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

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.

Locations

• Journal of the American Mathematical Society - View - PDF