Signs of Fourier coefficients of half-integral weight modular forms

Abstract

Let g be a Hecke cusp form of half-integral weight, level 4 and belonging to Kohnen’s plus subspace. Let c(n) denote the nth Fourier coefficient of g, normalized so that c(n) is real for all n ge 1. A theorem of Waldspurger determines the magnitude of c(n) at fundamental discriminants n by establishing that the square of c(n) is proportional to the central value of a certain L-function. The signs of the sequence c(n) however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that c(n) < 0 and respectively c(n) > 0 holds for a positive proportion of fundamental discriminants n. Moreover we show that the sequence {c(n)} where n ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of c(n) which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms g of level 4N with N odd, square-free.