Worked solution: Wiles's modularity proof via the Taylor–Wiles method (1994)

Step 9 of 11: Deformation ring

R

versus Hecke algebra

\mathbb{T}

: the

R = \mathbb{T}

formulation

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In plain words

Once the mod- $p$ representation $\rho_0 = \overline{\rho}_{E,p}$ is known to be modular (with $p = 3$ or $p = 5$ ), how do you prove that the full $p$ -adic representation $\rho_{E,p}$ of $E$ is also modular? Instead of testing $\rho_{E,p}$ in isolation, Barry Mazur and Andrew Wiles packaged every possible $p$ -adic lifting of $\rho_0$ with controlled ramification into a single master ring—the universal deformation ring $R_{\Sigma}$ .

On the other side, all modular forms whose mod- $p$ shadow equals $\rho_0$ are packaged into a second ring—the Hecke algebra $\mathbb{T}_{\Sigma}$ . Because every modular lifting is in particular a Galois lifting, there is a natural surjective map $\varphi_{\Sigma}: R_{\Sigma} \twoheadrightarrow \mathbb{T}_{\Sigma}$ . Proving that every valid lifting of $\rho_0$ comes from a modular form is equivalent to proving that $\varphi_{\Sigma}$ is an isomorphism: $R_{\Sigma} \cong \mathbb{T}_{\Sigma}$ , famous as the $R = \mathbb{T}$ theorem.

\varphi_{\Sigma}: R_{\Sigma} \twoheadrightarrow \mathbb{T}_{\Sigma}, \quad \text{every lifting of } \rho_0 \text{ is modular } \iff R_{\Sigma} \xrightarrow{\sim} \mathbb{T}_{\Sigma}

Detailed analysis

In Chapter 1 of Wiles (1995, p. 454–460), let $\Sigma$ be a finite set of primes containing $p$ , and let $\rho_0: \operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q}) \to \operatorname{GL}_2(k)$ be an irreducible modular representation over a finite field $k$ of characteristic $p$ . Using Barry Mazur's deformation theory (1989) in the ordinary case and Ravi Ramakrishna's theorem (1993) in the flat case, there exists a complete Noetherian local ring $R_{\Sigma}$ and a universal deformation $\rho_{\Sigma}: \operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q}) \to \operatorname{GL}_2(R_{\Sigma})$ classifying all liftings of $\rho_0$ with prescribed local behavior at $p$ and ramification inside $\Sigma$ .

In Chapter 2 of Wiles (1995), let $\mathbb{T}_{\Sigma}$ be the completion of the Hecke algebra generated by the Hecke operators $T_q$ ( $q \notin \Sigma$ ) acting on weight- $2$ cusp forms at the appropriate level, localized at the maximal ideal $\mathfrak{m}$ corresponding to $\rho_0$ . Because the modular Galois representation over $\mathbb{T}_{\Sigma}$ satisfies the deformation conditions, the universal property of $R_{\Sigma}$ induces a surjective homomorphism of local rings $\varphi_{\Sigma}: R_{\Sigma} \twoheadrightarrow \mathbb{T}_{\Sigma}$ (Wiles 1995, p. 450–451).

In the spring of 1991, inspired by a commutative algebra paper of Ernst Kunz and John Tate's account of Grothendieck duality, Wiles discovered a numerical criterion (Appendix to Wiles 1995) showing that a surjection $\varphi_{\Sigma}: R_{\Sigma} \twoheadrightarrow \mathbb{T}_{\Sigma}$ onto a Gorenstein ring $\mathbb{T}_{\Sigma}$ is an isomorphism of complete intersections if and only if a size bound holds between two invariants: the tangent-space size $|\mathfrak{p}_R / \mathfrak{p}_R^2|$ (which is dual to a Galois cohomology Selmer group $H^1_{\Sigma}$ ) and the congruence invariant $|\mathcal{O} / \eta_{\mathbb{T}}|$ measuring congruences between modular forms (Wiles 1995, p. 451–452).

Terms in this step

Universal deformation ring $R_{\Sigma}$: The complete local ring introduced by Barry Mazur whose homomorphisms $R_{\Sigma} \to A$ classify all $p$ -adic Galois representations lifting the fixed mod- $p$ representation $\rho_0$ with allowed ramification in $\Sigma$ .
Hecke algebra $\mathbb{T}_{\Sigma}$: The commutative ring generated by the Hecke operators $T_q$ acting on modular forms, completed at the maximal ideal corresponding to $\rho_0$ . Its quotients correspond to modular Galois representations lifting $\rho_0$ .
Selmer group: A Galois cohomology group carved out by local conditions at each prime that measures the tangent space $|\mathfrak{p}_R / \mathfrak{p}_R^2|$ of the deformation ring $R_{\Sigma}$ , and therefore controls how large $R_{\Sigma}$ can be.

Knowledge used in this step

Galois theory Commutative algebra Modular forms