On the rationality and holomorphy of Langlands-Shahidi L-functions over function fields

TL;DR

This paper establishes the rationality, polynomial nature after twists, and holomorphy of various Langlands-Shahidi automorphic L-functions over function fields, advancing the understanding of their analytic properties and functoriality.

Contribution

It proves the rationality of all Langlands-Shahidi L-functions over function fields and demonstrates their holomorphy and non-vanishing in key cases, completing previous functoriality results.

Findings

All Langlands-Shahidi automorphic L-functions over function fields are rational.

Twisted automorphic L-functions become polynomials after highly ramified twists.

Certain L-functions are holomorphic for Re(s) > 1 and have at most simple poles at s=1.

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 $\pi$ is a globally generic cuspidal automorphic representation of a split classical group or a unitary group ${\bf G}_n$ and $\tau$ a cuspidal (unitary) automorphic representation of a general linear group, then $L(s,\pi \times \tau)$ is holomorphic for $\Re(s) > 1$ 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 L-functions for $\Re(s) > 1$. Finally, we complete previous results on functoriality for the classical groups over function fields.