Search

From a result of Waldspurger [W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math.64 (1981) 175-198], it is known that the normalized Fourier coefficients a(m) of a half-integral weight holomorphic cusp eigenform 픣 are, up to a finite set of factors, one of $\pm \sqrt{L(\frac{1}{2}, f, \chi_m)}$ when m is square-free and f is the integral weight cusp form related to 픣 by the Shimura correspondence [G. Shimura, On modular forms of half-integral weight, Ann. of Math.97 (1973) 440-481]. In this paper we address a question posed by Kohnen: which square root is a(m)? In particular, if we look at the set of a(m) with m square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet series, we are able to show that this is so. [ABSTRACT FROM AUTHOR]