New approaches to Fermat’s Last Theorem.
(Nouvelles approches du “théorème” de Fermat.)

*(French)*Zbl 0668.10024
Sémin. Bourbaki, 40ème Année, Vol. 1987/88, Exp. No. 694, Astérisque 161/162, 165-186 (1988).

[For the entire collection see Zbl 0659.00006.]

Fermat’s Last “Theorem” says that the equation \(A^ n+B^ n+C^ n=0\) has no solutions in coprime, non-zero integers \(A, B, C\), whenever \(n\geq 3\). This problem has attracted the attention of many mathematicians over the past 300 years, with the result, for example, that the theorem is known for all \(n\leq 150 000\). In this well-written and extremely informative survey, the author describes some new approaches to the Fermat problem which have recently attracted considerable interest.

The initial observation, due to Frey, is that if \((A,B,C)\) is a solution to Fermat’s equation with prime exponent \(p\), then one should look at the elliptic curve \[ E: Y^ 2=X(X-A^ p)(X+B^ p). \] This is an elliptic curve, defined over \(\mathbb Q\), which has such amazing properties that one can hope to show that \(E\) does not exist.

For example, the discriminant of \(E\) is essentially \((ABC)^{2p}\), while its conductor is at most \(2ABC\). This would contradict a conjecture of Szpiro, and also the “abc-conjecture” of Masser and the author. The author describes the relationships between these conjectures, and shows how they would imply Fermat’s Theorem.

Next comes a description of Serre’s conjecture, which asserts that every continuous representation \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to \text{GL}_ 2(\overline{\mathbb F}_ p)\) which is absolutely irreducible and odd, is obtained from a cusp form for \(\Gamma_ 0(N)\) by a process described by Deligne. Further, Serre (conjecturally) describes the level, weight, and character of the cusp form in terms of the conductor, weight, and character of the representation. The author explains how the representation on the \(p\)- torsion subgroup of \(E\) and Serre’s conjecture would imply the existence of a cusp form of weight 2 for \(\Gamma_ 0(2)\). Since there are no such modular forms, Fermat’s Theorem follows. The author also describes some other consequences of Serre’s conjecture, including the conjecture of Taniyama-Weil that every elliptic curve over \(\mathbb Q\) is modular (i.e. is parametrized by modular functions).

Since both Fermat’s Theorem and the Taniyama-Weil conjecture are consequences of Serre’s conjecture, this suggests a relationship between the two. The final part of the paper describes theorems of Mazur and Ribet which give conditions under which a modular representation has lower level than one would expect. The more difficult case, which was recently resolved by Ribet, has as a consequence that the conjecture of Taniyama-Weil implies Fermat’s Theorem. The author gives a brief sketch of the proof of Mazur’s and Ribet’s result which should be helpful for anyone planning to study the details of these difficult theorems.

Fermat’s Last “Theorem” says that the equation \(A^ n+B^ n+C^ n=0\) has no solutions in coprime, non-zero integers \(A, B, C\), whenever \(n\geq 3\). This problem has attracted the attention of many mathematicians over the past 300 years, with the result, for example, that the theorem is known for all \(n\leq 150 000\). In this well-written and extremely informative survey, the author describes some new approaches to the Fermat problem which have recently attracted considerable interest.

The initial observation, due to Frey, is that if \((A,B,C)\) is a solution to Fermat’s equation with prime exponent \(p\), then one should look at the elliptic curve \[ E: Y^ 2=X(X-A^ p)(X+B^ p). \] This is an elliptic curve, defined over \(\mathbb Q\), which has such amazing properties that one can hope to show that \(E\) does not exist.

For example, the discriminant of \(E\) is essentially \((ABC)^{2p}\), while its conductor is at most \(2ABC\). This would contradict a conjecture of Szpiro, and also the “abc-conjecture” of Masser and the author. The author describes the relationships between these conjectures, and shows how they would imply Fermat’s Theorem.

Next comes a description of Serre’s conjecture, which asserts that every continuous representation \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to \text{GL}_ 2(\overline{\mathbb F}_ p)\) which is absolutely irreducible and odd, is obtained from a cusp form for \(\Gamma_ 0(N)\) by a process described by Deligne. Further, Serre (conjecturally) describes the level, weight, and character of the cusp form in terms of the conductor, weight, and character of the representation. The author explains how the representation on the \(p\)- torsion subgroup of \(E\) and Serre’s conjecture would imply the existence of a cusp form of weight 2 for \(\Gamma_ 0(2)\). Since there are no such modular forms, Fermat’s Theorem follows. The author also describes some other consequences of Serre’s conjecture, including the conjecture of Taniyama-Weil that every elliptic curve over \(\mathbb Q\) is modular (i.e. is parametrized by modular functions).

Since both Fermat’s Theorem and the Taniyama-Weil conjecture are consequences of Serre’s conjecture, this suggests a relationship between the two. The final part of the paper describes theorems of Mazur and Ribet which give conditions under which a modular representation has lower level than one would expect. The more difficult case, which was recently resolved by Ribet, has as a consequence that the conjecture of Taniyama-Weil implies Fermat’s Theorem. The author gives a brief sketch of the proof of Mazur’s and Ribet’s result which should be helpful for anyone planning to study the details of these difficult theorems.

Reviewer: Joseph H. Silverman (Providence)

##### MSC:

11D41 | Higher degree equations; Fermat’s equation |

11G05 | Elliptic curves over global fields |

11F80 | Galois representations |

14J25 | Special surfaces |