Squares of toric period integrals in higher cohomology

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.