Periods of modular forms and applications to the conjectures of Oda and of Prasanna-Venkatesh

TL;DR

This paper derives formulas connecting periods of modular forms on quaternion algebras to special L-values, providing evidence for conjectures related to elliptic curves using cohomological and explicit Waldspurger techniques.

Contribution

It introduces new formulas relating modular form periods to L-values and applies them to support conjectures of Oda and Prasanna-Venkatesh for elliptic curves.

Findings

Formulas relating periods and L-values are established.

Partial evidence for Oda and Prasanna-Venkatesh conjectures is provided.

Applicable to modular forms of general even positive weights.

Abstract

We establish several formulas relating periods of modular forms on quaternion algebras over number fields to special values of L-functions. Our main inputs are the cohomological techniques for working with periods introduced in [Mol21], along with explicit versions of the Waldspurger formula due to Cai-Shu-Tian. We work in general even positive weights; when specialized to parallel weight 2, our formulas provide partial evidence for the conjectures of Oda and of Prasanna-Venkatesh in the case of forms associated to elliptic curves.