Vol. 308, No. 2, 2020

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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
Abstract

We establish an “automorphic version” of Deligne’s conjecture for motivic L-functions in the case of Rankin–Selberg L-functions L(s,Π × Π) of GLn × GLm over arbitrary CM-fields F. Our main results are of two different kinds: Firstly, for arbitrary integers 1 m < n and suitable pairs (Π,Π) of cohomological automorphic representations, we relate critical values of L(s,Π × Π) with a product of Whittaker periods attached to Π and Π, 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
Authors
Harald Grobner
Fakultät für Mathematik
Universität Wien
Wien
Austria
Gunja Sachdeva
IISER Tirupati
Tirupati
India