Klingen Eisenstein series congruences and modularity

2025/01/17 by Tobias Berger, Jim Brown, Berger, Tobias +3
Mathematics · #11F33 #11F46 #11F67 #11F80 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2501.10327

openalex publication_date 2025/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a mod ℓ congruence between a Klingen Eisenstein series (associated to a classical newform ϕ of weight k) and a Siegel cusp form f with irreducible Galois representation. We use this congruence to show non-vanishing of the Bloch-Kato Selmer group H¹_f(Q, \textrmad⁰ρ_ϕ(2-k)⊗ Q_ℓ/Z_ℓ) under certain assumptions and provide an example. We then prove an R=dvr theorem for the Fontaine-Laffaille universal deformation ring of ρ_f under some assumptions, in particular, that the residual Selmer group H¹_f(Q, \textrmad⁰ρ_ϕ(k-2)) is cyclic. For this we prove a result about extensions of Fontaine-Laffaille modules. We end by formulating conditions for when H¹_f(Q, \textrmad⁰ρ_ϕ(k-2)) is non-cyclic and the Eisenstein ideal is non-principal.

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