Torsion of Rational Elliptic Curves over the $\mathbb{Z}_p$-Extensions of Quadratic Fields

TL;DR

This paper proves that the torsion subgroup of an elliptic curve over certain infinite extensions of quadratic fields remains unchanged for primes greater than 5, aiding classification of possible torsion groups.

Contribution

It establishes the invariance of torsion subgroups over $bZ_p$-extensions of quadratic fields for primes greater than 5, extending torsion classification results.

Findings

Torsion subgroup remains constant over $bZ_p$-extensions for $p>5$

Classification of possible torsion groups over these extensions

Extension of known torsion classification over quadratic fields

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. For a quadratic number field $K$ and an odd prime number $p$, let $L$ be a $\mathbb{Z}_p$-extension of $K$. We prove that $E(L)_{\text{tors}}=E(K)_{\text{tors}}$ when $p>5$. It enables us to classify the groups that can be realized as the torsion subgroup $E(L)_{\text{tors}}$, by using the classification of torsion subgroups over the quadratic fields.