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

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 Bella\"iche--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'apr\`es 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.