Families of symplectic Galois representations over small parabolic eigenvarieties for Siegel cuspforms of genus $2$
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| Format: | Preprint |
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2026
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| _version_ | 1866914492313501696 |
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| author | Manji, Muhammad Thøgersen, Frederick E. Wu, Ju-Feng |
| author_facet | Manji, Muhammad Thøgersen, Frederick E. Wu, Ju-Feng |
| contents | We construct small parabolic eigenvarieties for holomorphic Siegel cuspforms of genus $2$ and study families of Galois representations attached to them in the spirit of Bellaïche--Chenevier. In the course, we introduce the notion of $(φ, Γ)$-modules with $G$-structures and the notion of refined families of symplectic Galois representations by implementing the theory of symplectic Galois determinant d'après Moakher--Quast. We then prove an infinitesimal $R=\mathbb{T}$ theorem under mild hypotheses. As an application, we study the relationship between the geometry of the small parabolic eigenvarieties at the Saito--Kurokawa lifts for cuspidal eigenforms (both finite-slope and infinite-slope) and the Bloch--Kato Selmer groups of those eigenforms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_18433 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Families of symplectic Galois representations over small parabolic eigenvarieties for Siegel cuspforms of genus $2$ Manji, Muhammad Thøgersen, Frederick E. Wu, Ju-Feng Number Theory 11F33 11F46, 11F80 We construct small parabolic eigenvarieties for holomorphic Siegel cuspforms of genus $2$ and study families of Galois representations attached to them in the spirit of Bellaïche--Chenevier. In the course, we introduce the notion of $(φ, Γ)$-modules with $G$-structures and the notion of refined families of symplectic Galois representations by implementing the theory of symplectic Galois determinant d'après Moakher--Quast. We then prove an infinitesimal $R=\mathbb{T}$ theorem under mild hypotheses. As an application, we study the relationship between the geometry of the small parabolic eigenvarieties at the Saito--Kurokawa lifts for cuspidal eigenforms (both finite-slope and infinite-slope) and the Bloch--Kato Selmer groups of those eigenforms. |
| title | Families of symplectic Galois representations over small parabolic eigenvarieties for Siegel cuspforms of genus $2$ |
| topic | Number Theory 11F33 11F46, 11F80 |
| url | https://arxiv.org/abs/2604.18433 |