Determining Modular Forms of Half-Integral Weight by Central Values of Convolution L-Functions

Abstract

In this paper, we show that a Hecke eigenform g of half-integral weight on $\Gamma _0(4)$ which belongs to the Kohnen plus space is uniquely determined by the central values of the family of convolution (Rankin–Selberg) L-functions $L(s, f\otimes g)$, where f runs over an orthogonal basis of Hecke eigenforms of weight $k+1/2$ on $\Gamma _0(4)$ which lies in the Kohnen plus space with k varying over an infinite set of integers. This generalizes the work of Ganguly et al. (Math Ann 345:843–857, 2009) to the case of forms of half-integral weight in the Kohnen plus space.