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Torsion of elliptic curves with rational $j$-invariant over the maximal elementary abelian 2-extension of $\mathbb{Q}$

Lucas Hamada

TL;DR

The paper classifies the torsion subgroups of elliptic curves defined over the maximal elementary abelian 2-extension $\mathbb{Q}(2^{\infty})$ for curves with rational $j$-invariant not equal to $0$ or $1728$. It reduces the problem to a $p$-primary analysis for primes $p\in\{2,3,5,7,13\}$ by relating a given curve to a rational-$j$ curve via quadratic twists and leveraging Galois/abelian constraints, isogeny classifications, and known results (Fujita, Chou, Daniels). The main result is an explicit list of possible torsion structures for $E(\mathbb{Q}(2^{\infty}))_{\mathrm{tors}}$, together with a discussion of which structures occur and which are ruled out, including the two notable exceptions. This advances the Mazur-type program to an infinite extension, clarifying how the torsion subgroup can grow in the twisted, abelianized setting and highlighting when the twisting field remains constrained to abelian extensions. The findings rely on a careful interplay between twist theory, Weil pairings, isogeny theory, and detailed group-theoretic arguments to control extension degrees and Galois actions.

Abstract

In this paper, we classify the possible torsion subgroup structures of elliptic curves defined over the compositum of all quadratic extensions of the rational number field, whose $j$-invariant is a rational number not equal to 0 or 1728.

Torsion of elliptic curves with rational $j$-invariant over the maximal elementary abelian 2-extension of $\mathbb{Q}$

TL;DR

The paper classifies the torsion subgroups of elliptic curves defined over the maximal elementary abelian 2-extension for curves with rational -invariant not equal to or . It reduces the problem to a -primary analysis for primes by relating a given curve to a rational- curve via quadratic twists and leveraging Galois/abelian constraints, isogeny classifications, and known results (Fujita, Chou, Daniels). The main result is an explicit list of possible torsion structures for , together with a discussion of which structures occur and which are ruled out, including the two notable exceptions. This advances the Mazur-type program to an infinite extension, clarifying how the torsion subgroup can grow in the twisted, abelianized setting and highlighting when the twisting field remains constrained to abelian extensions. The findings rely on a careful interplay between twist theory, Weil pairings, isogeny theory, and detailed group-theoretic arguments to control extension degrees and Galois actions.

Abstract

In this paper, we classify the possible torsion subgroup structures of elliptic curves defined over the compositum of all quadratic extensions of the rational number field, whose -invariant is a rational number not equal to 0 or 1728.

Paper Structure

This paper contains 11 sections, 39 equations, 1 table.

Table of Contents

  1. Introduction
  2. Torsion points of elliptic curves over $\mathop{\mathrm{\mathbb{Q}}}\nolimits$
  3. Elliptic curves with rational $j$-invariant
  4. Technical lemmas from group theory
  5. Proof of the main theorem
  6. The 2-primary part of $E(\mathop{\mathrm{\mathbb{Q}}}\nolimits(2^\infty))_{\mathop{\mathrm{tors}}\nolimits}$
  7. The 3-primary part of $E(\mathop{\mathrm{\mathbb{Q}}}\nolimits(2^\infty))_{\mathop{\mathrm{tors}}\nolimits}$
  8. The 5-primary part of $E(\mathop{\mathrm{\mathbb{Q}}}\nolimits(2^\infty))_{\mathop{\mathrm{tors}}\nolimits}$
  9. The 7-primary part of $E(\mathop{\mathrm{\mathbb{Q}}}\nolimits(2^\infty))_{\mathop{\mathrm{tors}}\nolimits}$
  10. The 13-primary part of $E(\mathop{\mathrm{\mathbb{Q}}}\nolimits(2^\infty))_{\mathop{\mathrm{tors}}\nolimits}$
  11. Compatibility of $p$-Primary Torsion Subgroups in $E(\mathop{\mathrm{\mathbb{Q}}}\nolimits(2^{\infty}))_{\text{tors}}$

Theorems & Definitions (24)

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  • ...and 14 more