On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras

TL;DR

This paper investigates the structure and density of supercuspidal points in local deformation rings and their relation to global Hecke algebras, revealing conditions under which these points are dense in certain components.

Contribution

It establishes conditions where the Zariski closure of supercuspidal points equals a union of irreducible components in local deformation rings and explores analogous properties in global Hecke algebras.

Findings

Supercuspidal points are dense in certain local deformation components.

The Zariski closure of these points matches unions of irreducible components under specific conditions.

Analogous density results are obtained for global Hecke algebras.

Abstract

We study the Zariski closure of points in local deformation rings corresponding to potential semi-stable representations with certain prescribed $p$-adic Hodge theoretic properties. We show in favourable cases that the closure is equal to a union of irreducible components of the deformation space. We also study an analogous question for global Hecke algebras.