Radical characterizations of elliptic curves

TL;DR

This paper shows that for elliptic curves over number fields, the isogeny class can be determined by the radical of the number of points over primes of good reduction, simplifying Faltings' original criterion.

Contribution

It demonstrates that the radical of the group order at primes of good reduction suffices to determine the elliptic curve's isogeny class, refining Faltings' theorem.

Findings

The radical of the group order determines the isogeny class.

A set of primes with density one is sufficient for this determination.

Simplifies the criteria needed to classify elliptic curves over number fields.

Abstract

Let K be a number field, and let E be an elliptic curve over K. A famous result by Faltings of 1983 can be reformulated for elliptic curves as follows: if S is a set of primes of good reduction for E having density one, then the K-isogeny class of E is determined by the function which maps a prime in S to the size of the group of points over the residue field. In this paper, we prove that it suffices to look at the radical of the size.