The archimedean theory of the exterior square 𝐿-functions over ℚ

Miller, Stephen; Schmid, Wilfried

doi:10.1090/S0894-0347-2011-00719-4

The archimedean theory of the exterior square $L$-functions over $\mathbb {Q}$
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by Stephen D. Miller and Wilfried Schmid

J. Amer. Math. Soc. 25 (2012), 465-506

DOI: https://doi.org/10.1090/S0894-0347-2011-00719-4

Published electronically: December 6, 2011

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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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