Fourier coefficients of theta functions at cusps other than infinity
Volume 175 / 2016
Acta Arithmetica 175 (2016), 341-383
MSC: Primary 11F27; Secondary 11F30.
DOI: 10.4064/aa8278-4-2016
Published online: 5 October 2016
Abstract
We consider theta functions twisted by certain Dirichlet characters, and derive explicit formulae for their Fourier coefficients at cusps other than infinity. The method is based on expressing these theta functions in terms of explicit elements of the adelic Schwartz space and studying the action of the adelic metaplectic group on them. The formulae obtained are quite amenable to effective computations, in contrast to those available in the previous work of Goldfeld, Hundley and Lee on the integral weight case. In particular, we prove a conjecture of Goldfeld and Gunnells on twisted theta functions.
