Automorphic Forms with Anisotropic Periods on a Unitary Group

Abstract

Let G be an algebraic group over a global field F with ring of adeles, denote by Z the center of G and by C an algebraic subgroup of G over F such that the cycle CF‘\C ‘ has finite volume. Fix unitary characters ωx Z ‘/ZF‘ → 1 = unit circle in ב and ξx CF‘\C ‘ → 1, and denote by φx GF‘\G ‘ → a cusp form in the cuspidal representation π of G ‘, whose central character is ωπ = ω. By a cuspidal representation we mean an irreducible one. We say that π is C ‘-cyclic if it has a nonzero C ‘-period PC ‘φ‘ = ∫ CF‘\C ‘φc‘ξc‘dc. The overbar indicates complex conjugation. Studies of cyclic automorphic forms have applications to special values of L-functions (Waldspurger [W1, W2], Jacquet [J1, J2]), lifting problems [F], and studies of cohomology of symmetric spaces, in particular the Tate conjecture on algebraic cycles on some Shimura surfaces [FH]. The purpose of this paper—inspired by the applications to the Tate conjecture of [FH]—is to compare the notion of cyclicity by C ‘, with cyclicity by an inner form of C ‘. We let G be the quasi-split unitary group U2; 1‘ = U2; 1yE/F‘ in three variables defined by means of a quadratic separable extension E/F of global fields. The subgroup C is taken to be the