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 (u...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2019-02, Vol.291 (1-2), p.711-739
Main Author: Lomelí, Luis Alberto
Format: Article
Language:English
Subjects:
Mathematical analysis
Mathematicians
Mathematics
Mathematics and Statistics
Polynomials
Representations
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Summary: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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-018-2100-7