Modularity Theorem (Taniyama–Shimura–Weil)

Wiles proved modularity for semistable elliptic curves over $\mathbb{Q}$ in 1994–95 (with the final step, a numerical criterion for isomorphism between deformation rings and Hecke algebras — the '$R=T$ theorem' — established jointly with Richard Taylor). The strategy shows that the Galois representation on the $\ell$-adic Tate module of $E$, and the corresponding representation attached to a candidate modular form, live in the same deformation space; proving the deformation ring $R$ and the Hecke algebra $T$ acting on modular forms coincide forces every allowed Galois representation — in particular $E$'s — to come from a modular form. Since every semistable curve suffices to rule out a counterexample to Fermat's equation (any solution $a^n+b^n=c^n$ would yield a semistable Frey curve $y^2=x(x-a^n)(x+b^n)$ that Kenneth Ribet had shown, via his 1990 proof of $\varepsilon$-conjecture, cannot be modular), this proved Fermat's Last Theorem. The semistability restriction was later removed entirely, extending modularity to all elliptic curves over $\mathbb{Q}$, by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor in 2001.