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

This paper investigates the existence of rational elliptic curves with specific torsion subgroups ( $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2N\mathbb{Z}$ for $N=5,6$ ) over quadratic fields $K=\mathbb{Q}(\sqrt{-p})$ , proving that for infinitely many primes $p$ such curves neither exist nor exist in infinite quantities (the latter being conditional on the parity conjecture), while applying these findings to refine torsion classifications over broader field extensions.

Original authors: Omer Avci

Published 2026-02-10

📖 3 min read🧠 Deep dive

CC0 1.0

Original authors: Omer Avci

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about a very specific type of "mathematical shape" called an elliptic curve.

These curves are like complex, elegant loops in space. Mathematicians are obsessed with finding the "hidden patterns" inside them—specifically, the points on these curves that follow very strict, predictable rules. We call these patterns torsion subgroups.

Think of an elliptic curve as a high-security vault, and the torsion subgroups as the combination locks on that vault. For a long time, we’ve known what the standard locks look like (this is called Mazur’s Theorem). But mathematicians want to know: If we change the environment where the vault is kept, do new, weird, exotic locks appear?

In this paper, the "environment" is a Number Field (specifically quadratic fields). Instead of working with standard whole numbers, we are working in "extended" number worlds, like adding the square root of a prime number to our math toolkit.

Here is the breakdown of what Ömer Avci discovered:

1. The "No-Go" Zones (Existence and Non-Existence)

Imagine you are building a Lego set. You have certain special pieces (torsion subgroups like $Z/2Z \times Z/10Z$ ). You might assume that if you have enough space, you can eventually build anything.

Avci proved that some combinations are impossible. He found specific "mathematical climates" (certain prime numbers) where it is physically impossible for a rational elliptic curve to have certain exotic locks. It’s like saying, "In a room that is exactly 70 degrees, you will never find a block of ice." He proved that for certain types of numbers, those specific "ice block" torsion patterns simply cannot exist.

2. The "Infinite Possibilities" (Conditional Proofs)

On the flip side, he also looked for the opposite: When can we build infinitely many of these exotic shapes?

He found that in other specific "climates," you can actually create an infinite variety of these curves. However, there is a catch: he had to use a "mathematical bridge" called the Parity Conjecture.

The Analogy: Imagine you are trying to prove that a certain type of rare bird exists in a forest. You can't see them directly, but you can prove that if the wind blows a certain way (the Parity Conjecture), then those birds must be there in infinite numbers. Most mathematicians believe the wind blows that way, so his results are considered very strong.

3. The "Expanding Universe" (Kummer Extensions and $\mathbb{Z}_p$ -extensions)

The paper doesn't stop at small, simple environments. It looks at much larger, more complex "universes" called Kummer extensions and $\mathbb{Z}_p$ -extensions.

Think of these as zooming out from a single room to an entire galaxy. As the mathematical universe gets bigger and more complex, you’d expect the number of possible patterns to explode.

But Avci discovered something surprising: The patterns stay surprisingly stable. He proved that even as you expand into these massive, infinite mathematical galaxies, the "locks" on the curves often don't change at all. They stay exactly as they were in the small, simple room. This is a "classification" result—it tells us that even in the vastness of infinity, the rules of the game remain predictable.

Summary in a Nutshell

The paper is a map of a mathematical landscape. It tells us:

Where the "dead zones" are: (Where certain patterns can never exist).
Where the "gold mines" are: (Where we can find infinite patterns, assuming a famous theory is true).
How the landscape holds together: (Showing that even when we expand our mathematical world to infinity, the fundamental structures remain remarkably consistent).

Technical Summary: Existence and Non-Existence of Rational Elliptic Curves with Prescribed Torsion Subgroups over Quadratic Fields

Author: Ömer Avci
Subject Area: Arithmetic Geometry / Number Theory (Elliptic Curves)

1. The Problem

The central problem addressed in this paper is the classification of torsion subgroups $E(K)_{\text{tors}}$ for elliptic curves $E$ defined over $\mathbb{Q}$ when viewed over specific number fields $K$ (specifically quadratic fields, cyclotomic extensions, and $\mathbb{Z}_p$ -extensions).

While Mazur’s Theorem provides a complete classification for $E(\mathbb{Q})_{\text{tors}}$ , the behavior of these curves over larger fields is more complex. The author specifically investigates the "realizability" of certain torsion structures—namely $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/10\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/12\mathbb{Z}$ —over quadratic fields $K = \mathbb{Q}(\sqrt{-p})$ . The paper seeks to determine for which primes $p$ these groups can or cannot appear as the torsion subgroup of a rational elliptic curve.

