Counting Models of Genus One Curves

TL;DR

This paper investigates the classification and counting of minimal genus one models of degrees 2, 3, and 4 over local and global fields, relating their number to the Jacobian's Kodaira symbol.

Contribution

It provides a detailed analysis of the variation in the number of minimal genus one equations based on the Jacobian's Kodaira symbol and counts these models over certain number fields.

Findings

Number of minimal models varies with Kodaira symbol

Explicit counts of models over number fields of class number 1

Connection between minimal models and Jacobian invariants

Abstract

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