Ribet's theorem

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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 proven by Ken Ribet. The proof of the 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.

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 which gives rise to the same Galois representation).[1]

Statement

Let f be a weight 2 newform on Γ0(qN)–i.e. of level qN where q does not divide N–with absolutely irreducible 2-dimensional mod p Galois representation ρf,p unramified at q if qp and finite flat at q = p. Then there exists a weight 2 newform g of level N such that

${\displaystyle \rho _{f,p}\simeq \rho _{g,p}.}$

In particular, if E is an elliptic curve over ${\displaystyle \mathbb {Q} }$ with conductor qN, then the modularity theorem guarantees that there exists a weight 2 newform f of level qN such that the 2-dimensional mod p Galois representation ρf, p of f is isomorphic to the 2-dimensional mod p Galois representation ρE, p of E. To apply Ribet's Theorem to ρE, p, it suffices to check the irreducibility and ramification of ρE, p. Using the theory of the Tate curve, one can prove that ρE, p is unramified at qp and finite flat at q = p if p divides the power to which q appears in the minimal discriminant ΔE. Then Ribet's theorem implies that there exists a weight 2 newform g of level N such that ρg, pρE, p.

The result of level lowering

Note that Ribet's theorem does not guarantee that if one begins with an elliptic curve E of conductor qN, there exists an elliptic curve E' of level N such that ρE, pρE′, p. The newform g of level N 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

${\displaystyle E:y^{2}+xy+y=x^{3}-663204x+206441595}$

with conductor 43×97 and discriminant 437 × 973 does not level-lower mod 7 to an elliptic curve of conductor 97. Rather, the mod p Galois representation is isomorphic to the mod p Galois representation of an irrational newform g of level 97.

However, for p large enough compared to the level N 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 pNN1+ε, the mod p Galois representation of a rational newform cannot be isomorphic to that of an irrational newform of level N.[2]

Similarly, the Frey-Mazur conjecture predicts that for p large enough (independent of the conductor N), elliptic curves with isomorphic mod p 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 p (in particular p > 17).

History

In his thesis, Yves Hellegouarch [fr] came up with the idea of associating solutions (a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve.[3] If p is an odd prime and a, b, and c are positive integers such that

${\displaystyle a^{p}+b^{p}=c^{p},}$

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

${\displaystyle y^{2}=x(x-a^{p})(x+b^{p}).}$

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

In 1982 Gerhard Frey called attention to the unusual properties of the same curve as Hellegouarch, now called a Frey curve.[4] 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.[5] In 1985 Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof of this.[6][7] 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, Kenneth Alan Ribet proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implied Fermat's Last Theorem.[8]

Implications for Fermat's Last Theorem

Suppose that the Fermat equation with exponent p ≥ 5[8] had a solution in non-zero integers a, b, c. Let us form the corresponding Frey curve Eap,bp,cp. It is an elliptic curve and one can show that its minimal discriminant Δ is equal to 2−8 (abc)2p and its conductor N is the radical of abc, i.e. the product of all distinct primes dividing abc. By an elementary consideration of the equation ap + bp = cp, it is clear that one of a, b, c is even and hence so is N. By the Taniyama–Shimura conjecture, E is a modular elliptic curve. Since all odd primes dividing a,b,c in N appear to a pth power in the minimal discriminant Δ, by Ribet's theorem one can perform level descent modulo p repetitively to strip off all odd primes from the conductor. However, there are no newforms of level 2 as the genus of the modular curve X0(2) is zero (and newforms of level N are differentials on X0(N)).

Notes

1. ^ "The Proof of Fermat's Last Theorem". 2008-12-10.
2. ^ Silliman, Jesse; Vogt, Isabel (2015). "Powers in Lucas Sequences via Galois Representations". Proceedings of the American Mathematical Society. 143 (3): 1027–1041. arXiv:1307.5078. CiteSeerX 10.1.1.742.7591. doi:10.1090/S0002-9939-2014-12316-1. MR 3293720.
3. ^ Hellegouarch, Yves (1972). "Courbes elliptiques et equation de Fermat". Doctoral Dissertation.
4. ^ Frey, Gerhard (1982), "Rationale Punkte auf Fermatkurven und getwisteten Modulkurven" [Rational points on Fermat curves and twisted modular curves], J. Reine Angew. Math. (in German), 331 (331): 185–191, doi:10.1515/crll.1982.331.185, MR 0647382
5. ^ Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae, 1 (1): iv+40, ISSN 0933-8268, MR 0853387
6. ^ Serre, J.-P. (1987), "Lettre à J.-F. Mestre [Letter to J.-F. Mestre]", Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemporary Mathematics (in French), 67, Providence, RI: American Mathematical Society, pp. 263–268, doi:10.1090/conm/067/902597, ISBN 9780821850749, MR 0902597
7. ^ 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, MR 0885783
8. ^ a b Ribet, Ken (1990). "On modular representations of Gal(Q/Q) arising from modular forms" (PDF). Inventiones Mathematicae. 100 (2): 431–476. Bibcode:1990InMat.100..431R. doi:10.1007/BF01231195. MR 1047143.