Whittaker rational structures and special values of the Asai $L$-function

TL;DR

This paper establishes a deep connection between special values of the Asai L-function and rational structures from cohomology for automorphic representations over totally real and imaginary quadratic fields, generalizing previous results.

Contribution

It generalizes prior results by relating Asai L-values to cohomological rational structures for higher rank groups over more general number fields.

Findings

Relates residues and values of Asai L-functions to cohomological rational structures.

Extends previous results from quadratic fields to totally real and imaginary quadratic extensions.

Provides a framework for understanding special L-values via cohomology and Whittaker models.

Abstract

Let $F$ be a totally real number field and $E/F$ a totally imaginary quadratic extension of $F$. Let $\Pi$ be a cohomological, conjugate self-dual cuspidal automorphic representation of $GL_n(\mathbb A_E)$. Under a certain non-vanishing condition we relate the residue and the value of the Asai $L$-functions at $s=1$ with rational structures obtained from the cohomologies in top and bottom degrees via the Whittaker coefficient map. This generalizes a result in Eric Urban's thesis when $n = 2$, as well as a result of the first two named authors, both in the case $F = \mathbb Q$.