Unobstructed Hilbert modular deformation problems

TL;DR

This paper investigates conditions under which the deformation rings of certain Galois representations associated with Hilbert modular forms are unobstructed, extending prior work on classical modular forms to the Hilbert setting.

Contribution

It generalizes the unobstructed deformation problem to Hilbert modular forms, providing explicit criteria and examples for primes where the deformation rings are unobstructed.

Findings

Universal deformation rings are unobstructed for almost all primes under certain conditions.

Verification of local invariants is key to establishing unobstructedness.

Explicit examples illustrate how to determine primes with unobstructed deformation rings.

Abstract

Consider the semisimple mod p reduction of the Galois representation associated to a Hilbert newform f by Carayol and Taylor. This paper discusses how, under certain conditions on f, the universal ring for deformations of this residual representation with fixed determinant is unobstructed for almost all primes. We follow the approach of Weston, who carried out a similar program for classical modular forms in 2004. As such, the problem essentially comes down to verifying that various local invariants vanish at all places dividing p or the level of the newform. We conclude with an explicit example illustrating how one can in principle find a lower bound on p such that the universal ring for deformations of the residual representation attached to f with fixed determinant is unobstructed for all primes over p.