Ribet's theorem

Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT). As shown by Serre and Ribet, the Taniyama–Shimura conjecture (whose status was unresolved at the time) and the epsilon conjecture together imply that FLT is true. In mathematical terms, Ribet's theorem shows that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form that gives rise to the same representation). Let f be a weight 2 newform on Γ0(qN) – i.e. of level qN where q does not divide N – with absolutely irreducible 2-dimensional mod p Galois representation ρf,p unramified at q if q ≠ p and finite flat at q = p. Then there exists a weight 2 newform g of level N such that In particular, if E is an elliptic curve over with conductor qN, then the modularity theorem guarantees that there exists a weight 2 newform f of level qN such that the 2-dimensional mod p Galois representation ρf, p of f is isomorphic to the 2-dimensional mod p Galois representation ρE, p of E. To apply Ribet's Theorem to ρE, p, it suffices to check the irreducibility and ramification of ρE, p. Using the theory of the Tate curve, one can prove that ρE, p is unramified at q ≠ p and finite flat at q = p if p divides the power to which q appears in the minimal discriminant ΔE. Then Ribet's theorem implies that there exists a weight 2 newform g of level N such that ρg, p ≈ ρE, p. Ribet's theorem states that beginning with an elliptic curve E of conductor qN does not guarantee the existence of an elliptic curve E of level N such that ρE, p ≈ ρE, p. The newform g of level N may not have rational Fourier coefficients, and hence may be associated to a higher-dimensional abelian variety, not an elliptic curve.