Ribet's theorem

{{safesubst:#invoke:Unsubst||date=__DATE__|$B= Template:Ambox }} Template:More citations needed Template:Short description Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT). As shown by Serre and Ribet, the Taniyama–Shimura conjecture (whose status was unresolved at the time) and the epsilon conjecture together imply that FLT is true.

In mathematical terms, Ribet's theorem shows that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form that gives rise to the same representation).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Statement

Let Template:Math be a weight 2 newform on Template:Math – i.e. of level Template:Math where Template:Math does not divide Template:Math – with absolutely irreducible 2-dimensional mod Template:Math Galois representation Template:Math unramified at Template:Math if Template:Math and finite flat at Template:Math. Then there exists a weight 2 newform Template:Math of level Template:Math such that

<math> \rho_{f,p} \simeq \rho_{g,p}. </math>

In particular, if Template:Math is an elliptic curve over <math>\mathbb{Q}</math> with conductor Template:Math, then the modularity theorem guarantees that there exists a weight 2 newform Template:Math of level Template:Math such that the 2-dimensional mod Template:Math Galois representation Template:Math of Template:Math is isomorphic to the 2-dimensional mod Template:Math Galois representation Template:Math of Template:Math. To apply Ribet's Theorem to Template:Math, it suffices to check the irreducibility and ramification of Template:Math. Using the theory of the Tate curve, one can prove that Template:Math is unramified at Template:Math and finite flat at Template:Math if Template:Math divides the power to which Template:Math appears in the minimal discriminant Template:Math. Then Ribet's theorem implies that there exists a weight 2 newform Template:Math of level Template:Math such that Template:Math.

Level lowering

Ribet's theorem states that beginning with an elliptic curve Template:Math of conductor Template:Math does not guarantee the existence of an elliptic curve Template:Math of level Template:Math such that Template:Math. The newform Template:Math of level Template:Math may not have rational Fourier coefficients, and hence may be associated to a higher-dimensional abelian variety, not an elliptic curve. For example, elliptic curve 4171a1 in the Cremona database given by the equation

<math>E: y^2 + xy + y = x^3 - 663204x + 206441595</math>

with conductor Template:Math and discriminant Template:Math does not level-lower mod 7 to an elliptic curve of conductor 97. Rather, the mod Template:Math Galois representation is isomorphic to the mod Template:Math Galois representation of an irrational newform Template:Math of level 97.

However, for Template:Math large enough compared to the level Template:Math of the level-lowered newform, a rational newform (e.g. an elliptic curve) must level-lower to another rational newform (e.g. elliptic curve). In particular for Template:Math, the mod Template:Math Galois representation of a rational newform cannot be isomorphic to an irrational newform of level Template:Math.<ref>Template:Cite journal</ref>

Similarly, the Frey-Mazur conjecture predicts that for large enough Template:Math (independent of the conductor Template:Math), elliptic curves with isomorphic mod Template:Math Galois representations are in fact isogenous, and hence have the same conductor. Thus non-trivial level-lowering between rational newforms is not predicted to occur for large Template:Math.

History

In his thesis, Template:Interlanguage link multi originated the idea of associating solutions (a,b,c) of Fermat's equation with a different mathematical object: an elliptic curve.<ref>Template:Cite journal</ref> If p is an odd prime and a, b, and c are positive integers such that

<math>a^p + b^p = c^p,</math>

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

<math>y^2 = x(x - a^p)(x + b^p).</math>

This is a nonsingular algebraic curve of genus one defined over <math>\mathbb{Q}</math>, and its projective completion is an elliptic curve over <math>\mathbb{Q}</math>.

In 1982 Gerhard Frey called attention to the unusual properties of the same curve, now called a Frey curve.<ref>Template:Citation</ref> This provided a bridge between Fermat and Taniyama by showing that a counterexample to FLT would create a curve that would not be modular. The conjecture attracted considerable interest when Frey suggested that the Taniyama–Shimura conjecture implies FLT. However, his argument was not complete.<ref>Template:Citation</ref> In 1985 Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof.<ref>Template:Citation</ref><ref>Template:Citation</ref> This showed that a proof of the semistable case of the Taniyama–Shimura conjecture would imply FLT. Serre did not provide a complete proof and the missing bit became known as the epsilon conjecture or ε-conjecture. In the summer of 1986, Kenneth Alan Ribet proved the epsilon conjecture, thereby proving that the Modularity theorem implied FLT.<ref name="ribet">Template:Cite journal</ref>

The origin of the name is from the ε part of "Taniyama-Shimura conjecture + ε ⇒ Fermat's last theorem".

Implications

Suppose that the Fermat equation with exponent Template:Math<ref name="ribet"/> had a solution in non-zero integers Template:Math. The corresponding Frey curve Template:Math is an elliptic curve whose minimal discriminant Template:Math is equal to Template:Math and whose conductor Template:Math is the radical of Template:Math, i.e. the product of all distinct primes dividing Template:Math. An elementary consideration of the equation Template:Math, makes it clear that one of Template:Math is even and hence so is N. By the Taniyama–Shimura conjecture, Template:Math is a modular elliptic curve. Since all odd primes dividing Template:Math in Template:Math appear to a Template:Math power in the minimal discriminant Template:Math, by Ribet's theorem repetitive level descent modulo Template:Math strips all odd primes from the conductor. However, no newforms of level 2 remain because the genus of the modular curve Template:Math is zero (and newforms of level N are differentials on Template:Math.

See also

• ABC conjecture
• Wiles' proof of Fermat's Last Theorem

Notes

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References

• 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.
• Template:Cite journal
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• Frey Curve and Ribet's Theorem

External links

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