Squares of toric period integrals in higher cohomology

TL;DR

This paper extends the understanding of toric period integrals in higher cohomology by computing their squares, relating them to special values of Rankin-Selberg L-functions, and generalizing Waldspurger's formula.

Contribution

It provides a new computation of the square of toric period integrals in higher cohomology, advancing the connection between cohomological pairings and L-functions.

Findings

Computed the square of the cap-product in higher cohomology.

Related the square to special values of Rankin-Selberg L-functions.

Generalized Waldspurger's formula to higher cohomological settings.

Abstract

Thanks to the Harder-Eichler-Shimura isomorphism we can realize a quaternionic automorphic representation of a fixed weight in the cohomology space of certain arithmetic groups. For many interesting applications, it is convenient to consider the cap-product of a cohomology class in these spaces with a fundamental class associated to a maximal torus. In a recent paper, the author computes the absolute value of such a cap-product, and he relates it to special values of Rankin-Selberg L-functions. This provides a formula analogous to that of Waldspurger in higher cohomology. In this paper we compute the square of the cap-product instead of its absolute value.