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

Resumen

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 L-functions for ℜ(s)>1. Finally, we complete previous results on functoriality for the classical groups over function fields.