Nonvanishing modulo of Fourier Coefficients of Half-integral Weight Modular Forms

1. Introduction. Let k be an integer and N be a positive integer divisible by 4. If is a prime, denote by v a continuation of the usual-adic valuation on Q to a fixed algebraic closure. Let f be a modular form of weight k + 1/2 with respect to 0 (N) and Nebentypus character χ which has integral algebraic Fourier coefficients a(n), and put v (f) = inf n v (a(n)). Suppose that f is a common eigenform of all Hecke operators T (p 2) with corresponding eigenvalues λ p. In a recent paper, Ono and Skinner (under the additional assumption that f is " good ") proved the following theorem [OS]: For all but finitely many primes , there exist infinitely many square-free integers d for which v (a(d)) = 0. Their proof uses the theory of-adic Galois representations. Similar results were obtained by Jochnowitz in [J] by developing a theory of half-integral weight modular forms modulo analogous to the integral weight theory due to Serre, Swinnerton-Dyer, and Katz. Results of this type can be viewed as mod versions of a well-known theorem of Vignéras about the nonvanishing of Fourier coefficients of half-integral weight modular forms (see [V]). A new proof for this was given by the author (see [B]). In the present paper, we extend the method introduced in [B] to the modulo situation and thereby obtain a new approach to the above stated theorem and certain generalizations. We use an application of the q-expansion principle of arithmetic algebraic geometry (Lemma 1) and exploit the properties of various well-known operators defined on modular forms to infer our first result (Theorem 1). Roughly speaking, it states that if for a given prime p and a given ε ∈ {±1}, all Fourier coefficients a(n) with (n p) = ε vanish modulo , then the Hecke eigenvalue λ p satisfies a certain congruence modulo. Under the (obviously necessary) assumption that f is not a linear combination of elementary theta series of weight 1/2 or 3/2, one can deduce several nonvanishing theorems. For instance, in Theorem 4 we show that there exists a finite set A N (f) of primes that has an explicit description in terms of the eigenvalues λ p with the following property: For every prime with ((, N) = 1, v (f) = 0, and / ∈ A N (f), there are infinitely many square-free d such …