Ribet's Theorem

Ribet's theorem, proved by Ken Ribet, shows that if the Galois representation attached to an elliptic curve has certain properties then that curve cannot be modular. Ribet proved the result, originally conjectured by Jean-Pierre Serre as the epsilon conjecture, in the summer of 1986 and published the formal proof in 1990. Because the epsilon conjecture together with the Taniyama-Shimura conjecture implies Fermat's Last Theorem, Ribet's proof showed that establishing the modularity theorem would be enough to prove Fermat's Last Theorem, which Andrew Wiles did five years later in 1995.