𝑅=𝑇 theorems for weight one modular forms

Berger, Tobias; Klosin, Krzysztof

doi:10.1090/tran/9001

$R=T$ theorems for weight one modular forms
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by Tobias Berger and Krzysztof Klosin;

Trans. Amer. Math. Soc. 376 (2023), 8095-8128

DOI: https://doi.org/10.1090/tran/9001

Published electronically: August 22, 2023

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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 cuspidal Hida Hecke algebra acting on non-classical weight 1 forms. Under the additional assumption that no two cuspidal Hida families congruent to an Eisenstein series cross in weight 1 we show that $T$ is reduced. 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.

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