Elliptic curves over the rational numbers with semi-abelian reduction and two-division points

TL;DR

This paper classifies certain elliptic curves over the rationals with specific reduction properties and two-division points, providing a detailed understanding of their structure and associated moduli spaces.

Contribution

It offers a classification of elliptic curves with semi-abelian reduction at 2 and specific two-torsion properties, linking algebraic and geometric perspectives.

Findings

Classification of elliptic curves with semi-abelian reduction at 2

Description of curves with non-trivial two-torsion at p=2

Connection to Deligne--Mumford stacks of genus one curves

Abstract

We classify elliptic curves over the rationals whose N\'eron model over the integers is semi-abelian, with good reduction at p=2, and whose Mordell--Weil group contains an element of order two that stays non-trivial at p=2. Furthermore, we describe those curves where the element of order two is narrow, or where another element of order two exists, and also express our findings in terms of Deligne--Mumford stacks of pointed curves of genus one.