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We prove a theorem on the minimisation of genus one curves, generalising work of Birch and Swinnerton-Dyer [2], Cremona and Serf [4], [12] and Cremona and Stoll [5], [6]. The advantage of our approach over earlier methods is that we do not need to treat residue characteristics 2 and 3 as special cases. This work has applications to descent calculations on elliptic curves and to the study of the Tate-Shafarevich group.

A new approach to minimising binary quartics and ternary cubics   (21 pages)