Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions

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

$$ f(z)\, = \,\sum\nolimits_{n = 0}^\infty {a(n)q^n } $$

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

$$ a(lN)\, \equiv \,0\,\bmod \,l $$

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

$$ \left( {\frac{{ - N}} {l}} \right)\, = \,1 $$

. 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

$$ s\, = \,\left( {\frac{{1 - l}} {2}} \right) $$

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 suitable Bernoulli numbers. 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 more examples of half-integer weight forms with such congruence properties, whose coefficients are determined by the special values at the center of the critical strip for the quadratic twists of the modular L- functions associated to the modular form Δ of weight 12 and level 1, and to the unique form η ⁸(z)η⁸(2z) of weight 8 and level 2. We suggest a conceptual explanation for these congruences by remarking that the twists of the mod p Galois representations (p = 11 and 7 respectively) associated to these two forms are isomorphic to the Galois representations associated to certain elliptic curves of odd analytic rank.