Period relations for automorphic induction and applications, I

Abstract

Let $K$ be a quadratic imaginary field. Let $Π$ (resp. $Π'$) be a regular algebraic cuspidal representation of $GL_{n}(K)$ (resp. $GL_{n-1}(K)$) which is moreover cohomological and conjugate self-dual. In \cite{harris97}, M. Harris has defined automorphic periods of such a representation. These periods are automorphic analogues of motivic periods. In this paper, we show that automorphic periods are functorial in the case where $Π$ is a cyclic automorphic induction of a Hecke character $χ$ over a CM field. More precisely, we prove relations between automorphic periods of $Π$ and those of $χ$. As a corollary, we refine the formula given by H. Grobner and M. Harris of critical values for the Rankin-Selberg $L$-function $L(s,Π\times Π')$ in terms of automorphic periods. This completes the proof of an automorphic version of Deligne's conjecture in certain cases.

An abridged version is published in Comptes Rendus Mathématiques 353 (2015), pp. 95-100