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

TL;DR

The paper investigates torsion growth of rational elliptic curves over infinite $\mathbb{Z}_p$-extensions $L$ of quadratic fields $K$ for primes $p>5$. It proves $E(L)_{\text{tors}} = E(K)_{\text{tors}}$ in this setting and thereby classifies possible torsion subgroups realized over such $L$ by reducing to the known quadratic-base list; the proof employs Weil pairing, roots of unity constraints, and rational $m$-isogeny bounds to eliminate growth in finite subextensions, with careful treatment of exceptional primes $p=7,11$. The work also analyzes the exceptional $p=3$ and $p=5$ cases, providing cyclotomic and anticyclotomic classifications for $K_{\text{cyc}}$ and $K_{\text{anti}}$ and identifying some nonrealizable torsion structures. Overall, the results extend torsion classification in infinite Iwasawa-type towers and clarify which torsion structures can occur in $\mathbb{Z}_p$-extensions of 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.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 2.1: Mazur, Mazur
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5: Fricke, Kenku, Klein, Kubert, Ligozat, Mazur, and Ogg, among others
• Lemma 2.6: Chou, Chou
• Lemma 3.1
• proof
• Lemma 3.2: A., Lemma 3.1, Omer