Signs of Fourier coefficients of half-integral weight modular forms

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 n th Fourier coefficient of g , normalized so that c ( n ) is real for all n ≥ 1 n ≥ 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 $$c(n) < 0$$ c ( n ) < 0 and respectively $$c(n) > 0$$ c ( n ) > 0 holds for a positive proportion of fundamental discriminants n . Moreover we show that the sequence (n)\ { 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 4 N with N odd, square-free.