This is the web page of the DIAMANT / Mastermath course Elliptic Curves.

## Announcements

- The grades are now available.
- For more information on how to use pari and mwrank from home or your own university, see this page.

## Organization

Lectures: | G.L.M. Cornelissen (Utrecht) P. Stevenhagen (Leiden) |

Problem session: | Peter Bruin (Leiden) Marco Streng (Leiden) |

Location: | W&N Building S205, Vrije Universiteit, Amsterdam |

Time: | Tuesdays, February 5 – May 13, 2008, 10:15 – 13:00 Each class there will be three 45 minutes time slots: two lectures and one exercise class. |

## Aim

Elliptic curves are fundamental objects in a large part of mathematics. Along various historical paths, their origins can be traced to calculus, complex analysis and algebraic geometry, and their arithmetic aspects have made them key objects in modern cryptography and in Wiles' proof of Fermat's last theorem. This course is an introduction to the algebraic, geometric, complex analytic and number theoretical aspects of the theory of elliptic curves.

## Program

date | lecturer | subject |

February 5 | PS | intro, course rules, history, elliptic integrals, analogy sin/cos [notes, § 1 and 2] |

February 12 | PS | elliptic functions, Weierstrass isomorphism, group structure, endomorphisms [notes, § 2 and 3] |

February 19 | GC | diophantine equations in two variables, conics, chord and tangent construction [Silverman-Tate, I.1 and I.2] |

February 26 | GC | basic projective geometry, bezout, nine point theorem, [Silverman-Tate, I.3, II.1,3–5, A1, A2; read I.4 and A.3] |

March 4 | GC | torsion group/Lutz–Nagell theorem, heights [Silverman-Tate] |

March 11 | PS | Mordell's theorem [notes and Cassels, § 14] |

March 18 | PS |
Mordell–Weil theorem; computation of the group of rational
points of an elliptic curve over Q [notes and Cassels, § 14] |

March 25 | break | |

April 1 | GC | elliptic curves over finite fields, Hasse theorem, zeta function [Milne, Section IV.9] |

April 8 | PS | the elliptic curve factorization method (ECM), elliptic curve primality proving (ECPP), elliptic curve cryptography (ECC) [Silverman-Tate, Section IV.4] |

April 15 | GC | computer class with pari and Mwrank (in room P4.47). |

April 22 | break | |

April 29 | GC | elliptic curves over local fields, local-global principle [Cassels, §2--5, 10, 18] |

May 6 | PS | complex multiplication [notes, § 3 and Cohen-Stevenhagen] |

May 13 | PS | complex multiplication |

May 20 | GC | modularity and Fermat's Last Theorem |

## Homework

due date | homework |

February 12 | Hand in 4 exercises from the first 14 pages of the lecture notes [notes]. |

February 19 | Hand in 4 exercises from sections 2 and 3 of the lecture notes [notes]. |

February 26 | Do exercises 1–3 from the exercise sheet and hand in exercise 2. |

March 4 | Do exercises 4, 5, 8, 9 and 10 from the exercise sheet and hand in exercise 5. (Note: in an older version of these exercises, there was a typo in exercise 5.) From [Silverman-Tate], do exercises 1.17, 2.1, 2.10, 2.11, 2.12, and if you don't know projective geometry, A.1,4,5,8,9. |

March 11 | Do exercises 3.1, 3.2, 3.3 of [Silverman-Tate] and hand in exercise 3.3. (Here is a copy of the exercises in case you don't have the book. A solution of exercise 3.2(a,b) can be found in the proof of Theorem 6.2 in [Silverman1].) |

March 18 | Hand in 4 exercises from the following list: exercise 3.17, exercise 3.18 and section 4 from the [notes], and section 14 of [Cassels]. |

