Rational Structures on Automorphic Representations

TL;DR

This paper establishes the existence of rational structures on automorphic representations, particularly for GL(n), and explores their implications for periods and L-functions, while developing foundational theory for Harish-Chandra modules over arbitrary fields.

Contribution

It introduces a general framework for rational structures on automorphic representations and develops foundational tools for Harish-Chandra modules over arbitrary fields.

Findings

Existence of rational structures on cusp forms for general reductive groups.

Optimal rational structures identified for GL(n).

Implications for periods and special values of L-functions.

Abstract

This paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups. We identify cases where the constructed rational structures are optimal, which includes the case of GL($n$). As an application, we deduce the existence of a natural set of periods attached to cuspidal automorphic representations of GL($n$). This has consequences for the arithmetic of special values of $L$-functions that we discuss in subsequent articles. In the course of proving our results, we lay the foundations for a general theory of Harish-Chandra modules over arbitrary fields of characteristic $0$. In this context, a rational character theory, translation functors and an equivariant theory of cohomological induction are developed. We also study descent problems for Harish-Chandra modules in quadratic extensions, where we obtain a complete theory over number fields.