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

Lectures: | Peter Stevenhagen (Leiden) |

René Schoof (Leiden) | |

Problem session: | Chloe Martindale (Leiden) |

Abtien Javanpeykar (Leiden) | |

Yan Zhao (Leiden) | |

Weidong Zhuang (Leiden) | |

E-mail: | ec.mastermath.2013 (at) gmail.com. |

Location: | VU Amsterdam |

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"WN" means "Wis- en Natuurkundegebouw", Vrije
Universiteit, De Boelelaan 1081a, Amsterdam. The arrow on this map
points to an entrance. | |

"HG" means "Hoofdgebouw" (map),
Vrije Universiteit, De Boelelaan 1105, Amsterdam. | |

Time: | 10:15--13:00 |

Along various historical paths, the origins of elliptic curves 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's proof of Fermat's last theorem. This course is an introduction to both the theoretical and the computational aspects of elliptic curves.

The topics treated include a general discussion of elliptic curves and their group law, Diophantine equations in two variables, and Mordell's theorem. We will also discuss elliptic curves over finite fields with applications such as factoring integers, elliptic discrete logarithms, and cryptography. We will pursue both a theoretical and a computational approach.

The final grade will be based exclusively on homework, that is, the final grade is the average of all thirteen homework grades. The homework grades and the final grade can be found here.

Each week you must attempt four exercises from the exercise sheet, which will be posted following the lecture. These must be handed in by the beginning of the lecture the following week.

You are encouraged to use TeX or LaTeX to typeset your solutions, and to hand these in via e-mail to ec.mastermath.2013 (at) gmail.com. If you hand in your work via e-mail, you do not need to give us a second copy in the lecture.

The starred problems are more difficult, and you will be given credit for attempting these. If you wish to work together (which we encourage), please write up your answers individually. Almost identical answers will not be accepted.

Date | Subject | Homework | |
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1 | 10 September | Practical information, introduction to elliptic curves, history, examples, language of algebraic geometry (algebraic sets, varieties, coordinate ring, function field, dimension). | Four of these exercises. |

2 | 17 September | Projective space and varieties, rational maps, morphisms, tangent space, elliptic curves, Weierstrass form, group law on elliptic curves. | Four of these exercises. |

3 | 24 September | Algorithm III.2.3 of [Silverman1], discrete valuation rings, divisors, Riemann-Roch (see Chapter 2 (minus section 4) of [Silverman1]), Picard group, another definition of an elliptic curve, Proposition III.3.1 and Proposition III.3.4 of [Silverman1]. | Four of these exercises. |

4 | 1 October | Chapter 1 of the lecture notes for Peter Stevenhagen's lectures (from now on we will just refer to these as `the lecture notes'). | Four of Exercises 8, 10-18 of Chapter 1 of the lecture notes. |

5 | 8 October | Chapter 2 of the lecture notes. | Four of Exercises 8-18, 20-23 of Chapter 2 of the lecture notes. |

6 | 15 October | Chapter 3 of the lecture notes. | Four of Exercises 8-18 of Chapter 3 of the lecture notes. |

7 | 22 October | See these notes. | Four of these exercises. |

8 | 29 October | Torsion points of elliptic curves, isogenies, Frobenius morphism (see Chapter 3 of [Silverman1]). | Four of these exercises. |

5 November | No class | ||

9 | 12 November | Torsion points and dual isogenies, introduction to algorithms and ECC. | Four of the following exercises: the previous exercises and these new exercises apart from the ones you have already handed in and Exercise 15 and 16 which were done in the problem session. |

10 | 19 November | Complex multiplication on elliptic curves. | Four of these exercises. |

11 | 26 November | Mordell-Weil theorem. | Four of these exercises. |

12 | 3 December | SAGE workshop, see this. | This worksheet. |

13 | 10 December | Algorithms and calculations | The final assignment. |

14 | 17 December | BSD Conjecture | |

Group, ring and field theory (cf. the Leiden syllabi Algebra 1, 2 and 3 found here) and complex variables.

[notes] |
Lecture notes for Peter Stevenhagen's lectures:
P. Stevenhagen:
PDFElliptic 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, here of §§6–9, here of §§10–12,
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 following 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.
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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