Torsion of rational elliptic curves over cubic fields

TL;DR

This paper investigates how the torsion subgroup of a rational elliptic curve changes when extended from the rationals to cubic number fields, focusing on the number of such fields where the torsion subgroup differs.

Contribution

It provides a detailed analysis of the relationship between torsion subgroups over Q and cubic fields, including the count of cubic fields where the torsion subgroup enlarges.

Findings

Identifies conditions under which torsion subgroups grow over cubic fields.

Quantifies the number of cubic fields causing torsion growth.

Provides classifications of possible torsion subgroup changes.

Abstract

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors.