Abstract

This thesis reports the three articles written by the author and his collaborators. These three papers concern various arithmetic aspects of the algebraic group $\GSp_{2g}$, which are interrelated under the theme of eigenvarieties.

We first present a construction of sheaves of overconvergnet Siegel modular forms by using the perfectoid method, originally introduced by Chojecki–Hansen–Johansson for automorphic
forms on compact Shimura curves over $\Q$. These sheaves are then proven to be isomorphic to the ones constructed by Andreatta--Iovita--Pilloni. Using perfectoid methods, we establish an overconvergent Eichler--Shimura morphism for Siegel modular forms, generalising the result of Andreatta--Iovita--Stevens for elliptic modular forms. More precisely, we establish a Hecke- and Galois-equivariant morphism from the overconvergent cohomology groups associated with $\GSp_{2g}$ to the space of overconvergent Siegel modular forms.

It was asked by Andreatta--Iovita--Pilloni whether the classical points of the eigenvariety parametrising the finite-slope cuspidal Siegel eigenforms are étale over the weight space. Inspired by Kim's pairing presented in the book of Bellaïche, which allows one to study the ramification locus of the eigencurve, we generalise Kim's pairing to study the ramification locus of the cuspidal eigenvariety for $\GSp_{2g}$, providing some partial answer to the question asked by Andreatta--Iovita--Pilloni.

Finally, it is expected that such a pairing not only allows one to study the geometry of the eigenvariety but also carries interesting arithmetic information. Inspired by the book of Bellaïche--Chenevier, we study families of Galois representations over the cuspidal eigenvariety for $\GSp_{2g}$. Under some reasonable hypotheses as well as some conditions, we deduce the vanishing of the adjoint Selmer group associated with the Galois representation attached to a cuspidal eigenclass in the cohomology of the Siegel modular variety.