Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations

TL;DR

This paper establishes a $p$-adic divisibility relation between automorphic periods of $ ext{GL}_3$ over $Q$ and its base change to a real quadratic field, extending previous modular form results and exploring stable base change scenarios.

Contribution

It introduces a new automorphic period involving middle degree cohomology and generalizes divisibility results to $ ext{GL}_3$ and real quadratic fields, including stable base change cases.

Findings

Proves $p$-adic divisibility between automorphic periods of $ ext{GL}_3(Q)$ and its base change.

Defines a new automorphic period using middle degree cohomology of $ ext{GL}_3(E)$.

Provides results on $p$-adic divisibility in the context of Rogawski's stable base change.

Abstract

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel's base change to some real quadratic field $E$. This generalizes earlier works of Tilouine-Urban and of Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods, defined using the middle degree of the cuspidal cohomology of $\mathrm{GL}_3(E)$, instead of the top or bottom degrees. We also investigate the Rogawski's stable base change from the quasi-split unitary group $U_E$ associated with $E$ to $\mathrm{GL}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.