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Torsion of $\mathbb Q$ -curves over number fields of small odd prime degree

This paper completes the classification of torsion subgroups for $\mathbb{Q}$ -curves over number fields of prime degree by determining all such groups for degrees 3, 5, and 7 and proving that they coincide with the torsion subgroups of elliptic curves with rational $j$ -invariants.

Original authors: Ivan Novak

Published 2026-02-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ivan Novak

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 Elliptic Curves. In the world of mathematics, these aren't smooth, round circles like a hula hoop. They are wavy, squiggly lines defined by specific equations.

On these curves, there are special "points" that you can add together like numbers. If you keep adding a specific point to itself, eventually you might loop back to the starting point (zero). The number of steps it takes to get back to zero is called the order of that point. The collection of all these "looping" points forms a group called the Torsion Subgroup.

The Big Question

Mathematicians have long asked: What are all the possible shapes (groups) that these torsion subgroups can take?

The Rational Case (The Easy Street): If the curve is defined over the Rational Numbers (fractions like 1/2, 3/4), we know the answer perfectly. There are only 15 possible shapes. This was solved by Barry Mazur decades ago.
The Harder Streets: What if the curve is defined over a more complex "neighborhood" of numbers, like a Number Field?
- If the neighborhood is a Quadratic Field (degree 2, like adding $\sqrt{2}$ to the rationals), we know the answer.
- If the neighborhood is a Prime Degree Field (degree 3, 5, 7, 11, etc.), it gets much harder.

The Special Suspects: Q-Curves

This paper focuses on a specific type of curve called a Q-curve.

Analogy: Imagine a Q-curve is a "chameleon." It lives in a complex neighborhood, but it has a secret connection to the "Rational World." It is "isogenous" (mathematically related via a specific transformation) to a curve that does have a rational number as its defining characteristic (its $j$ -invariant).
The Discovery: The author, Ivan Novak, proves that for neighborhoods of degree 3, 5, and 7, these chameleons (Q-curves) don't actually have any new torsion shapes that the Rational curves don't already have.

The Detective Work: How He Solved It

Novak didn't just guess; he used a process of elimination, like a detective ruling out suspects.

1. The "Isogeny Graph" (The Family Tree)
Imagine every elliptic curve is a person. An "isogeny" is a family relationship.

If Curve A is related to Curve B, they are connected by a line.
Novak looked at the "family trees" of these curves. He knew that if a Q-curve has a point of a certain order (say, 11), then its "rational cousin" must also have a relationship to a point of that order.

2. The "Galois Representation" (The Security Camera)
Mathematicians use something called "Galois representations" to watch how these points move when you swap the numbers in the field.

The Metaphor: Imagine the points on the curve are dancers. The Galois group is the DJ. The DJ can spin the music (swap numbers), and the dancers move in specific patterns.
Novak looked at the "footprints" (the matrices) left by the DJ. He proved that for degrees 3, 5, and 7, the footprints of a Q-curve are always compatible with the footprints of a Rational curve.

3. Ruling Out the "Impossible" Shapes
There were some shapes (like a group of size 11, 15, 16, or 20) that can appear in general cubic fields (degree 3), but Novak had to prove they cannot appear in Q-curves.

The Logic: He said, "If a Q-curve had a point of order 11, its rational cousin would have to have a specific 'security camera' pattern. But we checked all the rational cousins, and none of them have that pattern. Therefore, the Q-curve cannot have that point."
He did this for every suspicious group size (11, 15, 16, 20, etc.) and proved they were all impossible for Q-curves in these specific neighborhoods.

The Final Verdict

The paper completes a massive puzzle.

Degree 2 (Quadratic): We already knew the list (it has some extra shapes like 13, 15, 16).
Degree 3, 5, 7: Novak proved the list is exactly the same as the list for Rational curves.
Degree > 7: Others had already solved this.

The Big Takeaway:
If you are looking for a Q-curve in a number field of degree 3, 5, or 7, you will never find a torsion subgroup that you couldn't find on a curve defined over the simple Rational numbers. The "chameleons" in these neighborhoods don't change their shape; they just wear the same clothes as their rational relatives.

Why Does This Matter?

In the world of math, knowing the "limits" of what is possible is crucial. It helps mathematicians understand the fundamental structure of numbers. By proving that Q-curves in these specific fields behave just like Rational curves, Novak has closed the book on this specific classification problem. It's like finishing the last piece of a jigsaw puzzle, revealing the complete picture of how these mathematical objects can behave in the "small odd prime" neighborhoods.

1. Problem Statement

The paper addresses the classification of torsion subgroups of $\mathbb{Q}$ -curves defined over number fields of odd prime degree ( $p = 3, 5, 7$ ).

