Irreducibility of limits of Galois representations of Saito-Kurokawa\n type

2020/03/26 by Tobias Berger, Berger, Tobias, Krzysztof Klosin +1
Mathematics · #11F80 #11R34 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2003.11822

openalex publication_date 2020/03/26 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

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

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