On the period of the Ikeda lift for U(m,m)

Abstract

be an imaginary quadratic field with discriminant −D, and χ the Dirichlet character corresponding to the extension K/Q. Let m = 2n or 2n + 1 with n a positive integer. Let f be a primitive form of weight 2k + 1 and character χ for Γ 0 (D), or a primitive form of weight 2k for SL 2 (Z) according as m = 2n, or m = 2n + 1. For such an f let Im(f ) be the lift of f to the space of modular forms of weight 2k +2n and character det −k−n for the Hermitian modular group Γ (m) K constructed by Ikeda. We then express the period Im(f ), Im(f ) of Im(f ) in terms of special values of the adjoint L-function of f and its twist by the character χ. This proves the conjecture concerning the period of the Hermitian Ikeda lift proposed by Ikeda. Period,