Distinguished representations and quadratic base change for 𝐺𝐿(3)

Abstract

Let E / F E/F be a quadratic extension of number fields. Suppose that every real place of F F splits in E E and let H H be the unitary group in 3 variables. Suppose that Π \Pi is an automorphic cuspidal representation of G L ( 3 , E A ) GL(3,E_{\mathbb {A}}) . We prove that there is a form ϕ \phi in the space of Π \Pi such that the integral of ϕ \phi over H ( F ) ∖ H ( F A ) H(F)\setminus H(F_{\mathbb {A}}) is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.