Notes from the course have been typeset by Steve Alberts, and are available here.

Mondays, 13:45 in room 402.

It will be useful to have a list of your email addresses etc, to let you know about changes to the lecture times, homework sheets etc. To register informally for the course, please send an email to

ellipticcurvesleiden at gmail dot com

with the subject line `registration', and containing your name, student number, which year you are in (Bachelor/1st year master/other), and which university you are studying at if not Leiden. Please note this is separate from (and does not replace) registing on USIS.

Elliptic curves have been studied for at least 2,500 years, and are everywhere in algebraic geometry and number theory - from Fermat's last theorem to cryotography. This course will give something of an overview of a few of the places elliptic curves appear. In particular, we will study some of:

- rational points (and prove the Mordell-Weil theorem in moderate generality);

- integral points (Thue's theorem);

- elliptic curves over finite fields (and applications to factorisation);

- complex multiplication;

- ...

We will try to cover a range of topics rather than working in maximal generality. Hopefully this will serve the two aims of

- giving some nice applications of the commutative algebra etc you have already learnt;

- motivate you to keep studying commutative algebra etc, and to take a course in algebraic geometry, so that you can see how to do the things I cover in this course more naturally and in greater generality.

We WILL assume familiarity with some basic category theory, at least up to the level of natural transformations and equivalences, and probably more later on. If you have never seen these before (or want some extra reading) you could read the relevant pages on wikipedia, or look at the first 3 chapters of this set of notes and attempt some of the exercises.

- it contains some nice backgound material which we won't have time for in the lectures;

- it contains extra exercises (in case I don't set enough);

- it gives an alternative point of view, especially on algebraic varieties, from the lectures, which may be helpful. I prefer the approach I will take (or else I would not take it), but perhaps it helps to see another approach.

Pages 1-8

Pages 9-16

Pages 17-22

Pages 23-45

Pages 46-59

Pages 60-79

Pages 80-86

Sheet for week 1 (due 9/2/2015)

Sheet for week 2 (due 16/2/2015)

Sheet for week 3 (due 23/2/2015)

Sheet for week 4 (due 2/3/2015)

Sheet for week 5 (due 9/3/2015)

Sheet for week 6 (due 16/3/2015, by email only)

Sheet for week 7 (due 23/3/2015, by email only)

Sheet for week 8 (due 13/4/2015)

Sheet for week 9 (due 20/4/2015)

Sheet for week 10 (due 27/4/2015)

Sheet for week 11 (due 11/5/2015)

Sheet for week 12 (due 18/5/2015). Final sheet.

Handout on localisation of commutative rings. May be helpful for doing your homework.

Studyguide page

David Holmes <holmesdst@math.leidenuniv.nl>