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 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.
The other papers in this series are Paper II. Geometry and Paper III. Algorithms.