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On the local converse theorem and the descent theorem in families

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Abstract

In this paper, we prove an analogue of Jacquet’s conjecture on the local converse theorem for \(\ell \)-adic families of co-Whittaker representations of \({\mathrm {GL}}_n(F)\), where F is a finite extension of \({\mathbb {Q}}_p\) and \(\ell \ne p\). We also prove an analogue of Jacquet’s conjecture for a descent theorem, which asks for the smallest collection of gamma factors determining the subring of definition of an \(\ell \)-adic family. These two theorems are closely related to the local Langlands correspondence in \(\ell \)-adic families.

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Acknowledgements

Baiying Liu is partially supported by NSF Grants DMS-1702218, DMS-1848058, and by start-up funds from the Department of Mathematics at Purdue University. Gilbert Moss was supported by the FSMP postdoctoral fellowship in France. The authors would like to thank Matthew Emerton, David Helm, Hervé Jacquet, Dihua Jiang, and Shaun Stevens for their interest in this work, encouragement, and helpful conversations and suggestions, and Mahdi Asgari for helpful discussions on stability. The authors also would like to thank the referee for the careful reading of our manuscript and also the valuable comments and suggestions.

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  1. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907, USA

    Baiying Liu

  2. Department of Mathematics, University of Utah, 155 S 1400 E, Room 233, Salt Lake City, UT, 84112, USA

    Gilbert Moss

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  1. Baiying Liu
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  2. Gilbert Moss
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Correspondence to Gilbert Moss.

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Liu, B., Moss, G. On the local converse theorem and the descent theorem in families. Math. Z. 295, 463–483 (2020). https://doi.org/10.1007/s00209-019-02350-5

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