Ribet's theorem

In mathematics, Ribet's theorem (earlier called the epsilon conjecture or ε-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 Fermat's Last Theorem is true.

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 Πp^δ_p of prime powers p^δ_p and similarly the conductor N of E is the product Πp^n_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 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 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.

History

In his thesis, Yves Hellegouarch came up with the idea of associating solutions (a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve.^{[citation needed]}. If ℓ is an odd prime and a, b, and c are positive integers such that

$a^\ell + b^\ell = c^\ell,\$

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

$y^2 = x(x - a^\ell)(x + b^\ell).\$

This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q. In 1982 Gerhard Frey called attention to the unusual properties of the same curve as Hellegouarch, now called a Frey curve. This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular. The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem. However, his argument was not complete. In 1985, Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof of this. This showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Serre did not provide a complete proof and what was missing became known as the epsilon conjecture or ε-conjecture. In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implied Fermat's Last Theorem.

Implication of 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.

See also

• abc conjecture
• Modularity theorem
• Wiles' proof of Fermat's Last Theorem

References

• Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae 1 (1): iv+40, ISSN 0933-8268, MR853387 
• Frey, Gerhard (1982), "Rationale Punkte auf Fermatkurven und getwisteten Modulkurven", J.reine u.angew.Math 331: 185–191 
• J.F Mestre and J.P Serre (1987). "Current Trends in Arithmetical Algebraic Geometry". Contemporary Mathematics 67: 263–268. 
• Serre, Jean-Pierre (1987), "Sur les représentations modulaires de degré 2 de Gal(Q/Q)", Duke Mathematical Journal 54 (1): 179–230, doi:10.1215/S0012-7094-87-05413-5, ISSN 0012-7094, MR885783 
• Ken Ribet (1990). "On modular representations of $\scriptstyle\operatorname{Gal}(\bar{Q}/Q)$ arising from modular forms" (PDF). Inventiones mathematicae 100 (2): 431–471. http://math.berkeley.edu/~ribet/Articles/invent_100.pdf. 
• Kenneth Ribet, From the Taniyama-Shimura conjecture to Fermat's last theorem. Annales de la faculté des sciences de Toulouse Sér. 5, 11 no. 1 (1990), p. 116–139.
• Andrew Wiles (May 1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics 141 (3): 443–551. doi:10.2307/2118559. http://math.stanford.edu/~lekheng/flt/wiles.pdf. 
• Richard Taylor and Andrew Wiles (May 1995). "Ring-theoretic properties of certain Hecke algebras" (PDF). Annals of Mathematics (Annals of Mathematics) 141 (3): 553–572. doi:10.2307/2118560. ISSN 0003486X. JSTOR 2118560. OCLC 37032255. Zbl 0823.11030. http://math.stanford.edu/~lekheng/flt/taylor-wiles.pdf. 
• Frey Curve and Ribet's Theorem

External links

• Ken Ribet and Fermat's Last Theorem by Kevin Buzzard June 28, 2008