April 1 | Hand in 4 exercises from section 4 (Weak Mordell-Weil)
from the
[notes] (not the section on
Elliptic curves over fields
from an older version) and section 14 of
[Cassels]. Do not hand in exercise 5
from the notes or the first curve of exercise 1 from Cassels. In
exercise 3 from the notes and exercise 1 from Cassels, each curve
counts as a separate exercise. |

April 8 | Do exercises 4.1, 4.2, 4.3 (a) and (b), 4.5, (4.6*), 4.8, 4.12 of [Silverman-Tate] and exercise IV 5.6* of [Silverman 1]. Hand in exercise 9.13 of [Milne, Chapter IV]. (* means hard exercise) |

April 15 | Hand in 4 exercises from the following set of exercises in [Silverman-Tate]. 4.1 -- 4.12, 4.13 (without using a computer), 4.17, 4.20 (for part (a), use the method of 4.19), 4.21 (especially recommended). |

April 29 | Complete the computer class and hand in the exercise mentioned in bold. For information on using the software at home or your own university, see this page. |

May 6 | Do as many exercises as possible from §2 of
[Cassels] and do exercise 1(i) of §10.
Prove that Selmer's curve 3x^{3} +
4y^{3} + 5z^{3} = 0 has points
in Q_{p} for all primes p (including the
‘infinite’ case of R), and do the same for the
curve x^{4} − 17 = 2y^{2} of Lind
and Reichardt. Hand in your solution for Selmer's curve. |

May 13 | See the exercise sheet. References: [notes, § 3] and [Cohen-Stevenhagen]. |

May 20 | See the new exercise sheet. |

## Description

The topics treated include a general discussion of elliptic integrals and functions, elliptic curves and their group law; Diophantine equations in two variables; Mordell's theorem, with computational aspects; Computer class (pari, MWrank); Elliptic curves over finite fields with applications (factoring integers, elliptic discrete logarithms and cryptography); complex multiplication; a survey of modularity of elliptic curves.

## Examination

The final grade will be based exclusively on homework.

## Prerequisites

Linear algebra, groups, rings, fields, complex variables.

## Literature

[notes] |
Lecture notes for Peter Stevenhagen's lectures:
P. Stevenhagen:
PDF, PSElliptic Curves. |

[Cassels] | J.W.S. Cassels: §§2–5 for the local-global
principle, and §14 for
2-descent. Here is a scanned copy
of §§2–6, 10 and 18,
and here is one of §14.Lectures on
Elliptic Curves |

[Cohen-Stevenhagen] | H. Cohen and P. Stevenhagen -
Computational class field theory. Chapter 15
in the upcoming book
on algorithmic number theory. See pages 518--519 for how to enumerate all lattices having CM by a given
ring. |

[Milne] | J.S. Milne: is electronically available
online and (according to
the book's web page)
the paperback version costs only $17.
Section IV.9 is a good reference for the Zeta function of a curve.
The book replaces Milne's lecture notes
that we linked to earlier: chapter 19 of the notes corresponds to section IV.9 of the book,
exercise 19.8(b) of the notes
corresponds to exercise 9.13 of chapter IV of the book.
Elliptic Curves |

[Silverman-Tate] |
Newcomers to the subject are suggested to buy the book
J.H. Silverman and J. Tate:
Undergraduate Texts in Mathematics, Springer-Verlag, Corr. 2nd printing,
1994, ISBN: 978-0-387-97825-3:
it contains a lot of the material treated in the course.Rational Points on Elliptic Curves. |

[Silverman1] |
Advanced students with a good knowledge of algebraic geometry
are recommended to (also) buy J.H. Silverman: Corrected
reprint of the 1986 original. Graduate Texts
in Mathematics, 106. Springer-Verlag, New York,
1992. ISBN: 0-387-96203-4.
The arithmetic of elliptic curves. |

[Silverman2] | Further references: J.H. Silverman:
Graduate Texts in Mathematics 151, Springer-Verlag, 1994.
ISBN: 0-387-94328-5.Advanced
topics in the arithmetic of elliptic curves. |

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