Minimal genus one curves

Abstract

In this paper we consider genus one equations of degree $n$, namely a (generalised) binary quartic when $n=2$, a ternary cubic when $n=3$, and a pair of quaternary quadrics when $n=4$. A new definition for the minimality of genus one equations of degree $n$ over local fields is introduced. The advantage of this definition is that it does not depend on invariant theory of genus one curves. We prove that this definition coincides with the classical definition of minimality for all $n\le4$. As an application, we give a new proof for the existence of global minimal genus one equations over number fields of class number 1.