Waldspurger formulas in higher cohomology

TL;DR

This paper extends Waldspurger formulas to higher cohomology, connecting automorphic forms, Shimura varieties, and special L-values, thus broadening the scope of classical automorphic period formulas.

Contribution

It introduces a Waldspurger-type formula in higher cohomology, linking cocycles in etale cohomology with special values of Rankin-Selberg L-functions.

Findings

Derived a formula relating cap-products of cocycles and fundamental classes to L-values.

Connected automorphic forms in higher cohomology with arithmetic invariants.

Extended classical Waldspurger formulas to a higher cohomological setting.

Abstract

The classical Waldspurger formula, which computes periods of quaternionic automorphic forms in maximal torus, has been used in a wide variety of arithmetic applications, such as the Birch and Swinnerton-Dyer conjecture in rank 0 situations. This is why this formula is considered the rank 0 analogue of the celebrated Gross-Zagier formula. On the other hand, Eichler-Shimura correspondence allows us to interpret this quaternionic automorphic form as a cocycle in higher cohomology spaces of certain arithmetic groups. In this way we can realize the corresponding automorphic representation in the etale cohomology of certain Shimura varieties. In this work we find a formula, analogous to that of Waldspurger, which relates cap-products of this cocycle and fundamental classes associated with maximal torus with special values of Rankin-Selberg L-functions.