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

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 $G{L}_n({\mathbb{A}}_E)$. 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 $F=\mathbb{Q}$.

This version of the paper has been entirely reworked. Its results are a "basis-free" variant of the previous version

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Cited by in corpus (7)

• Whittaker periods, motivic periods, and special values of tensor product L-functions
• Minimal modularity lifting for non-regular symplectic representations
• Deligne's conjecture for automorphic motives over CM-fields
• Algebraicity of the near central non-critical value of symmetric fourth $L$-functions for Hilbert modular forms
• Special values of $L$-functions and the refined Gan-Gross-Prasad conjecture
• Factorization of arithmetic automorphic periods
• On Deligne's conjecture for symmetric fourth $L$-functions of Hilbert modular forms