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

We prove that all Langlands–Shahidi automorphic L-functions over function fields are rational; after twists by highly ramified characters they become polynomials; and, if π is a globally generic cuspidal automorphic representation of a split classical group or a unitary group and τ is 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 with applications to the Ramanujan Conjecture and Riemann Hypothesis.