The joys and pains of elliptic curves over number fields

Nils Bruin

Recently it has become feasible to determine the rational solutions to
equations of the form y^2=F(x) (hyperelliptic curves), where F is some
polynomial in x with rational coefficients. The methods used for solving
these equations need, amongst other things, the group of rational points
on elliptic curves (Mordell-Weil groups). These elliptic curves will in
general only be defined over number fields, even if the equation we start
out with is defined over Q.

A tool in determining Mordell-Weil groups goes back to the "descent
infini" of Fermat. In the simplest form, the so called 2-descents, we need
to compute in the multiplicative group of a number field modulo squares.

In this talk, we will sketch how one arrives at studying elliptic curves
over field extensions starting from hyperelliptic curves, quickly review
2-isogeny descents, and describe the computational problems that one
encounters in performing these computations.