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

TL;DR

This paper constructs small parabolic eigenvarieties for genus 2 Siegel cuspforms, introduces new notions of Galois modules, and proves an infinitesimal R=T theorem, linking geometry and Selmer groups.

Contribution

It introduces $( heta, au)$-modules with $G$-structures and refined symplectic Galois families, advancing the understanding of Galois representations in this setting.

Findings

Construction of small parabolic eigenvarieties for genus 2 Siegel cuspforms.

Introduction of $( heta, au)$-modules with $G$-structures.

Proof of an infinitesimal R=T theorem under mild hypotheses.

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.