A local global question in automorphic forms

TL;DR

This paper explores the analogue of two significant theorems related to period integrals of automorphic forms, focusing on the $ ext{SL}(2)$ case, extending known results from $ ext{GL}(2)$.

Contribution

It establishes the $ ext{SL}(2)$ analogue of key theorems concerning period integrals, expanding the understanding of automorphic forms in this setting.

Findings

Derived the $ ext{SL}(2)$ analogue of Harder-Langlands-Rapoport theorem.

Extended Waldspurger's theorem to the $ ext{SL}(2)$ context.

Provided new insights into period integrals for automorphic forms on $ ext{SL}(2)$.

Abstract

In this paper, we consider the $\SL(2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\GL(2)$: one due to Harder-Langlands-Rapoport, and the other due to Waldspurger.