Obstruction criteria for modular deformation problems

TL;DR

This paper refines obstruction criteria for modular deformation problems, removing previous restrictions and establishing conditions linking unobstructed deformations to level congruences of newforms.

Contribution

It improves bounds on primes for unobstructed deformation problems and removes the squarefree level restriction, also relating unobstructed deformations to congruences with lower level forms.

Findings

Explicit bounds on primes outside which deformation problems are unobstructed.

Removal of the squarefree level hypothesis in previous results.

Unobstructed deformation implies no congruence to lower level newforms.

Abstract

For a newform $f=\sum a_n q^n$ of weight $k \geq 3$ and a prime $\lambda$ of $\mathbf{Q}(a_n)$, the deformation problem for its associated mod $\lambda$ Galois representation is unobstructed for all primes outside some finite set. Previous results gave an explicit bound on this finite set for $f$ of squarefree level; we modify this bound and remove the squarefree hypothesis. We also show that if the $\lambda$-adic deformation problem for $f$ is unobstructed, then $f$ is not congruent mod $\lambda$ to a newform of lower level.