Ribet's theorem

Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory.

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.

In particular for p ≫ NN1+ε, the mod p Galois representation of a rational newform cannot be isomorphic to an irrational newform of level N.[2] Similarly, the Frey-Mazur conjecture predicts that for large enough p (independent of the conductor N), elliptic curves with isomorphic mod p Galois representations are in fact isogenous, and hence have the same conductor.

In his thesis, Yves Hellegouarch [fr] originated the idea of associating solutions (a,b,c) of Fermat's equation with a different mathematical object: an elliptic curve.

[4] This provided a bridge between Fermat and Taniyama by showing that a counterexample to FLT would create a curve that would not be modular.

An elementary consideration of the equation ap + bp = cp, makes it clear that one of a, b, c is even and hence so is N. By the Taniyama–Shimura conjecture, E is a modular elliptic curve.