Computing all elliptic curves over an arbitrary number field with prescribed primes of bad reduction

TL;DR

This paper develops a method to find all elliptic curves over any number field with specified primes of bad reduction by solving $S$-unit equations, providing explicit examples over $Q$ and quadratic fields.

Contribution

It introduces a systematic approach to classify elliptic curves over arbitrary number fields with prescribed bad reduction primes using $S$-unit equations.

Findings

Explicit examples of elliptic curves over $Q$ and quadratic fields.

A general method for determining elliptic curves with specified bad reduction.

Solution techniques for $S$-unit equations in this context.

Abstract

In this paper we study the problem of how to determine all elliptic curves defined over an arbitrary number field $K$ with good reduction outside a given finite set of primes $S$ of $K$ by solving $S$-unit equations. We give examples of elliptic curves over $\mathbb Q$ and quadratic fields.