Epsilon conjecture

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The epsilon conjecture is a statement in number theory concerning properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proved by Ken Ribet. The proof of 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 the Fermat's Last Theorem is true.

1 Statement of the epsilon conjecture
2 Application of the epsilon conjecture to Fermat's Last Theorem
- 2.1 Taniyama–Shimura plus epsilon implies Fermat's Last Theorem
- 2.2 Coda
3 See also
4 References

[edit] Statement of the epsilon conjecture

Let E be an elliptic curve with integer coefficients in a global minimal form. Denote by δ_p, respectively, n_p, the exponent with which a prime p appears in the prime factorization of the discriminant Δ of E, respectively, the conductor N of E. 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 p^δ_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 and congruent modulo ℓ to the coefficients of the L-series of E' .

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

[edit] Application of the epsilon conjecture to Fermat's Last Theorem

In his thesis, Yves Hellegouarch defined an object that is now called the Frey curve. If ℓ is an odd prime and a, b, and c are positive integers such that

a^ℓ + b^ℓ = c^ℓ,

then a corresponding Frey curve is an algebraic curve given by the equation

y² = x(x − a^ℓ)(x + b^ℓ).

This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q. Gerhard Frey suggested that any such curve would have peculiar properties, and in particular, will not be modular. In the early 1980s, Jean-Pierre Serre gave a reformulation in terms of Galois representations, and proved "all but ε" to show that Frey had been correct and that a Frey curve cannot be modular. The remaining ε is the epsilon conjecture.

[edit] Taniyama–Shimura plus epsilon implies Fermat's Last Theorem

Suppose that the Fermat equation with exponent ℓ ≥ 3 had a solution in non-zero integers a, b, c. Let us form the corresponding Frey curve E. It is an elliptic curve and one can show that its discriminant Δ is equal to 16 (abc)^2ℓ and its conductor N is the radical of abc, i.e. the product of all distinct primes dividing abc. By the Taniyama–Shimura conjecture, E is a modular elliptic curve. Since N is square-free, by the epsilon conjecture one can perform level descent modulo ℓ. Repeating this procedure, we will eliminate all odd primes from the conductor and reach the modular curve X₀(2) of level 2. However, this curve is not an elliptic curve since it has genus zero, resulting in a contradiction.

[edit] Coda

In 1994, Andrew Wiles and Richard Taylor completed a proof of a big part of the Taniyama–Shimura conjecture concerning the modularity of the semistable elliptic curves, which is sufficient to yield Fermat's Last Theorem. Their papers were published in 1995 in the Annals of Mathematics.