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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On period relations for automorphic $L$-functions I
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by Fabian Januszewski PDF
Trans. Amer. Math. Soc. 371 (2019), 6547-6580

Abstract:

This paper is the first in a series of two dedicated to the study of period relations of the type \begin{equation*} L\Big (\frac {1}{2}+k,\Pi \Big )\;\in \;(2\pi i)^{d\cdot k}\Omega _{(-1)^k}\textrm {\bf Q}(\Pi ),\quad \frac {1}{2}+k\;\text {critical}, \end{equation*} 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
  • MR Author ID: 893898
  • Email: januszewski@kit.edu
  • 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
  • © Copyright 2018 by the author
  • 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
  • MathSciNet review: 3937337