# Elliptic Curves, Spring 2015

## Teacher: David Holmes.

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

Mondays, 13:45 in room 402.

## Registering

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.

## Aims

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.

## Prerequisites

We will assume familiarity with Galois theory and (commutative) algebra, but will NOT assume any knowledge of algebraic geometry. We will spend the first week or so introducing algebraic varieties, intersection theory etc via functors of points. Extra (optional) exercises will be available for those who already know some algebraic geometry to illustrate how this fits into the general framework of schemes, but this will not be used in the course.

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.

## Assesment

There will be homework problems set each week, which will be handed in and graded, and will count for 20% towards the final grade. The other 80% will come from a written exam at the end of the course.

## Literature

The course will be loosely based on the book `Rational points on elliptic curves' by Silverman and Tate. You do not need a copy of the book, and reading it will not be a good substitute to attending the lectures (or at least reading someone elses notes), as we will not use the same basic definition. This may make the book seem pointless, but
- 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.

## Lecture notes

I hope to scan and post here my lecture notes, maybe even before the corresponding lectures. Please me aware that I may forget/not have time, so it is not a good idea to rely on these notes - if you can't make it to a lecture, please ask a friend to take notes for you!

## Homework sheets

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.

Handouts

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

## Office hours

My office is 233, drop by any time or email me to make an appointment. If this starts to get out of hand I will pick a formal office hour.