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ISSN 1088-6834(e) ISSN 0894-0347(p)

The archimedean theory of the exterior square -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
Posted: December 6, 2011
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Abstract: The analytic properties of automorphic -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 -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 -functions on , by constructing a pairing which we compute as a product of this -function times an explicit ratio of Gamma functions. We use this to deduce that exterior square -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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