Klingen Eisenstein series congruences and modularity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916936520040448 |
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| author | Berger, Tobias Brown, Jim Klosin, Krzysztof |
| author_facet | Berger, Tobias Brown, Jim Klosin, Krzysztof |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Klingen Eisenstein series congruences and modularity Berger, Tobias Brown, Jim Klosin, Krzysztof Number Theory 11F33, 11F46, 11F80, 11F67 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. |
| title | Klingen Eisenstein series congruences and modularity |
| topic | Number Theory 11F33, 11F46, 11F80, 11F67 |
| url | https://arxiv.org/abs/2501.10327 |