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Journal of the American Mathematical Society

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

   
 

 

The archimedean theory of the exterior square $ L$-functions over $ \mathbb{Q}$


Authors: 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
Published electronically: December 6, 2011
MathSciNet review: 2869024
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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.


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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

DOI: https://doi.org/10.1090/S0894-0347-2011-00719-4
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
Article copyright: © Copyright 2011 by Stephen D. Miller and Wilfried Schmid