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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The archimedean theory of the exterior square $L$-functions over $\mathbb {Q}$
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by Stephen D. Miller and Wilfried Schmid PDF
J. Amer. Math. Soc. 25 (2012), 465-506

Abstract:

The analytic properties of automorphic $L$-functions have historically been obtained either through integral representations (the “Rankin-Selberg method”) or properties of the Fourier expansions of Eisenstein series (the “Langlands-Shahidi method”). We introduce a method based on pairings of automorphic distributions that appears to be applicable to a wide variety of $L$-functions, including all which have integral representations. In some sense our method could be considered a completion of the Rankin-Selberg method because of its common features. We consider a particular but representative example, the exterior square $L$-functions on $GL(n)$, by constructing a pairing which we compute as a product of this $L$-function times an explicit ratio of Gamma functions. We use this to deduce that exterior square $L$-functions, when multiplied by the Gamma factors predicted by Langlands, are holomorphic on $\mathbb {C}-\{0,1\}$ with at most simple poles at 0 and 1, proving a conjecture of Langlands which has not been obtained by the existing two methods.
References
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Additional Information
  • Stephen D. Miller
  • Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019
  • Email: miller@math.rutgers.edu
  • Wilfried Schmid
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • Email: schmid@math.harvard.edu
  • Received by editor(s): November 24, 2010
  • Received by editor(s) in revised form: September 12, 2011
  • Published electronically: December 6, 2011
  • Additional Notes: The first author was partially supported by NSF grant DMS-0901594 and an Alfred P. Sloan Foundation Fellowship
    The second author was partially supported by DARPA grant HR0011-04-1-0031 and NSF grant DMS-0500922

  • Dedicated: In memory of Joseph Shalika
  • © Copyright 2011 by Stephen D. Miller and Wilfried Schmid
  • Journal: J. Amer. Math. Soc. 25 (2012), 465-506
  • MSC (2010): Primary 11F55, 11F66
  • DOI: https://doi.org/10.1090/S0894-0347-2011-00719-4
  • MathSciNet review: 2869024