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Licensed Unlicensed Requires Authentication Published by De Gruyter March 31, 2015

Critical values of Rankin–Selberg L-functions for GLn × GLn-1 and the symmetric cube L-functions for GL2

A. Raghuram
From the journal Forum Mathematicum

Abstract

In a previous article we had proved an algebraicity result for the central critical value for L-functions for GLn × GLn-1 over ℚ assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize that result for all critical values for L-functions for GLn × GLn-1 over any number field F while using certain period relations proved by Freydoon Shahidi and the author, and some additional inputs as will be explained below. Thanks to some recent work of Binyong Sun, the nonvanishing hypothesis has now been proved. The results of this article are unconditional. Applying this to GL3 × GL2, new unconditional algebraicity results for the special values of symmetric cube L-functions for GL2 over F have been proved. Previously, algebraicity results for the critical values of symmetric cube L-functions for GL2 have been known only in special cases by the works of Garrett–Harris, Kim–Shahidi, Grobner–Raghuram, and Januszewski.

Funding source: National Science Foundation (NSF)

Award Identifier / Grant number: DMS-0856113

Funding source: Alexander von Humboldt Foundation

Award Identifier / Grant number: Research Fellowship

Funding statement: This work is partially supported by the National Science Foundation (NSF), award number DMS-0856113, and an Alexander von Humboldt Research Fellowship.

It is a pleasure to thank Michael Harris and Fabian Januszewski for their comments on a preliminary version of this article. I thank the referee for pointing out a piquant phenomenon concerning critical points that manifests itself when the base field has a complex place; in an earlier version of this manuscript this subtle issue was glossed over, but it is now addressed in Section 2.4.5.5.

Received: 2014-3-13
Revised: 2015-2-10
Published Online: 2015-3-31
Published in Print: 2016-5-1

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