Congruences for Fourier coefficients of integer weight modular forms have been the focal point of... more 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) = P ∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coefficients. If is prime, then we shall be interested in congruences of the form a( N ) ≡ 0 mod where N is any quadratic residue (resp. non-residue) modulo . For every prime > 3 we exhibit a natural holomorphic weight 2 + 1 modular form whose coefficients satisfy the congruence a( N ) ≡ 0 mod for every N satisfying`− N ´= 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 = 1− 2 which are expressed as generalized Bernoulli numbers whose numerators we show are multiples of . ¿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 −primary parts of Shafarevich-Tate groups of certain infinite families of quadratic twists of modular elliptic curves with conductors 11 and 14.
Congruences for Fourier coefficients of integer weight modular forms have been the focal point of... more 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) = P ∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coefficients. If is prime, then we shall be interested in congruences of the form a( N ) ≡ 0 mod where N is any quadratic residue (resp. non-residue) modulo . For every prime > 3 we exhibit a natural holomorphic weight 2 + 1 modular form whose coefficients satisfy the congruence a( N ) ≡ 0 mod for every N satisfying`− N ´= 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 = 1− 2 which are expressed as generalized Bernoulli numbers whose numerators we show are multiples of . ¿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 −primary parts of Shafarevich-Tate groups of certain infinite families of quadratic twists of modular elliptic curves with conductors 11 and 14.
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Papers by Antal Balog