## Explicit 8-descent on elliptic curves

- In this thesis I describe an explicit method for performing an 8-descent on an elliptic curve E over the rational numbers. An 8-descent is the next step after a 2- and a 4-descent. The method of 2-descent belongs to the classical tools in arithmetic geometry. More recent is the exploration of higher 2-descent. In 1996 Merriman, Siksek, and Smart found a method for doing 4-descent, and a different method for 4-descent was presented in 1998 by Cassels. I have been working on extending these methods to an 8-descent. I present a method for computing the 8-Selmer group of E as an abstract group, and next, for representing its elements as 8-coverings of E. With this method I was able to compute some examples, one of which is a curve of order 8 in the Shafarevich-Tate group of an elliptic curve. This is the first explicit example of a curve of such high order in Sha.