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

TL;DR

This paper introduces a new pairing-based method for analyzing automorphic L-functions, specifically proving Langlands' conjecture on the holomorphicity of exterior square L-functions over Q, which was previously unproven.

Contribution

It develops a novel approach based on automorphic distributions pairings, extending the Rankin-Selberg method to establish properties of L-functions that were previously inaccessible.

Findings

Proves holomorphicity of exterior square L-functions on C-{0,1}

Establishes simple poles at 0 and 1 for these L-functions

Validates a conjecture of Langlands using a new method

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 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.