The joys and pains of elliptic curves over number fields
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.