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

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.

The definition of the periods has been adjusted to apply only in the lowest and highest cohomological degrees. No other substantial changes