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I have recently been reading into the proof of modularity of semistable elliptic curves, in particular (what is now known as) the Taylor-Wiles patching argument used to prove the $R=T$ theorem in the ...

Let $R$ be a deformation ring and $M$ be a finitely generated $R$-module. A strategy for proving the theorems $R=T$ is to associate with $(R,M)$ a Taylor-Wiles system denoted $(R_{Q},M_{Q})$. Here I'm ...

This question is about comparing Hecke algebras and endomorphism algebras. Let $\mathbf{A}_f$ be the ring of finite adèles of $\mathbf{Q}$ and let $K$ be a compact open subgroup of $\mathrm{GL}_2(\...

Let $h^{n,ord}(Np^\infty)$ be the cuspidal nearly ordinary Hecke algebra of tame level $N$. For $N \geq 4$, we know that the Hecke algebra is the generic fibre of the Hecke-Hilbert Eigenvariety and so ...

Let $\mathbb{T}_{\mathbb{Z}}$ be a $\mathbb{Z}$-module generated by Hecke operators $T_n$ acting on the space of cups forms $S_{k}(\Gamma,\mathbb{C})$ for the congruent subgroup satisfying $\Gamma_1(...