Modularity theorem

Remarks

Equivalently, the $L$-function of $E$ equals the $L$-function of $f$, or there is a nonconstant morphism $X_0(N) \to E$ defined over $\Q$ from the modular curve $X_0(N)$. Wiles, with a key step completed jointly with Taylor, proved the theorem for semistable curves in 1995; Breuil, Conrad, Diamond and Taylor completed the general case in 2001. The proof shows that the $\ell$-adic Galois representation of $E$ is modular. It lifts the modularity of the reduction modulo $3$, known by the Langlands–Tunnell theorem, with a modularity lifting theorem. A switch to the prime $5$ handles the cases where the reduction modulo $3$ is not irreducible. Example: $y^2 + y = x^3 - x^2$ has conductor $11$, and it corresponds to $f = q \prod_{n \ge 1} (1 - q^n)^2 (1 - q^{11n})^2$. The semistable case, combined with the earlier work of Frey, Serre and Ribet, implies Fermat's Last Theorem.

Going deeper

For a student

Modularity gives the analytic continuation and functional equation of $L(E, s)$, which are needed even to state the Birch and Swinnerton-Dyer conjecture. The parametrisation $X_0(N) \to E$ lets one construct Heegner points on $E$, the input of the Gross–Zagier theorem. It is a case of the reciprocity conjectures of the Langlands program.