Diophantine Equations and Integral Points on Curves (formerly: Rational Points on Elliptic Curves)

An equation for which we seek integer solutions is called Diophantine. The most famous such equation is probably the one in Pythagoras' theorem. We shall see how to parameterize the solutions of equations like this and ones with different exponents. When solving Diophantine equations in two variables we are finding integral points on curves. We shall pay special attention to elliptic curves. Elliptic curves are of great importance in mathematics; they were used in the proof of Fermat's last theorem, gave rise to the Birch and Swinnerton-Dyer conjecture and have applications in Cryptography.

Example

Show that if x^3-y^2=2, where x,y are nonzero pairwise coprime integers, then x=3 and y=5 or -5.

Email: J.M.Reynolds@uu.nl

COURSE BOOK

Henri Cohen, Number Theory: Volume I: Tools and Diophantine Equations (Springer Graduate Texts in Mathematics 239)
(Only parts of the book will be covered and there may be a small amount of material in the lectures which is not in the book!)

LECTURES/CLASSES

This course will run in Block 3 and 4 (09/02-08/06/2010). There will be a 2 hour lecture followed by a 2 hour exercise class every Tuesday starting at 09:00 (except weeks 11(16/03), 16(20/04), 21(25/05), 24 and 25).

LECTURE NOTES

These will be updated as the course proceeds. There will be exercises after each subsection; you should try them all.

EXAMS

There will be one "take home" exam which will be given out week 15(13/04) and handed in week 17(27/04). There will be one "open book" exam in week 25. The course will be marked out of 10, each exam will be worth 5.