Ribet's Theorem

In mathematics, Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is a statement in number theory concerning properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof of the epsilon conjecture was a significant step towards the proof of Fermat's Last Theorem. As shown by Serre and Ribet, the Taniyama–Shimura conjecture (whose status was unresolved at the time) and the epsilon conjecture together imply that Fermat's Last Theorem is true.

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 ρ_f,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 ${\displaystyle \mathbb {Q} }$ 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 ρ_{f, p} of f is isomorphic to the 2-dimensional mod p Galois representation ρ_{E, p} of E. To apply Ribet's Theorem to ρ_{E, p}, it suffices to check the irreducibility and ramification of ρ_{E, p}. Using the theory of the Tate curve, one can prove that ρ_{E, 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 Δ_E. Then Ribet's theorem implies that there exists a weight 2 newform g of level N such that ρ_{g, p} ≈ ρ_{E, p}.

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