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

Step 6 of 11: Ribet's level-lowering theorem: the Frey curve cannot be modular

In plain words

If the Frey curve $E_{a,b,c}$ were modular, it would correspond to a cusp form of weight $2$ and level $N = \operatorname{rad}(abc)$ . Because the discriminant $\Delta = (abc)^{2p}$ is a perfect $p$ -th power at every odd prime factor $q \mid N$ , the curve's mod- $p$ arithmetic is artificially smooth at $q$ .

Jean-Pierre Serre conjectured (the $\varepsilon$ -conjecture, 1985) and Ken Ribet proved in the summer of 1986 (published 1990) that you can strip away one such prime factor $q$ after another from the level $N$ without changing the mod- $p$ shadow of the modular form. Stripping away every odd prime factor of $N = \operatorname{rad}(abc)$ drives the level all the way down to $N = 2$ —forcing a nonzero weight- $2$ cusp form of level $2$ to exist, when the space $S_2(\Gamma_0(2))$ is known to be completely empty.

E_{a,b,c} \text{ modular of level } N = \operatorname{rad}(abc) \xRightarrow{\text{Ribet 1990}} \exists\, f \in S_2(\Gamma_0(2)) = \{0\}

Detailed analysis

In 1985, Jean-Pierre Serre formulated the precise mechanism linking Frey's curve to the Taniyama–Shimura–Weil conjecture as the $\varepsilon$ -conjecture, and Ken Ribet proved it in the summer of 1986 (Ribet, Inventiones Mathematicae 100, 1990; Wiles 1995, p. 443 and p. 448; Qiu et al. 2025, Section 2.2 and Section 3.2.3).

Ribet's level-lowering theorem states that if $f$ is a weight- $2$ eigenform of square-free level $N$ , and the mod- $p$ Galois representation $\overline{\rho}_{f,p}$ is irreducible and unramified (finite flat) at an odd prime $q \mid N$ , then there is another weight- $2$ eigenform $g$ of level $N/q$ whose mod- $p$ representation is isomorphic to $\overline{\rho}_{f,p}$ (using a theorem of Mazur when $q = p$ ; see Qiu et al. 2025, Section 3.2.3). For the Frey curve $E_{a,b,c}$ with $p \ge 5$ , the exponent of every odd prime $q \mid N = \operatorname{rad}(abc)$ in the minimal discriminant is a multiple of $p$ , which makes $\overline{\rho}_{E_{a,b,c},p}$ finite flat at every odd prime $q \mid N$ . Iterating Ribet's theorem removes every odd prime from $N$ , leaving a weight- $2$ eigenform $g \in S_2(\Gamma_0(2))$ of level $2$ .

By a classical dimension calculation on the modular curve $X_0(2)$ , the space $S_2(\Gamma_0(2))$ of weight- $2$ cusp forms at level $2$ has dimension $0$ . Therefore $E_{a,b,c}$ cannot be modular. Since $E_{a,b,c}$ is semistable (Step 3), Ribet's theorem reduces Fermat's Last Theorem to a single statement about elliptic curves: prove that every semistable elliptic curve over $\mathbb{Q}$ is modular (Wiles 1995, Theorem 0.4 $\implies$ Theorem 0.5, p. 448).

Terms in this step

Serre's $\varepsilon$ -conjecture (Ribet's theorem): The theorem proved by Ken Ribet (1986, published 1990) showing that if the mod- $p$ Galois representation of a weight- $2$ modular form of level $N$ is unramified at a prime $q \mid N$ , the level can be lowered from $N$ to $N/q$ . Applied to the Frey curve, it lowers the level to $2$ , where no weight- $2$ cusp form exists.

Knowledge used in this step

Fermat's Last Theorem Modular forms