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

Abstract

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 $$\pi $$ is a globally generic cuspidal automorphic representation of a split classical group or a unitary group and $$\tau $$ is a cuspidal (unitary) automorphic representation of a general linear group, then $$L(s,\pi \times \tau )$$ is holomorphic for $$\mathfrak {R}(s) > 1$$ 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 $$\mathfrak {R}(s) > 1$$ . Finally, we complete previous results on functoriality for the classical groups over function fields with applications to the Ramanujan Conjecture and Riemann Hypothesis.