Relations of rationality for special values of Rankin–Selberg $L$-functions of $\mathrm{GL}_n\times \mathrm{GL}_m$ over CM-fields

Harald Grobner and Gunja Sachdeva

Vol. 308 (2020), No. 2, 281–305

DOI: 10.2140/pjm.2020.308.281

Abstract

We establish an “automorphic version” of Deligne’s conjecture for motivic $L$-functions in the case of Rankin–Selberg $L$-functions $L\left(s,\Pi \times {\Pi }^{\prime }\right)$ of ${GL}_n\times {GL}_m$ over arbitrary CM-fields $F$. Our main results are of two different kinds: Firstly, for arbitrary integers $1\le m<n$ and suitable pairs $\left(\Pi ,{\Pi }^{\prime }\right)$ of cohomological automorphic representations, we relate critical values of $L\left(s,\Pi \times {\Pi }^{\prime }\right)$ with a product of Whittaker periods attached to $\Pi$ and ${\Pi }^{\prime }$, Blasius’s CM-periods of Hecke-characters and certain nonzero values of standard $L$-functions. Secondly, these relations lead to quite broad generalizations of fundamental rationality-results of Waldspurger, Harder and Raghuram, and others.

Keywords

periods, rationality, special values, L-function, Rankin–Selberg, GL(n)

Mathematical Subject Classification

Primary: 11F67

Secondary: 11F70, 11G18, 11R39, 22E55

Milestones

Received: 27 December 2019

Revised: 23 May 2020

Accepted: 14 June 2020

Published: 9 December 2020