TY - JOUR

T1 - Rationality and holomorphy of Langlands–Shahidi L-functions over function fields

AU - Lomelí, Luis Alberto

N1 - Funding Information:
Acknowledgements I would like to thank Günter Harder for enlightening conversations that we held during the time the article was written. I thank Guy Henniart and Freydoon Shahidi for their encouragement to work on this project. I would like to thank W. Casselman for useful discussions concerning the rationality of the Langlands–Shahidi local coefficient. I also thank R. Ganapathy, V. Heiermann, B. Lemaire, D. Prasad and S. Varma for mathematical communications. The anonymous referee made helpful comments and suggestions, for which I am grateful. The Mathematical Sciences Research Institute and the Max-Planck Institute für Mathematik provided excellent working conditions while a Postdoctoral Fellow, when a preliminary version of this article was written. Work was concluded at the Instituto de Matemáticas PUCV, where I have found a great home institution. Work on this article was supported in part by MSRI NSF Grant DMS 0932078, Project VRIEA/PUCV 039.367/2016 and FONDECYT Grant 1171583.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/2/11

Y1 - 2019/2/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85047943644&partnerID=8YFLogxK

U2 - 10.1007/s00209-018-2100-7

DO - 10.1007/s00209-018-2100-7

M3 - Article

AN - SCOPUS:85047943644

VL - 291

SP - 711

EP - 739

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1-2

ER -