Torsion points on elliptic curves over number fields of small degree

TL;DR

This paper classifies the possible prime orders of rational points on elliptic curves over number fields of degrees 4 through 7, expanding understanding of torsion structures in these settings.

Contribution

It explicitly determines the sets of prime torsion orders for elliptic curves over degree 4 to 7 number fields, a previously unresolved classification.

Findings

Identifies the sets S(4), S(5), S(6), S(7) of prime torsion orders.

Provides a comprehensive classification for these degrees.

Enhances understanding of torsion points over small degree number fields.

Abstract

We determine the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$, for $d = 4$, $5$, $6$, and $7$.