Families of symplectic Galois representations over small parabolic eigenvarieties for Siegel cuspforms of genus 2

2026/04/20 by Muhammad Manji, Frederick E. Thøgersen, Ju-Feng Wu
#math.NT

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Abstract

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. Such families of symplectic Galois representations provide two applications: In the first application, we show that the small parabolic eigenvarieties are smooth at non-critical points by proving an infinitesimal R=\mathbbT theorem. In the second application, we study the relationship between the geometry of the small parabolic eigenvarieties at the Saito--Kurokawa lifts for cuspidal eigenforms (both finite- and infinite-slope) and the Bloch--Kato Selmer groups of those eigenforms.

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