On automorphic points in polarized deformation rings

TL;DR

This paper proves that in certain polarized deformation spaces of Galois representations, every component contains automorphic points, extending previous results and establishing their density in specific cases.

Contribution

It generalizes B"ockle's result to polarized Galois representations over CM and totally real fields, showing all components contain automorphic points under certain conditions.

Findings

Every irreducible component of the polarized deformation space contains an automorphic point.

Automorphic points are Zariski dense in the polarized deformation space in dimension three.

The results apply under assumptions like a small R = T theorem and local mod p conditions.

Abstract

For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, B\"{o}ckle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouv\^{e}a-Mazur. We generalize B\"{o}ckle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under assumptions necessary to apply a small $R = \mathbb{T}$ theorem and an assumption on the local mod $p$ representation, we prove that every irreducible component of the universal polarized deformation space contains an automorphic point. When combined with work of Chenevier, this implies new results on the Zariski density of automorphic points in polarized deformation space in dimension three.