Counting models of genus one curves

Abstract

AbstractLetCbe a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation definingCup to isomorphism. In this paper we consider genus one equations of degreendefiningC, namely a (generalised) binary quartic whenn= 2, a ternary cubic whenn= 3 and a pair of quaternary quadrics whenn= 4. In general, minimal genus one equations of degreenare not unique up to isomorphism. We explain how the number of these equations varies according to the Kodaira symbol of the Jacobian ofC. Then we count these equations up to isomorphism over a number field of class number 1.