_{14} be the elliptic curve X_{0}(14). It is the (unique) modular elliptic curve of conductor 14, and its associated eigenform f_{14} ∈ S_{2}(Γ_{0}(11)) is

f_{14} = η(z)η(2z)η(7z)η(14z) = q ∏(1 − q^{n})(1 − q^{2n})(1 − q^{7n})(1 − q^{14n}).

Let M_{E_{14},χ,1} be the motive satisfying L(M_{E_{14},χ,1}, 0) = L(E_{14}, χ, 1), where χ is the Dirichlet character corresponding to the quadratic imaginary field ℚ(\sqrt{−N}). Then we have:

Lemma 4. The motives M_{f,D,4} and M_{E_{11},χ,1} are congruent modulo 7.

Proof: We have

f = q ∏(1 − q^{n})^{8}(1 − q^{2n})^{8} ≡ q ∏(1 − q^{n})(1 − q^{2n})(1 − q^{7n})(1 − q^{14n}) = f_{14} (mod 14).

The lemma follows directly from this, and from the fact that χ_{D}(p)p^{−3} ≡ χ(p) (mod 7).

Lemma 5. If 2 and 7 are both split or both inert in the field ℚ(\sqrt{−N}), then

L(M_{E_{14},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 example [gross.zagier].) Since 1 is the symmetry point for the functional equation, it follows that

L(E_{14}, χ, 1) = L(M_{E_{14},χ,1}, 0) = 0.

Corollary. Assume conjecture C. Then L(f, D, 4) ≡ 0 (mod 7) whenever 2 and 7 are both split or both inert in ℚ(\sqrt{−N}).

Proof: By lemma 4, the motives M_{f,D,4} and M_{E_{14},χ,1} are congruent modulo 7. Conjecture C implies the congruence

L(f, D, 4) = L(M_{f,D,4}, 0) ≡ L(M_{E_{14},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 computations which were required for this paper.

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Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Current address: Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 127, Budapest 1364, Hungary

  E-mail address: h1165bal@ella.hu

Deaprtment of Mathematics, McGill University, Montréal, Canada PQ H3A 2K6 Current address: Department of Mathematics, Princeton University, Princeton, New Jersey 08540

  E-mail address: darmon@math.princeton.edu

Department 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