Period identities of CM forms on quaternion algebras

TL;DR

This paper proves an explicit identity relating torus periods of automorphic forms on quaternion algebras, connecting their norms directly and extending Waldspurger's formula to a new setting.

Contribution

It provides a direct proof of an explicit identity between torus periods on different quaternion algebras, expanding understanding of period relations and automorphic L-functions.

Findings

Explicit identity between torus periods on different quaternion algebras

Connection between period norms and automorphic L-functions

Extension of Waldspurger's formula to new cases

Abstract

Waldspurger's formula gives an identity between the norm of a torus period and an L-function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding L-functions agree, (the norms of) these periods---which occur on different quaternion algebras---are closely related. In this paper, we give a direct proof of an explicit identity between the torus periods themselves.