The theory of 4-descent on elliptic curves has been developed in the PhD theses of Siksek [18], Womack [21] and Stamminger [20]. Prompted by our use of 4-descent in the search for generators of large height on elliptic curves of rank at least 2, we explain how to cut down the number of class group and unit group calculations required, by using the group law on the 4-Selmer group.

In this thesis I will describe an explicit method for performing an 8-descent on elliptic curves. First I will present some basics on descent, in particular I will give a generalization of the definition of n-coverings, which suits the needs of higher descent. Then I will sketch the classical method of 2-descent, and the two methods that are known for doing a second 2-descent, also called 4-des...

This is the first in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The methods we describe are practical in the case n = 3 for elliptic curves over the rationals, and have been implemented in MAGMA .

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Zpextension D∞ of the CM field K, where p is a prime of good, supersingular reduction for E. Our main result yields an asymptotic formula for the corank of the p-primary Selmer group of E along the extension D∞/K.

In this paper we explain how to bound the p-Selmer group of an elliptic curve over K, a number eld. Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number elds of degree at most p 1. Our method is practical for p = 3 but for larger values of p becomes impractical with current computing power. In the examples we ha...

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Zpextension D∞ of the CM field K, where p is a prime of good, supersingular reduction for E. Our main result yields an asymptotic formula for the corank of the p-primary Selmer group of E along the extension D∞/K.

We study the Iwasawa μand λ-invariants of the plus/minus Selmer groups of elliptic curves with the same residual representation using the ideas of [8]. As a result we find a family of elliptic curves whose plus/minus Selmer groups have arbitrarily large λ-invariants.