Rationality and holomorphy of Langlands–Shahidi L-functions over function fields

Abstract

We prove three main results: all Langlands-Shahidi automorphic L-functions over function fields are rational; after twists by highly ramified characters our automorphic L-functions become polynomials; and, if π is a globally generic cuspidal automorphic representation of a split classical group or a unitary group Gn and τ a cuspidal (unitary) automorphic representation of a general linear group, then L(s, π × τ ) is holomorphic for ℜ(s) > 1 and has at most a simple pole at s = 1. We also prove the holomorphy and non-vanishing of automorphic exterior square, symmetric square and Asai Lfunctions for ℜ(s) > 1. Finally, we complete previous results on functoriality for the classical groups over function fields.2010 Mathematics Subject Classification. Primary 11F70, 22E50, 22E55.

Incorrigible Representations

“…The Langlands-Shahidi method has been developed by Lomelí over function fields. Section 7 of [Lo18] contains details about the Langlands-Shahidi local factors for classical groups; note that special care has to be taken in characteristic 2. Since we have no information a priori about generic representations (the results of [GV] on the tempered packet conjecture rely on Arthur's results in [A], which we have deliberately chosen not to use), the Langlands-Shahidi method is not available to us.…”

“…The assumption of characteristic zero was only made in [HII08] because the proofs given there in specific examples were based on methods that at the time were only available for p-adic fields. (Presumably the article [Lo18] allows for an extension of the proofs in [HII08] to positive characteristic. )…”

Grupos ortogonales sobre cuerpos de característica positiva

“…Estos grupos siguen siendo importantes en el estudio de la teoría de representaciónes, con importantes aplicaciones recientes en la teoría de formas automórficas y la functorialidad de Langlands (e.g. [10,19,14,15,5]).…”