The archimedean theory of the Exterior Square L-functions over Q

Details The archimedean theory of the Exterior Square L-functions over Q The archimedean theory of the Exterior Square L-functions over Q

2011-09-20

arxiv.org/abs/1109.4190v1

Mathematics

Number Theory

41 pages, dedicated in memory of Joseph Shalika, to appear in Jour. Amer. Math. Soc

Scientific paper

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 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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