Odd Coefficients of Weakly Holomorphic Modular Forms

is a weakly holomorphic modular form of integral or half-integral weight w2 on the congruence subgroup Γ1(N). By a weakly holomorphic modular form we mean a function f(z) which is holomorphic on the upper half-plane, meromorphic at the cusps, and which transforms in the usual way under the action of Γ1(N) on the upper half-plane (see, for example, [13] for generalities on modular forms of half-integral weight). We denote the space of such forms by Mw 2 (Γ1(N)). Now, suppose that L is an algebraic number field, v is a place of L over 2, and Ov is the local ring at v. We assume that the coefficients a(n) in (1.1) belong to Ov , and if mv is the maximal ideal of Ov , then we write (mod v) to mean (mod mv). We will consider the question of estimating the number of integers n for which a(n) 6≡ 0 (mod v). For a well-studied example, let p(n) be the ordinary partition function. Many authors have considered the problem of estimating the number of odd values of p(n). Among other references, one may see [1], [5], [15], [16], [17], [18], [19], [22], or [24]. To see the connection to the general situation above, we recall the definition of the Dedekind eta-function: