Local models for Galois deformation rings and applications

TL;DR

This paper constructs geometric models for Galois deformation rings, revealing their structure and applying these insights to prove cases of the Breuil-Mézard and Serre's conjectures in number theory.

Contribution

It introduces new geometric models for tamely potentially crystalline Galois deformation rings and derives significant consequences for key conjectures in number theory.

Findings

Models exhibit unibranch property at special points

Irreducible components described via representation theory

Proves Breuil-Mézard conjecture in arbitrary dimension

Abstract

We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular Hodge-Tate weights. We establish several significant facts about their geometry including a unibranch property at special points and a representation theoretic description of the irreducible components of their special fibers. We derive from these geometric results a number of local and global consequences: the Breuil-M\'ezard conjecture in arbitrary dimension for tamely potentially crystalline deformation rings with small Hodge-Tate weights (with appropriate genericity conditions), the weight part of Serre's conjecture for $U(n)$ as formulated by Herzig (for global Galois representations which satisfy the Taylor-Wiles hypotheses and are sufficiently generic at $p$), and an unconditional formulation of the weight part of Serre's conjecture for wildly ramified representations.