Stable base change from unitary groups in three variables and integral relation of automorphic periods

Abstract

Let $E$ be a real quadratic field and let $U_E$ be the quasi-split unitary group in three variables associated with $E$. We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation ${\pi }_U$ of $U_E$ and the periods of its Rogawski stable base change to ${\mathrm{GL}}_3(E)$. This generalizes earlier works of Tilouine-Urban and Hida in the case of ${\mathrm{GL}}_2$. The divisibility we prove involves a new kind of automorphic periods for self-conjugate cuspidal automorphic representations of ${\mathrm{GL}}_3(E)$, defined within the middle degree of the cuspidal cohomology rather than the top or bottom degrees. Moreover, we prove an à la Hida adjoint $L$-value formula for ${\mathrm{GL}}_3(E)$, relating these newly defined middle-degree periods to the usual top and bottom automorphic periods. Finally, we also prove a similar formula for $U_E$, which is the first instance of such a result for quasi-split unitary groups.