Odd coefficients of weakly holomorphic modular forms

Nonzero coefficients of half-integral weight modular forms mod $$\ell $$ ℓ

“…Theorem 1 improves, by a factor of about log X, the earlier work of Ahlgren and Boylan [2]. Here are some sample applications of Theorem 1.…”

“…We describe this deduction in Sect. 2 for the sake of completeness, but we should note that closely related arguments appear already in the work of Ahlgren and Boylan [2]. It will follow from the work of Sect.…”

“…It will follow from the work of Sect. 2 (and this is also implicit in [2]) that the sumset of the set of squares and the set of nonzero Fourier coefficients of f (mod ) contains all numbers of the form up for a fixed integer u and p lying in a positive density subset of the primes. Our improvement over previous work comes from a more careful analysis of this problem in analytic number theory/additive combinatorics.…”

“…For most forms f of half-integral weight however (specifically for f (q) = η −1 1 (q) = n p(n)q 24n−1 , and perhaps for all forms that are not congruent mod λ to a onevariable theta series), it is expected that f (mod λ) is not lacunary, which is to say that #{n ≤ X, a n ≡ 0 (mod λ)} X. We thank Scott Ahlgren for kindly drawing our attention to [2] and the referees for a careful reading.…”

“…Note that −4a is a discriminant, but it need not be a fundamental discriminant. We denote the associated (negative) fundamental discriminant by a, so that −4a = aa 2 2 for a suitable natural number a 2 .…”