Theorems Taylor–Wiles $R = \mathbb{T}$ Theorem Attributions

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Statement

Let $\mathcal{O}$ be a complete discrete valuation ring with maximal ideal $\mathfrak{p}_{\mathcal{O}}$, residue field $k = \mathcal{O}/\mathfrak{p}_{\mathcal{O}}$, and fraction field of characteristic $0$. Let

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\begin{align*} \bar{\rho}: G_{\mathbb{Q}} \to GL_2(k) \end{align*}

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be a continuous, absolutely irreducible, odd, modular residual representation. Fix a finite set of primes $S$ containing the prime $\ell$, all primes at which $\bar{\rho}$ is ramified, and all primes at which deformation conditions are imposed. Fix a determinant character

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\begin{align*} \delta: G_{\mathbb{Q}} \to \mathcal{O}^{\times} \end{align*}

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lifting $\det \bar{\rho}$.

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For each $v \in S$, let $\mathcal{D}_v$ be a representable local deformation condition for $\bar{\rho}|_{G_{\mathbb{Q}_v}}$ with determinant $\delta|_{G_{\mathbb{Q}_v}}$, including the prescribed ordinary or finite-flat condition at $v=\ell$ and the prescribed minimal or semistable condition at primes $v \neq \ell$. Let $\mathcal{D}$ be the resulting global fixed-determinant deformation problem: for every complete Noetherian local $\mathcal{O}$-algebra $A$ with residue field $k$, $\mathcal{D}(A)$ is the set of strict equivalence classes of continuous lifts

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\begin{align*} \rho_A: G_{\mathbb{Q}} \to GL_2(A) \end{align*}

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of $\bar{\rho}$ such that $\det \rho_A = \delta$, $\rho_A$ is unramified outside $S$, and $\rho_A|_{G_{\mathbb{Q}_v}}$ satisfies $\mathcal{D}_v$ for every $v \in S$. Suppose that $\mathcal{D}$ is represented by a complete local $\mathcal{O}$-algebra $R$.

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Let $\mathbb{T}_{\mathfrak{m}}$ be the complete local Hecke algebra obtained by localising at the maximal ideal $\mathfrak{m}$ cut out by $\bar{\rho}$ on the space of weight $2$ modular forms of the level and local type matching the deformation problem. Assume that there exists a continuous Galois representation

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\begin{align*} \rho_{\mathbb{T}}: G_{\mathbb{Q}} \to GL_2(\mathbb{T}_{\mathfrak{m}}) \end{align*}

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lifting $\bar{\rho}$, satisfying the same fixed determinant and local deformation conditions, and satisfying

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\begin{align*} \operatorname{tr}\bigl(\rho_{\mathbb{T}}(\operatorname{Frob}_p)\bigr)=T_p \end{align*}

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for every prime $p \notin S$ at which the Hecke operator $T_p$ is defined. Assume moreover that $\mathbb{T}_{\mathfrak{m}}$ is topologically generated as an $\mathcal{O}$-algebra by these Hecke operators $T_p$.

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If the Taylor–Wiles auxiliary prime construction applies to this deformation problem and produces a patched deformation ring $R_{\infty}$, a patched Hecke algebra $\mathbb{T}_{\infty}$, a patched Hecke module $M_{\infty}$, and a continuous homomorphism

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\begin{align*} \varphi_{\infty}: R_{\infty} \to \mathbb{T}_{\infty} \end{align*}

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such that:

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1. $\varphi_{\infty}$ is compatible with the natural homomorphism

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\begin{align*} \varphi: R \to \mathbb{T}_{\mathfrak{m}} \end{align*}

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induced by $\rho_{\mathbb{T}}$ after quotienting by the Taylor–Wiles augmentation ideal;

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2. the action of $R_{\infty}$ on $M_{\infty}$ factors through $\mathbb{T}_{\infty}$ via $\varphi_{\infty}$; 3. $M_{\infty}$ is faithful as an $R_{\infty}$-module; 4. the patched descent identifies $R$ and $\mathbb{T}_{\mathfrak{m}}$ with the corresponding quotients of $R_{\infty}$ and $\mathbb{T}_{\infty}$ by the same Taylor–Wiles augmentation ideal;

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then the natural homomorphism

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\begin{align*} \varphi: R \to \mathbb{T}_{\mathfrak{m}} \end{align*}

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is an isomorphism of complete local $\mathcal{O}$-algebras.

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