Irreducibility of limits of Galois representations of Saito-Kurokawa type

Abstract

We prove (under certain assumptions) the irreducibility of the limit σ2 of a sequence of irreducible essentially self-dual Galois representations σk:GQ→GL4(Q¯p) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to 1⊕ρ⊕χ with ρ irreducible, two-dimensional of determinant χ, where χ is the mod p cyclotomic character. More precisely, we assume that σk are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as k→2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to ρ) which we assume are non-zero.

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Item Type:	Article
Authors/Creators:	Berger, T. https://orcid.org/0000-0002-5207-6221 Klosin, K.
Copyright, Publisher and Additional Information:	© The Author(s) 2021. Open Access: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Keywords:	Galois representations; The paramodular conjecture; p-adic Siegel modular forms
Dates:	Published: 4 June 2021 Published (online): 4 June 2021 Accepted: 20 April 2021
Institution:	The University of Sheffield
Academic Units:	The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User:	Symplectic Sheffield
Date Deposited:	26 Apr 2021 12:57
Last Modified:	07 Jun 2021 09:45
Status:	Published
Publisher:	Springer Verlag
Refereed:	Yes
Identification Number:	10.1007/s40993-021-00265-x
Related URLs:	arXiv URL
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:173259

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Irreducibility of limits of Galois representations of Saito-Kurokawa type

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