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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

On period relations for automorphic $ L$-functions I


Author: Fabian Januszewski
Journal: Trans. Amer. Math. Soc. 371 (2019), 6547-6580
MSC (2010): Primary 11F67; Secondary 11F41, 11F70, 11F75, 22E55
DOI: https://doi.org/10.1090/tran/7527
Published electronically: September 24, 2018
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Abstract: This paper is the first in a series of two dedicated to the study of period relations of the type

$\displaystyle L\Big (\frac {1}{2}+k,\Pi \Big )\;\in \;(2\pi i)^{d\cdot k}\Omega _{(-1)^k}{\rm\bf Q}(\Pi ),\quad \frac {1}{2}+k\;$$\displaystyle \text {critical},$    

for certain automorphic representations $ \Pi $ of a reductive group $ G.$ In this paper we discuss the case $ G=\mathrm {GL}(n+1)\times \mathrm {GL}(n).$ The case $ G=\mathrm {GL}(2n)$ is discussed in part two. Our method is representation theoretic and relies on the author's recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation $ \Pi $ under consideration. The new period relations we prove are in accordance with Deligne's Conjecture on special values of $ L$-functions, and the author expects this method to apply to other cases as well.

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Additional Information

Fabian Januszewski
Affiliation: Institute of Algebra and Geometry, Karlsruhe Institute of Technology, Englerstr. 2, D-76128 Karlsruhe, Germany
Email: januszewski@kit.edu

DOI: https://doi.org/10.1090/tran/7527
Received by editor(s): October 22, 2017
Received by editor(s) in revised form: January 30, 2018, and January 31, 2018
Published electronically: September 24, 2018
Article copyright: © Copyright 2018 by the author