# Clay Institute Summer School

# The arithmetic of curves

From July 24 to August 4 I will be running a 15-hour mini-course on the
arithmetic of curves.

This mini-course is part of the
Clay Institute Summer School
that is taking place
in Gottingen, Germany, from July 17 to August 11, 2006.

The mini-course will be given in two parts, in two consecutive weeks,
and the syllabus is
roughly as follows.

## Week 1: The key finiteness theorems (8 hours)

1. Overview

2. The Mordell-Weil theorem

3. Faltings' Theorem

4. Modular curves and Mazur's Theorem

5. Fermat curves and Wiles' Theorem

## Week 2: Rational points on elliptic curves (7 hours)

1. Heegner points

2. The theorems of Gross-Zagier and Kolyvagin

3. Shimura curve parametrisations

4. p-adic uniformisation

5. Stark-Heegner points