November 8, 2025Open Access

On the torsion growth in quadratic number fields for elliptic curves defined over the rationals

Key Points

Torsion subgroup growth was linked to primes dividing the conductor of the curve, enhancing understanding of field relations.
The study found an explicit relationship between the conductor of the extension and the assigned elliptic curve, advancing knowledge in number theory.
An observational analysis was conducted on elliptic curves with base changes to quadratic number fields for comprehensive insight.
The results may enable further exploration of torsion subgroup properties in varying number fields.

Abstract

Given an elliptic curve defined over the field of rational numbers, it is known how its torsion subgroup may grow when we make a base change to a quadratic number field. In this paper we consider the inverse question: if we have the elliptic curve defined over the rationals and we know how the torsion subgroup grows, what can we say about the field? Our main result gives an explicit relationship between the primes dividing the conductor of the curve and the conductor of the extension as a first approach to a better understanding of this problem.

On the torsion growth in quadratic number fields for elliptic curves defined over the rationals

Key Points

Abstract

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