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arXiv · 2501.10327

Klingen Eisenstein series congruences and modularity

Tobias Berger·Jim Brown·Krzysztof Klosin
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Abstract

We construct a mod $\ell$ 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 $H^1_f(\mathbf{Q}, \textrm{ad}^0ρ_ϕ(2-k)\otimes \mathbf{Q}_{\ell}/\mathbf{Z}_{\ell})$ under certain assumptions and provide an example. We then prove an $R=dvr$ theorem for the Fontaine-Laffaille universal deformation ring of $\overlineρ_f$ under some assumptions, in particular, that the residual Selmer group $H^1_f(\mathbf{Q}, \textrm{ad}^0\overlineρ_ϕ(k-2))$ is cyclic. For this we prove a result about extensions of Fontaine-Laffaille modules. We end by formulating conditions for when $H^1_f(\mathbf{Q}, \textrm{ad}^0\overlineρ_ϕ(k-2))$ is non-cyclic and the Eisenstein ideal is non-principal.

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Tobias Berger, Jim Brown, Krzysztof Klosin. 2025-09-05. Klingen Eisenstein series congruences and modularity. https://arxiv.org/abs/2501.10327

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