Ribet's Theorem - Statement

Statement

Let E be an elliptic curve with integer coefficients in a Neron minimal model. Suppose that the discriminant Δ of E is written as the product Πpδ_p of prime powers pδ_p and similarly the conductor N of E is the product Πpn_p of prime powers. Suppose that E is a modular elliptic curve, then we can perform a level descent modulo primes ℓ dividing one of the exponents δ_p of a prime dividing the discriminant. If pδ_p is an odd prime power factor of Δ and if p divides N only once (i.e. n_p=1), then there exists another elliptic curve E', with conductor N' = N/p, such that the coefficients of the L-series of E are congruent modulo ℓ to the coefficients of the L-series of E' .

The epsilon conjecture is a relative statement: assuming that a given elliptic curve E over Q is modular, it predicts the precise level of E.