Epsilon theorem

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In number theory, Serre's epsilon conjecture stated a property of Galois representations associated with modular forms which was proven by Ken Ribet in the summer of 1986 (and hence is the epsilon theorem). It is a significant step towards the proof of Fermat's Last Theorem: the theorem proves that if the Taniyama-Shimura-Weil conjecture is true (which it is), Fermat's Last Theorem is also true.

1 Statement of the epsilon theorem
2 The Frey curve
3 Application of epsilon to the Frey curve
4 Fermat's Last Theorem
5 See also
6 References

[edit] Statement of the epsilon theorem

The epsilon theorem, boiled down to its fundamentals, states the following:

Suppose E is an elliptic curve with integer coefficients in global minimal form. If it has discriminant Δ a product of primes p with exponents δ_p, and conductor N a product of primes p with exponents n_p, and if E is a modular curve (now known to be true), 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, then we can descend to another elliptic curve E' with conductor N' = N/p, such that the L-series coefficients for L(s, E) and those for L(s, E') are congruent modulo ℓ.

[edit] The Frey curve

In his thesis Yves Hellegouarch defined what is now called the Frey curve. Frey then suggested that such curves would have peculiar properties, and in particular not be modular. Serre reformulated the question in terms of Galois representations, and proved all but "ε" to show that Frey was correct and that a Frey curve was not modular.

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

y² = x(x-a^ℓ)(x+b^ℓ),

[edit] Application of epsilon to the Frey curve

It is not too difficult to show that the discriminant of the Frey curve is 16 (abc)^2ℓ, and the conductor N is the product of all distinct primes dividing abc. Since each prime which divides N divides it only once, we can perform level descent modulo ℓ, but then we will divide out all the odd primes and reach X₀(2) of genus zero, so there is no elliptic curve, a contradiction.

[edit] Fermat's Last Theorem

In 1994 Fermat's Last Theorem was proven by Andrew Wiles and Richard Taylor in two separate papers published in 1995 in the Annals of Mathematics by showing that in fact, semistable elliptic curves, which includes the Frey curves, are modular and by doing so proved Fermat's Last Theorem.