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

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 $(\varphi, \Gamma)$-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 $R=\mathbb{T}$ 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.