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Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields

Omer Avci

TL;DR

The paper tackles the problem of classifying rational elliptic curves with prescribed torsion over quadratic fields, focusing on $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/10\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/12\mathbb{Z}$-torsion. It employs a $2$-descent analysis on genus-1 modular curves $X_1(2,10)$ and $X_1(2,12)$ to relate quadratic points to quadratic-torsion realizations, obtaining congruence-based nonexistence results for various primes and conditional existence results via the parity conjecture. The work extends to broader torsion classifications over cyclotomic, Kummer, and $\mathbb{Z}_p$-extensions, providing elimination criteria and new classification theorems that refine and extend prior results. Under parity or $\Sha$-finiteness assumptions, it yields infinite families of curves realizing certain torsion patterns and demonstrates that much torsion over metabelian and $\mathbb{Z}_p$-extensions is already defined over base fields. Overall, the paper advances torsion-subgroup classifications across a range of infinite abelian extensions, linking modular curves, descent techniques, and Galois-theoretic elimination in a unified framework.

Abstract

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2N\mathbb{Z}$ over $K$ for $N=5,6$. We also prove that there exist infinitely many primes $p$ for which there are infinitely many elliptic curves $E/\mathbb{Q}$ with this torsion structure, conditional on the parity conjecture. Using these results, we obtain new torsion classification results over Kummer extensions of cyclotomic fields and over composites of $\mathbb{Z}_p$-extensions of number fields, refining and extending our previous work.

Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields

TL;DR

The paper tackles the problem of classifying rational elliptic curves with prescribed torsion over quadratic fields, focusing on and -torsion. It employs a -descent analysis on genus-1 modular curves and to relate quadratic points to quadratic-torsion realizations, obtaining congruence-based nonexistence results for various primes and conditional existence results via the parity conjecture. The work extends to broader torsion classifications over cyclotomic, Kummer, and -extensions, providing elimination criteria and new classification theorems that refine and extend prior results. Under parity or -finiteness assumptions, it yields infinite families of curves realizing certain torsion patterns and demonstrates that much torsion over metabelian and -extensions is already defined over base fields. Overall, the paper advances torsion-subgroup classifications across a range of infinite abelian extensions, linking modular curves, descent techniques, and Galois-theoretic elimination in a unified framework.

Abstract

Let be a quadratic field for an odd prime . We show that there exist infinitely many primes for which no elliptic curve has torsion subgroup over for . We also prove that there exist infinitely many primes for which there are infinitely many elliptic curves with this torsion structure, conditional on the parity conjecture. Using these results, we obtain new torsion classification results over Kummer extensions of cyclotomic fields and over composites of -extensions of number fields, refining and extending our previous work.
Paper Structure (5 sections, 36 theorems, 204 equations)

This paper contains 5 sections, 36 theorems, 204 equations.

Table of Contents

  1. Introduction and Notation
  2. Finding Quadratic Points of two Modular Curves
  3. Conditional Proofs of our Conjectures
  4. Eliminating Possible Torsion
  5. More Classification Results

Key Result

Theorem 1.1

Let $E/\mathbb{Q}$ be an elliptic curve, and let $p > 3$ be a prime such that $p - 1$ is not divisible by $3$, $4$, or $5$. Then $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ is either one of the groups from Mazur’s theorem, or one of the following groups: Moreover, $E(\mathbb{Q}(\zeta_p))_{\text{tors}} = E(\mathbb{Q}(\sqrt{-p}))_{\text{tors}}$ holds.

Theorems & Definitions (63)

  • Theorem 1.1: A., Theorem 1.1, Omer
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Theorem 2.1: Najman, Kamienny, Theorem 2-8-9-15, najmankamienny
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • ...and 53 more