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

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.

This submission has been substantially revised and expanded into two separate articles, which have been submitted to arXiv as new submissions. The present version is therefore withdrawn to avoid duplication