Minimal Genus One Curves

TL;DR

This paper introduces a new invariant-theory-independent definition of minimality for genus one equations of degrees 2, 3, and 4, providing a unified approach and new proofs of classical results.

Contribution

It proposes a novel, invariant-theory-independent minimality definition for genus one equations of degrees 2, 3, and 4, aligning with classical definitions and enabling new proofs.

Findings

New minimality definition for genus one equations

Equivalence with classical minimality for degrees ≤ 4

Proof of existence of global minimal equations over certain number fields

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 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 when n <= 4. As an application, we give a new proof for the existence of global minimal genus one equations over number fields of class number 1.