A divisibility of automorphic periods for the real quadratic base change of $\mathrm{GL}_3$

Abstract: We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation $π$ of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel base change to a real quadratic field $E$. As a corollary, we establish a divisibility predicted by the Bloch-Kato conjecture for the adjoint motive of $π$ twisted by an even quadratic character. This generalizes earlier works of Tilouine-Urban and Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods for conjugate self-dual cuspidal automorphic representations of $\mathrm{GL}_3(E)$, defined within the middle degree of the cuspidal cohomology instead of the top or bottom degrees. Moreover, we prove an à la Hida adjoint $L$-value formula for $\mathrm{GL}_3(E)$ that relates these newly defined middle-degree periods to the usual top and bottom automorphic periods.