Elements of Computational Algebra

Starting on 27 February, there will be a series of four talks on the topic: "Rational Points on Curves and their Jacobians", designed to be accessible to a general audience from all branches of Mathematics, including students. These four talks are intended collectively to provide an overview of the main themes of the area. People are welcome to attend all four talks, or simply individual talks as preferred; each talk will be self contained in its own right and will include a summary of anything need from previous talks (for the benefit of anyone who is attending that talk on its own).

The emphasis of the talks will be on constructive techniques that have been developed in the last 10 years, which have broadened our ability to describe the rational points of a given curve, and to prove that we have described all of them. There is still no known algorithm for doing this, but there have been considerable recent advances.

Themes will include: the group law on an elliptic curve, a description of the Jacobian of a hyperelliptic curve and its group law, the Hasse principle, local methods, the use of the Jacobian to obtain information about the curve, and the application of these techniques to solve previously intractable problems.

The dates, speakers and titles of the individual talks are as follows.