
joint with John Cremona, Catherine O'Neil, Denis Simon and Michael Stoll
This is the first in a series of papers in which we study the nSelmer 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.
The other papers in this series are Paper II. Geometry and Paper III. Algorithms.