
The joys and pains of elliptic curves over number fieldsNils 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 (MordellWeil 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 MordellWeil groups goes back to the "descent infini" of Fermat. In the simplest form, the so called 2descents, 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 2isogeny descents, and describe the computational problems that one encounters in performing these computations. 