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

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of ${\mathrm{G}\mathrm{L}}_3(\mathbb{Q})$ 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 ${\mathrm{G}\mathrm{L}}_3(E)$, 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 ${\mathrm{G}\mathrm{L}}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.