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, in in a significant step towards the proof of Fermat's Last Theorem. To meet Wikipedias quality standards, this article or section may require cleanup. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense... Modular form - Wikipedia /**/ @import /skins-1. ... Kenneth Alan Ken Ribet is an American mathematician, currently a professor of mathematics at the University of California, Berkeley. ... Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats last theorem (edition of 1670). ...

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. In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as HΓ where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices. ...

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. In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense...

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),

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.

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. Sir Andrew John Wiles (April 11, 1953) is a British-American mathematician. ... There are several people called Richard Taylor: Richard Taylor (UK politician), independent Member of Parliament Richard Taylor (mathematician), involved in completing the proof of Fermats Last Theorem Richard Taylor (physicist), Canadian winner of the 1990 Nobel Prize Richard Taylor (movies), head of Weta Workshop Richard Taylor (musician), former member... The Annals of Mathematics (ISSN 0003-486X), often just called Annals, is a bimonthly mathematics research journal published by Princeton University and the Institute for Advanced Study. ... In mathematics, a semistable elliptic curve in diophantine geometry is an elliptic curve that has bad reduction only of multiplicative type. ... In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as HΓ where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices. ...

See also

• Abc conjecture

The abc conjecture in number theory was first formulated by Joseph Oesterlé and David Masser in 1985. ...

References

• Anthony W. Knapp, Elliptic Curves, Princeton, 1992
• Ken Ribet (1990). "On modular representations of Gal (.../Q) arising from modular forms". Inventiones mathematicae 100 (2): 431-471.
• Andrew Wiles (May 1995). "Modular elliptic curves and Fermat's Last Theorem". Annals of Mathematics 141 (3): 443-551.