$R=T$ theorems for weight one modular forms

TL;DR

This paper establishes $R=T$ theorems for certain weight one modular forms, connecting Galois representations and Hecke algebras, and extends modularity results to non-classical forms under specific conditions.

Contribution

It proves new $R=T$ theorems for weight one forms, including non-classical cases, and clarifies the structure of deformation rings for specific Galois representations.

Findings

Proves modularity of certain 2-dimensional Galois representations.

Establishes $R=T$ results for non-classical weight 1 forms.

Shows isomorphism between deformation rings and Hecke algebras for quadratic characters.

Abstract

We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $\chi$. For certain non-quadratic $\chi$ we prove an $R=T$ result for $T$ the weight 1 specialisation of the Hida Hecke algebra acting on non-classical weight 1 forms, under the assumption that no two Hida families congruent to an Eisenstein series cross in weight 1. For quadratic $\chi$ we prove that the quotient of $R$ corresponding to deformations split at $p$ is isomorphic to the Hecke algebra acting on classical CM weight 1 modular forms.