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**Elliptic curves: Progress and applications.
(Elliptische Kurven: Fortschritte und Anwendungen.)**
*(German)*
Zbl 0708.14019

The author reports the recent progress on the structure of the natural group consisting of the rational points of an elliptic curve \(y^2=x^3+ax+b\), with \(a,b\) integers such that \(4a^3+27b^2\ne 0\). This group was proved to be finitely generated by Mordell in 1922, and is called the Mordell-Weil group; the problem is to determine the structure of its torsion subgroup and the value of the rank \(r\). The first problem is completely solved, due to the deep results of B. Mazur (1977) showing that there are exactly 15 possibilities for this group. Much less is known for the possible values of \(r\); it is conjectured that it can be arbitrarily large, but up to now one has not found cases in which \(r>14\). The main conjecture was formulated around 1960 by Birch and Swinnerton-Dyer; they considered the \(L\)-function attached to the elliptic curve, conjectured that \(L\) is defined and holomorphic for \(s=1\), and considered the first term \(C_0(s-1)^{r'}\) of its Taylor series; then their main conjecture is that \(r=r'\), and \(C_0\) is equal to \(C(r) | Ш |\), where \(C(r)\) is explicitly known is a function of \(r\), and \(Ш\) is the Tate-Shafarevich group associated to the curve.

Progress on this conjecture started in 1977 and is due to the works of Coates-Wiles (1977), Greenberg (1983), Gross-Zagier (1983), Rubin (1987) and Kolyvagin (1987). It only concerns modular elliptic curves, i.e. those which can be parametrized by modular functions relative to a subgroup \(\Gamma_0(N)\) of \(\mathrm{SL}(2,\mathbb{Z})\), the group of matrices \(\begin{pmatrix} A&B \\ C&D \end{pmatrix} \) with \(C\equiv 0\pmod N\). For these curves, \(L\) is an entire function and satisfies a functional equation \(L(2 - s)=\Lambda(s)L(z)\), where \(\Lambda(s)\) is explicitly known (and depends on \(N\)); a conjecture of Taniyama and Weil is that all elliptic curves are modular. At present, the most far-reaching results for these curves are the fact that if \(r'=1\), then \(r\ge 1\) (Gross-Zagier), and if \(r'=0\) then \(r=0\) and the Tate-Shafarevich group \(Ш\) is finite (Kolyvagin); but the first example of a curve with \(Ш\) finite had been given slightly earlier by Rubin.

A remarkable fact is that from theorems on elliptic curves many results may be deduced for problems in number theory which seem remote from the Mordell-Weil group. For instance Satgé has proved in 1986 that each integer \(n\) such that \(n/2\) is prime and \(n\equiv 4\pmod 9\) is the sum \(x^3+y^3\) with \(x, y\) rational. Another application is an effective lower bound for the number \(h(d)\) of classes in the imaginary quadratic field \(\mathbb{Q}(\sqrt{d})\) for \(d<0\). But the most surprising is the connection with the Fermat problem, discovered by Frey: if the Taniyama-Weil conjecture is true, then there are no solutions of \(A^n+B^n=C^n\) in integers \(\ne 0\) for \(n>2\); this is due to the fact that the assumption of existence of that solution would show that the elliptic curve \(y^2=x(x-A^n)(x-C^n)\), considered by Hellegouarch, would not be modular, as was proved by K. Ribet.

Progress on this conjecture started in 1977 and is due to the works of Coates-Wiles (1977), Greenberg (1983), Gross-Zagier (1983), Rubin (1987) and Kolyvagin (1987). It only concerns modular elliptic curves, i.e. those which can be parametrized by modular functions relative to a subgroup \(\Gamma_0(N)\) of \(\mathrm{SL}(2,\mathbb{Z})\), the group of matrices \(\begin{pmatrix} A&B \\ C&D \end{pmatrix} \) with \(C\equiv 0\pmod N\). For these curves, \(L\) is an entire function and satisfies a functional equation \(L(2 - s)=\Lambda(s)L(z)\), where \(\Lambda(s)\) is explicitly known (and depends on \(N\)); a conjecture of Taniyama and Weil is that all elliptic curves are modular. At present, the most far-reaching results for these curves are the fact that if \(r'=1\), then \(r\ge 1\) (Gross-Zagier), and if \(r'=0\) then \(r=0\) and the Tate-Shafarevich group \(Ш\) is finite (Kolyvagin); but the first example of a curve with \(Ш\) finite had been given slightly earlier by Rubin.

A remarkable fact is that from theorems on elliptic curves many results may be deduced for problems in number theory which seem remote from the Mordell-Weil group. For instance Satgé has proved in 1986 that each integer \(n\) such that \(n/2\) is prime and \(n\equiv 4\pmod 9\) is the sum \(x^3+y^3\) with \(x, y\) rational. Another application is an effective lower bound for the number \(h(d)\) of classes in the imaginary quadratic field \(\mathbb{Q}(\sqrt{d})\) for \(d<0\). But the most surprising is the connection with the Fermat problem, discovered by Frey: if the Taniyama-Weil conjecture is true, then there are no solutions of \(A^n+B^n=C^n\) in integers \(\ne 0\) for \(n>2\); this is due to the fact that the assumption of existence of that solution would show that the elliptic curve \(y^2=x(x-A^n)(x-C^n)\), considered by Hellegouarch, would not be modular, as was proved by K. Ribet.

Reviewer: Jean Dieudonné (Paris)

### MSC:

14H52 | Elliptic curves |

11G05 | Elliptic curves over global fields |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

11D25 | Cubic and quartic Diophantine equations |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

14G05 | Rational points |