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

Step 7 of 11: Galois representations on

p

-torsion points and Langlands–Tunnell at

p = 3

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In plain words

On an elliptic curve $E$ , the $m$ -torsion points $E[m] = \{Q \in E : mQ = \mathcal{O}\}$ are the points whose $m$ -fold sum is the identity $\mathcal{O}$ (like dividing a clock face into $m$ equal parts, except in two independent directions, giving $m^2$ points over $\overline{\mathbb{Q}}$ ). Any symmetry of the algebraic numbers $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ permutes the coordinates of these $m^2$ points linearly, so for a prime $p$ it acts as a $2 \times 2$ matrix mod $p$ .

To prove that an elliptic curve $E$ is modular, Wiles needed a foothold: at least one prime $p$ where the mod- $p$ shadow $\overline{\rho}_{E,p}$ is already known to come from a modular form. Because $2 \times 2$ matrices over $\mathbb{F}_3$ form a small solvable group ( $\operatorname{PGL}_2(\mathbb{F}_3) \cong S_4$ ), a deep theorem of Robert Langlands and Jerrold Tunnell guarantees that every irreducible mod- $3$ representation $\overline{\rho}_{E,3}$ is automatically modular!

\overline{\rho}_{E,3}: \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}_2(\mathbb{F}_3) \;\text{ irreducible } \xRightarrow{\text{Langlands–Tunnell}} \overline{\rho}_{E,3} \text{ is modular}

Detailed analysis

For an integer $m \ge 1$ , the $m$ -torsion subgroup $E[m] = \{Q \in E(\overline{\mathbb{Q}}) : mQ = \mathcal{O}\}$ is isomorphic to $(\mathbb{Z}/m\mathbb{Z})^2$ (Qiu et al. 2025, Section 3.3.1, Definition 3.2). For a prime $p$ , the absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on $E[p] \cong \mathbb{F}_p^2$ , giving the mod- $p$ Galois representation $\overline{\rho}_{E,p}: \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}_2(\mathbb{F}_p)$ , and compatibly across all powers $E[p^n] \cong (\mathbb{Z}/p^n\mathbb{Z})^2$ to give the $p$ -adic Galois representation $\rho_{E,p}: \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}_2(\mathbb{Z}_p)$ (Wiles 1995, p. 444; Qiu et al. 2025, Section 3.3.2). For every prime $q \nmid Np$ , $\operatorname{trace}\rho_{E,p}(\operatorname{Frob}_q) = a_q(E)$ , so $E$ is modular if and only if $\rho_{E,p}$ comes from a modular form.

As Wiles explains in his introduction (Wiles 1995, p. 444–447 and p. 449), after several months studying the $2$ -adic representation in 1986, his first breakthrough was switching to $p = 3$ . The projective group $\operatorname{PGL}_2(\mathbb{F}_3)$ is isomorphic to the symmetric group $S_4$ , which is solvable, so the theorem of Langlands (1980) and Tunnell (1981) on Artin's conjecture for octahedral representations implies that whenever $\overline{\rho}_{E,3}$ is irreducible, $\overline{\rho}_{E,3}$ is modular (arising from a weight- $1$ form, and hence congruent to a weight- $2$ eigenform).

This gives Wiles his starting base case at the bottom of the $3$ -adic tower $\mathbb{Z}/3\mathbb{Z} \leftarrow \mathbb{Z}/3^2\mathbb{Z} \leftarrow \cdots \leftarrow \mathbb{Z}_3$ : if $\overline{\rho}_{E,3}$ is irreducible, one already knows the mod- $3$ shadow is modular and only needs to prove that every semistable $3$ -adic lifting $\rho_{E,3}$ of $\overline{\rho}_{E,3}$ remains modular (Wiles 1995, Theorem 0.2 and Theorem 0.3, p. 447). What if $\overline{\rho}_{E,3}$ happens to be reducible? That requires the $3$ – $5$ switch in Step 8.

Terms in this step

$m$ -torsion points $E[m]$: The set of points $Q \in E(\overline{\mathbb{Q}})$ satisfying $mQ = \mathcal{O}$ under the chord-and-tangent addition law. Over $\overline{\mathbb{Q}}$ , $E[m]$ forms a group isomorphic to $(\mathbb{Z}/m\mathbb{Z}) \times (\mathbb{Z}/m\mathbb{Z})$ of size $m^2$ .
Galois representation ( $\overline{\rho}_{E,p}$ and $\rho_{E,p}$ ): The homomorphism recording how symmetries in $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ permute the $p$ -torsion points $E[p]$ (giving $2 \times 2$ matrices over $\mathbb{F}_p$ , written $\overline{\rho}_{E,p}$ ) or the entire $p$ -power torsion tower $E[p^n]$ (giving $2 \times 2$ matrices over the $p$ -adic integers $\mathbb{Z}_p$ , written $\rho_{E,p}$ ).
Langlands–Tunnell theorem: A theorem of Robert Langlands (1980) and Jerrold Tunnell (1981) proving Artin's conjecture for two-dimensional representations whose projective image is solvable (such as $S_4 \cong \operatorname{PGL}_2(\mathbb{F}_3)$ ). It guarantees that every irreducible mod- $3$ representation $\overline{\rho}_{E,3}$ of an elliptic curve over $\mathbb{Q}$ is modular.

Knowledge used in this step

Galois theory Group theory Modular forms