Elliptic Curves - DIAMANT / Mastermath, Fall 2013

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
Week 37: WN-F612
Week 38: WN-M143
Week 39: WN-F123
Week 40: WN-P647, time changed to 11.00 -- 13.30
Week 41-42: WN-M639
Week 43: WN-F123
Week 44: HG-02A05, time changed to 11.00 -- 13.30
Week 45: no class
Week 46: K.02 in the Bungehuis of the UvA; it is a 15 minute walk from Amsterdam Central Station, see this!
Week 47: HG-08A05
Week 48: HG-02A05
Week 49: WN-S101, WN-S103, (WN-S353)
Week 50-51: K.02 in the Bungehuis of the UvA; it is a 15 minute walk from Amsterdam Central Station, see this!
"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.


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.


110 SeptemberPractical information, introduction to elliptic curves, history, examples, language of algebraic geometry (algebraic sets, varieties, coordinate ring, function field, dimension). Four of these exercises.
217 SeptemberProjective space and varieties, rational maps, morphisms, tangent space, elliptic curves, Weierstrass form, group law on elliptic curves. Four of these exercises.
324 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.
41 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.
58 October Chapter 2 of the lecture notes. Four of Exercises 8-18, 20-23 of Chapter 2 of the lecture notes.
615 October Chapter 3 of the lecture notes. Four of Exercises 8-18 of Chapter 3 of the lecture notes.
722 October See these notes. Four of these exercises.
829 October Torsion points of elliptic curves, isogenies, Frobenius morphism (see Chapter 3 of [Silverman1]). Four of these exercises.
5 November No class
912 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.
1019 November Complex multiplication on elliptic curves. Four of these exercises.
1126 November Mordell-Weil theorem. Four of these exercises.
123 December SAGE workshop, see this. This worksheet.
1310 December Algorithms and calculations The final assignment.
1417 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: Elliptic Curves. PDF
[Cassels]J.W.S. Cassels: Lectures on Elliptic Curves §§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.
[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: Elliptic Curves 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.
[Silverman-Tate] Newcomers to the subject are suggested to buy the book J.H. Silverman and J. Tate: Rational Points on Elliptic Curves. 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.
[Silverman1] Advanced students with a good knowledge of algebraic geometry are recommended to (also) buy J.H. Silverman: The arithmetic of elliptic curves. Corrected reprint of the 1986 original. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1992. ISBN: 0-387-96203-4.
[Silverman2]Further references: J.H. Silverman: Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics 151, Springer-Verlag, 1994. ISBN: 0-387-94328-5.

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