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

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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.

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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.

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  1. Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile

    Luis Alberto Lomelí

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Lomelí, L.A. Rationality and holomorphy of Langlands–Shahidi L-functions over function fields. Math. Z. 291, 711–739 (2019). https://doi.org/10.1007/s00209-018-2100-7

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