# Semistable models of elliptic curves over residue characteristic 2

## TL;DR

This paper introduces an algorithm to determine semistable models of elliptic curves over rings of mixed characteristic (0,2), based solely on the valuation of the parameter in Legendre form, with applications to 2-torsion.

## Contribution

It provides a novel valuation-dependent algorithm for constructing semistable models of elliptic curves in characteristic 2, with practical examples and implications for torsion points.

## Key findings

- Algorithm depends only on valuation of λ
- Examples illustrate the method's effectiveness
- Corollary relates to 2-torsion points

## Abstract

Given an elliptic curve $E$ in Legendre form $y^2 = x(x - 1)(x - \lambda)$ over the fraction field of a Henselian ring $R$ of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of $E$ over $R$ which depends only on the valuation of $\lambda$. We provide several examples along with an easy corollary concerning $2$-torsion.