Michael Stoll
Rational Points on Curves

Prerequisites for this course are a good knowledge of algebraic number theory (including Galois cohomology) and of the algebraic geometry of curves.

The main goal of the course is to explain methods that can be used to determine the (finite, by Falting's Theorem) set of rational points on a (smooth projective) algebraic curve over Q of genus at least 2. My article [1] gives an overview of the subject and the various topics I would like to discuss.

Here is an outline of the course. We will start with an overview, including a sketch of the procedure one can follow to try to determine the set of rational points. Then we will look into various parts of this procedure in some detail. This includes topics like descent (on the curve and on its Jacobian variety), the Mordell-Weil Sieve and Chabauty's Method.

Descent on the curve can in some cases show that there are no rational points (and give some potentially helpful information even when there are), whereas descent on the Jacobian is a good method for obtaining an upper bound on the rank of the Mordell-Weil group. Since sufficient knowledge of this group is important for the other methods one would like to use, a successful descent computation (alternatively, one can use L-series if one is willing to accept the conjecture of Birch and Swinnerton-Dyer) is a prerequisite for the success of the whole procedure.

The Mordell-Weil Sieve uses the Mordell-Weil group and local (i.e., p-adic) information to restrict the possible location of (rational) points on the curve within the Mordell-Weil group. This can lead to a proof that the curve does not possess rational points, but also to a very large lower bound for the height of any unknown rational point on the curve. This kind of bound can in some cases be combined with (large) upper bounds for the height of integral points coming from Baker's theory of linear forms in logarithms; in this way the set of integral points can often be determined.

Chabauty's Method was originally invented by Chabauty to give a proof of Mordell's Conjecture (finiteness of the set of rational points) in the case that the rank of the Mordell-Weil group is less than the genus of the curve (the ``Chabauty condition''). It has been developed (by Coleman and others) into a method that can give tight upper bounds on the number of rational points, which are often good enough to know that all the points are already known. It can be combined with the Mordell-Weil Sieve to give a method for finding all the rational points (if the Chabauty condition holds) that yields a correct result if it terminates. Its termination relies on some plausible conjectures whose proof is likely to be difficult, but in practice it always seems to work.

References: Michael Stoll: Rational points on curves, Journal de Théorie des Nombres de Bordeaux 23, 257Ð-277 (2011). Also available on the arXiv or from my homepage.