2. Methodology

The paper employs a multi-faceted approach combining classical algebraic number theory with modern computational methods:

Modular Curves and Quadratic Twists: The author relates the existence of torsion points over a quadratic field $K = \mathbb{Q}(\sqrt{d})$ to the existence of $K$ -rational points on specific modular curves, such as $X_1(2, 10)$ and $X_1(2, 12)$ . Since these curves have genus 1, they are themselves elliptic curves. The problem is thus reduced to studying the Mordell-Weil rank of the quadratic twists of these modular curves.
2-Descent: To determine the rank of these twists, the author utilizes the method of 2-descent (specifically via 2-isogenies). This involves solving local-to-global equations to bound the size of the Selmer group, which in turn provides bounds on the rank.
Parity Conjecture: For cases where the rank is not explicitly zero, the author uses the Parity Conjecture (which relates the algebraic rank to the root number of the $L$ -function) to prove the existence of infinitely many curves with prescribed torsion, conditional on the finiteness of the 2-primary part of the Tate-Shafarevich group ( $\text{Ш}$ ).
Galois Representations and Kummer Theory: To extend results to infinite extensions (like $\mathbb{Z}_p$ -extensions and Kummer extensions), the author analyzes the image of the Galois representation $\rho_{E, \ell}$ and uses the properties of the Weil pairing and the structure of the Galois groups of cyclotomic and $p$ -adic extensions.

3. Key Contributions and Results

A. Non-existence Results (Elimination of Torsion):
The author identifies specific congruence classes of primes $p$ for which certain torsion structures are impossible for any $E/\mathbb{Q}$ :

Theorem 1.2: If $p \equiv 3, 7 \pmod{20}$ , then $E(\mathbb{Q}(\sqrt{-p}))_{\text{tors}} \not\cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/10\mathbb{Z}$ .
Theorem 1.3: If $p \equiv 11 \pmod{24}$ , then $E(\mathbb{Q}(\sqrt{-p}))_{\text{tors}} \not\cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/12\mathbb{Z}$ .
Corollary 1.4: For $p \equiv 83, 107 \pmod{120}$ , neither $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/10\mathbb{Z}$ nor $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/12\mathbb{Z}$ can be realized.

B. Conditional Existence Results:
The author proves that for other families of primes, these torsion groups appear infinitely often, provided the Parity Conjecture holds:

Conjecture 1.5 & 1.6 (Proven Conditionally): For $p \equiv 11, 19 \pmod{20}$ , there are infinitely many $E/\mathbb{Q}$ such that $E(\mathbb{Q}(\sqrt{-p}))_{\text{tors}} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/10\mathbb{Z}$ . A similar result is provided for the $\mathbb{Z}/12\mathbb{Z}$ case.

C. Extensions to Infinite Abelian Fields:
The paper significantly advances the classification of torsion over much larger fields:

Kummer/Cyclotomic Extensions: The author refines the classification of $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ and $E(L)_{\text{tors}}$ for certain metabelian extensions $L$ , showing that the torsion remains highly constrained.
$\mathbb{Z}_p$ -extensions: A major result (Theorem 5.5) shows that for an imaginary quadratic field $K$ and a prime $p > 5$ , the torsion subgroup over the compositum of all $\mathbb{Z}_p$ -extensions ( $K_{\mathbb{Z}_p}$ ) is no larger than the torsion over the base field $K$ itself ( $E(K_{\mathbb{Z}_p})_{\text{tors}} = E(K)_{\text{tors}}$ ).

4. Significance

This paper is significant for several reasons:

Refinement of Torsion Classification: It moves beyond the classification of torsion over fixed number fields to the study of torsion over infinite towers of fields, a much more difficult regime.
Bridging Local and Global Properties: By using 2-descent and the parity conjecture, the work connects the arithmetic of modular curves (local/rank properties) to the global classification of elliptic curves.
Computational Validation: The use of SageMath to verify ranks and identify specific $j$ -invariants provides a robust computational foundation for the theoretical proofs.
Generalization: The results regarding $\mathbb{Z}_p$ -extensions provide a powerful tool for researchers studying the growth of Mordell-Weil groups in Iwasawa theory.