On the Fourier coefficients of modular forms of half-integral weight

Abstract

We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over ℚ. This formula has applications to proving the nonvanishing of this lift and to an explicit version of the Rallis inner product formula.