Torus periods of automorphic functions and the meromorphic continuation of related Dirichlet Series

TL;DR

This paper studies twisted periods of modular functions for PSL(2,Z) associated with a quadratic field and proves their related Dirichlet series can be meromorphically continued.

Contribution

It establishes the meromorphic continuation of Dirichlet series generated by twisted torus periods of modular functions, linking automorphic forms and number field data.

Findings

Meromorphic continuation of the Dirichlet series is achieved.

Twisted periods relate to quadratic number fields and automorphic forms.

Results extend understanding of automorphic L-functions and periods.

Abstract

We consider modular functions (i.e., the Eisenstein series and Hecke-Maass forms) for the group PSL(2,Z). We fix a quadratic number field E. This gives rise to twisted (by a Hecke character of the field E) periods of a modular function along the torus corresponding to E. We prove meromorphic continuation for a Dirichlet series generated by these twisted periods.