Let ≥ 5 be prime, let m ≥ 1 be an integer, and let p(n) denote the partition function. Folsom, Kent, and Ono recently proved that there exists a positive integer b (m) of size roughly m 2 such that the module formed from the Z/ m Z-span of generating functions for p b n+1 24 with odd b ≥ b (m) has finite rank. The same result holds with "odd" b replaced by "even" b. Furthermore, they proved an upper bound on the ranks of these modules. This upper bound is independent of m; it is +12 24. In this paper, we prove, with a mild condition on , that b (m) ≤ 2m − 1. Our bound is sharp in all computed cases with ≥ 29. To deduce it, we prove structure theorems for the relevant Z/ m Z-modules of modular forms. This work sheds further light on a question of Mazur posed to Folsom, Kent, and Ono.
Abstract. If F (z) is a newform of weight 2λ and D is a fundamental discriminant, then let L(F ⊗ χ D , s) be the usual twisted L-series. We study the algebraic parts of the central critical values of these twisted L-series modulo primes . We show that if there are two D (subject to some local conditions) for which the algebraic part of L(F ⊗ χ D , λ) is not 0 (mod ), then there are infinitely many such D. These results depend on precise nonvanishing results for the Fourier coefficients of half-integral weight modular forms modulo , which are of independent interest.
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