Context: A $\mathbb{Q}$ -curve is an elliptic curve $E$ defined over a number field $K$ that is isogenous to all its Galois conjugates.
Background:
- The torsion of elliptic curves over $\mathbb{Q}$ is classified by Mazur's Theorem (15 groups).
- The torsion of $\mathbb{Q}$ -curves over quadratic fields ( $p=2$ ) was solved by Le Fourn and Najman.
- The torsion of $\mathbb{Q}$ -curves over number fields of prime degree $p > 7$ was solved by Cremona and Najman.
- The cases for $p=3, 5, 7$ remained open.
Specific Goal: Determine all possible groups $E(K)_{\text{tors}}$ where $E$ is a $\mathbb{Q}$ -curve over a number field $K$ of degree $p \in \{3, 5, 7\}$ .
Hypothesis: The author aims to prove that for these degrees, the torsion subgroups of $\mathbb{Q}$ -curves are identical to the torsion subgroups of elliptic curves with rational $j$ -invariants defined over the same fields.

2. Methodology

The proof relies on the structural relationship between $\mathbb{Q}$ -curves and elliptic curves with rational $j$ -invariants, combined with the analysis of Galois representations.

A. Reduction to Rational $j$ -invariants

The author utilizes a key result by Cremona and Najman (Proposition 1.3):

If $E$ is a non-CM $\mathbb{Q}$ -curve over an odd-degree number field $K$ , then $E$ is $K$ -isogenous to an elliptic curve $E'$ with a rational $j$ -invariant.

This allows the problem to be reduced to studying the torsion of curves $E'$ (with $j(E') \in \mathbb{Q}$ ) and their quadratic twists, rather than arbitrary $\mathbb{Q}$ -curves.

B. Galois Representations and Isogenies

The core technical machinery involves analyzing how the image of the mod- $N$ Galois representation, $\rho_{E,N}(G_K)$ , changes under isogenies.

Lemma 2.2 & Corollary 2.3: If an elliptic curve $E_1$ has a $K$ -rational point of prime order $p$ , and $\phi: E_1 \to E_2$ is a $K$ -isogeny, then the image of the Galois representation on $E_2[p]$ must have eigenvalues equal to 1 (or -1 for twists).
Proposition 2.5 & Corollary 2.6: These generalize the above to composite orders and isogenies with specific kernels, describing the matrix structure of the Galois image (e.g., upper triangular matrices with specific diagonal entries).
Isogeny Graphs: The paper uses the structure of isogeny graphs (specifically $p$ -primary components) to limit the possible degrees of isogenies. For odd primes $p$ , these graphs are trees (often line graphs), which restricts how torsion can propagate through isogeny chains.

C. Elimination Strategy

For each candidate torsion group that appears for general elliptic curves over $K$ but not for rational $j$ -invariant curves, the author attempts to derive a contradiction:

Assume a $\mathbb{Q}$ -curve $E$ has a specific torsion subgroup (e.g., $\mathbb{Z}/11\mathbb{Z}$ ).
Use the isogeny to a rational $j$ -invariant curve $E'$ .
Analyze the quadratic twist $E''$ of $E'$ defined over $\mathbb{Q}$ .
Show that the existence of the torsion on $E$ forces $E''$ to have a Galois image structure (e.g., specific subgroups of $GL_2(\mathbb{Z}/N\mathbb{Z})$ ) that is mathematically impossible for a rational curve with that $j$ -invariant.
This often involves checking specific LMFDB curves and their known Galois images using computational algebra (Magma).

3. Key Contributions and Results

Main Theorems

The paper establishes the following classifications for non-CM $\mathbb{Q}$ -curves:

Cubic Fields ( $p=3$ ): The torsion subgroups are exactly the 20 groups listed in Theorem 1.5 (rational $j$ $j$ -invariant case).
- Excluded Groups: $\mathbb{Z}/11\mathbb{Z}$ , $\mathbb{Z}/15\mathbb{Z}$ , $\mathbb{Z}/16\mathbb{Z}$ , $\mathbb{Z}/20\mathbb{Z}$ , $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/10\mathbb{Z}$ , $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/12\mathbb{Z}$ .
Quintic Fields ( $p=5$ ): The torsion subgroups are exactly the 17 groups listed in Theorem 1.6.
- Excluded Groups: $\mathbb{Z}/13\mathbb{Z}$ , $\mathbb{Z}/14\mathbb{Z}$ , $\mathbb{Z}/15\mathbb{Z}$ , $\mathbb{Z}/16\mathbb{Z}$ , $\mathbb{Z}/20\mathbb{Z}$ , $\mathbb{Z}/21\mathbb{Z}$ , $\mathbb{Z}/25\mathbb{Z}$ , $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/10\mathbb{Z}$ , $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/12\mathbb{Z}$ , $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ , $\mathbb{Z}/50\mathbb{Z}$ .
Septic Fields ( $p=7$ ): The torsion subgroups are exactly the 15 groups listed in Theorem 1.7 (which are the same as over $\mathbb{Q}$ $Q$ ).
- Excluded Groups: All groups appearing in cubic/quintic extensions but not over $\mathbb{Q}$ (e.g., $\mathbb{Z}/13\mathbb{Z}$ , $\mathbb{Z}/14\mathbb{Z}$ , etc.).

