# Ramification of Hilbert eigenvarieties at classical points

## TL;DR

This paper investigates the local geometric structure of Hilbert eigenvarieties at classical points, linking ramification over the weight space to Galois representation properties, and provides new bounds on tangent space dimensions.

## Contribution

It computes lower bounds for tangent space dimensions at classical points and characterizes ramification points via Galois representation splitting behavior.

## Key findings

- Lower bounds for tangent space dimensions at classical points.
- Characterization of ramified points over the weight space.
- Connection between ramification and local Galois splitting behavior.

## Abstract

Andreatta-Iovita-Pilloni constructed eigenvarieties for cuspidal Hilbert modular forms. The eigenvariety has a natural map to the weight space, called the weight map. At a classical point, we compute a lower bound of the dimension of the tangent space of the fiber of the weight map using Galois deformation theory. Along with the classicality theorem due to Tian-Xiao, this enables us to characterize the classical points of the eigenvariety which are ramified over the weight space, in terms of the local splitting behavior of the associated Galois representation.