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

Lomelí, Luis Alberto

doi:10.1007/s00209-018-2100-7

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

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Lomelí, Luis Alberto
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s00209-018-2100-7
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arXiv:1505.06043.pdf
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Citation

Lomelí, L. A. (2019). Rationality and holomorphy of Langlands-Shahidi L-functions over function fields. Mathematische Zeitschrift, 291(1-2), 711-739. doi:10.1007/s00209-018-2100-7.

Cite as: https://hdl.handle.net/21.11116/0000-0004-F6E7-2

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 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.