Corollary 1.13 (The Main Conclusion)

For any odd prime $p$ , if a group $T$ is the torsion subgroup of a $\mathbb{Q}$ -curve over a number field of degree $p$ , then $T$ is also the torsion subgroup of an elliptic curve with a rational $j$ -invariant over a number field of degree $p$ .

Significance: This completes the classification of torsion for $\mathbb{Q}$ -curves over all number fields of prime degree. The "new" torsion groups that appear in higher degree extensions are entirely explained by the existence of rational $j$ -invariants; no "exotic" $\mathbb{Q}$ -curves (those not isogenous to rational $j$ -invariants) introduce new torsion structures in these degrees.

Theorem 1.14 (Extension to Isogenous Curves)

The paper proves a stronger statement: If an elliptic curve $E$ over a number field of degree $p$ is isogenous to a curve with a rational $j$ -invariant, its torsion subgroup is the same as that of a curve with a rational $j$ -invariant.

Note: This holds for $p=2$ as well, provided one excludes the specific cases of $\mathbb{Z}/14\mathbb{Z}$ and $\mathbb{Z}/18\mathbb{Z}$ which appear for general curves but not for those isogenous to rational $j$ -invariants (Proposition 5.1).

4. Technical Highlights of the Proofs

Elimination of $\mathbb{Z}/11\mathbb{Z}$ (Cubic): The author shows that if a cubic field $K$ has a $\mathbb{Q}$ -curve with 11-torsion, the associated rational curve $E''$ must have an 11-isogeny over $K$ . However, the Galois image of the 11-torsion for rational curves with 11-isogenies (specifically $j = -11^2$ and $j = -11 \cdot 131^3$ ) does not allow for a subgroup of index 3 that satisfies the eigenvalue constraints required by the isogeny.
Elimination of $\mathbb{Z}/16\mathbb{Z}$ (Cubic): The proof uses the structure of 2-isogeny graphs. It demonstrates that if a cubic field supports a 16-torsion $\mathbb{Q}$ -curve, the associated rational curve would require a 4-isogeny over $K$ that is not defined over $\mathbb{Q}$ . This forces the 2-adic Galois image to be a specific subgroup ( $H_{20}$ ) which, when reduced modulo 4, contradicts the existence of the required torsion structure.
Degree Constraints on Isogenies: A recurring theme is that for $p=5, 7$ , the degree of the field extension does not divide the order of certain general linear groups (e.g., $|GL_2(\mathbb{Z}/3\mathbb{Z})|$ or $|GL_2(\mathbb{Z}/5\mathbb{Z})|$ ). This forces isogenies defined over $K$ to actually be defined over $\mathbb{Q}$ , collapsing the problem back to the rational case.

5. Significance

Completeness: The paper closes the gap in the classification of $\mathbb{Q}$ -curves over prime degree fields, unifying the results for $p=2$ (Le Fourn/Najman), $p=3,5,7$ (Novak), and $p>7$ (Cremona/Najman).
Structural Insight: It confirms that for odd prime degrees, the phenomenon of "new" torsion appearing in extensions is strictly tied to the existence of rational $j$ -invariants. There are no "hidden" torsion structures arising solely from the $\mathbb{Q}$ -curve property without a rational $j$ -invariant ancestor.
Methodological Advancement: The paper provides a robust framework for analyzing torsion via Galois representation images under isogeny, utilizing isogeny graphs and specific matrix subgroup constraints. This approach is applicable to other problems in the arithmetic of elliptic curves.
Computational Verification: The proofs rely on explicit computations of Galois images for specific LMFDB curves, bridging theoretical number theory with computational algebra (Magma).

In summary, Novak proves that for number fields of odd prime degree, the torsion of $\mathbb{Q}$ -curves is indistinguishable from the torsion of elliptic curves with rational $j$ -invariants, thereby completing a major classification program in the arithmetic of elliptic curves.