1. Summary.- 2. Heights.- 3. Normalised heights.- 4. The Mordell-Weil theorem.- 5. Mordell's conjecture.- 6. Local calculation of normalised heights.- 7. Siegel's method.- 8. Baker's method.- 9. Hilbert's irreducibility theorem.- 10. Construction of Galois extensions.- 11. Construction of elliptic curves of large rank.- 12. The large sieve.- 13. Applications of the large sieve to thin sets.- Appendix: The class number 1 problem and integral points on modular curves.
