Period relations and special values of Rankin-Selberg $L$-functions

TL;DR

This survey reviews recent advances in understanding the special values of Rankin-Selberg L-functions for cohomological automorphic representations over CM fields, connecting automorphic periods with motivic periods and Deligne's conjecture.

Contribution

It synthesizes recent automorphic and motivic results, illustrating how automorphic periods relate to Deligne's conjecture on critical L-values over CM fields.

Findings

Automorphic methods express critical values as period integrals.

Langlands functoriality links automorphic and motivic periods.

Recent results connect automorphic periods with Deligne's conjecture.

Abstract

This is a survey of recent work on values of Rankin-Selberg $L$-functions of pairs of cohomological automorphic representations that are {\it critical} in Deligne's sense. The base field is assumed to be a CM field. Deligne's conjecture is stated in the language of motives over $\QQ$, and express the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by automorphic methods express certain critical values as (twisted) period integrals of automorphic forms. Using Langlands functoriality between cohomological automorphic representations of unitary groups, which can be identified with the de Rham cohomology of Shimura varieties, and cohomological automorphic representations of $GL(n)$, the automorphic periods can be interpreted as motivic periods. We report on recent results of the two authors, of the first-named author with Grobner, and of Guerberoff.