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Torsion groups of elliptic curves that appear infinitely often over septic fields

This paper determines the complete set of Abelian groups that occur as torsion subgroups for infinitely many elliptic curves over number fields of degree 7.

Original authors: Filip Najman

Published 2026-04-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Filip Najman

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 have a magical garden called the World of Elliptic Curves. In this garden, every plant is a specific type of curve, and growing on each plant are tiny, special fruits called points.

Mathematicians are obsessed with counting these fruits. Specifically, they want to know how many "torsion" fruits exist. These are special fruits that, if you keep adding them to themselves, eventually disappear back into the soil (the "zero" point). The collection of these special fruits forms a Torsion Group.

Now, imagine you can change the soil in which these plants grow.

If you plant them in Standard Soil (the rational numbers, $\mathbb{Q}$ ), there are only a few specific shapes of fruit baskets you can get.
If you plant them in Quadratic Soil (degree 2), you get a few more shapes.
If you plant them in Septic Soil (degree 7), the soil is much richer and more complex.

The Big Question

The paper asks: "In this rich Septic Soil, which shapes of fruit baskets can appear in infinite numbers?"

It's not enough to find just one plant with a weird fruit basket. The mathematician wants to know which baskets are so common that you can find infinitely many different plants growing them in this specific soil.

The Previous Detective Work

Before this paper, mathematicians had already solved this mystery for:

Standard Soil (Degree 1): Solved by Mazur in the 1970s.
Quadratic Soil (Degree 2): Solved by Kamienny.
Cubic (Degree 3) and Quartic (Degree 4) Soils: Solved by various teams recently.
Degree 5 and 6: Solved by Derickx and Sutherland.

But Degree 7 (Septic) was the missing piece of the puzzle. No one knew the full list of "infinitely common" fruit baskets for this specific soil.

The Main Discovery

The author, Filip Najman, has finally cracked the code for Septic Soil. He determined the exact list of fruit baskets (Torsion Groups) that appear infinitely often.

Here is the simple breakdown of his findings:

Single-Row Baskets (Cyclic Groups): You can find infinitely many plants with baskets containing $n$ $n$ fruits, where $n$ $n$ is any number from 1 to 30, except for 25 and 29.
- Analogy: It's like saying you can find infinite plants with 1 fruit, 2 fruits, up to 30 fruits, but if you try to grow a basket with exactly 25 or 29 fruits, you will never find a second one, let alone an infinite number.
Double-Row Baskets (Product Groups): You can find infinitely many plants with baskets that look like two rows of fruits, specifically $(2, 2n)$ $(2, 2 n)$ , where $n$ $n$ goes from 1 to 10.
- Analogy: These are baskets with 2 fruits in the first row and an even number in the second. This pattern works up to a certain size, but stops after $n=10$ .

How Did He Solve It?

To prove this, Najman had to act like a detective checking the "depth" of the soil.

The "Depth" Test: He used a mathematical concept called gonality. Think of this as measuring how "deep" or "complex" the soil is. If the soil is too shallow (low depth), you can't grow certain complex fruit baskets. If the soil is deep enough, you can.
The Ruling Out: He proved that for baskets larger than the ones listed above (like $n=11$ to $15$ in the double-row category), the Septic Soil simply isn't deep enough to support them infinitely. He used powerful computer calculations to measure the "depth" of the curves associated with these baskets and found they were too shallow.
The Ruling In: For the baskets that do work, he relied on code written by a colleague (Maarten Derickx) that explicitly constructed examples, proving they exist infinitely.

The "Magic" Tools

The paper mentions using some heavy-duty mathematical tools:

Modular Curves: Think of these as the "blueprints" or "maps" that tell you where these fruit baskets can grow.
The Euler Server: This is a supercomputer at the University of Zagreb. Najman used it to run complex calculations for about 48 minutes to prove that certain baskets (specifically for $n=11$ to $14$) simply cannot grow infinitely in Septic Soil.
Castelnuovo-Severi Inequality: This is a mathematical rule of thumb that says, "If you try to map a very complex shape to a simple shape, you can't do it unless the shape is huge." He used this to quickly rule out the $n=15$ case without needing a long computer run.

The Bottom Line

This paper is a completion of a massive puzzle. It tells us exactly which "shapes" of torsion groups are possible infinitely often when elliptic curves are grown in number fields of degree 7. It confirms that while Septic Soil is very fertile, it still has strict limits on what can grow there infinitely, specifically ruling out baskets of size 25, 29, and any double-row baskets larger than $(2, 20)$ .

1. Problem Statement

The paper addresses a fundamental problem in the arithmetic of elliptic curves: determining the set of possible torsion subgroups for elliptic curves defined over number fields of a specific degree.

Context: By the Mordell-Weil theorem, the group of rational points $E(K)$ on an elliptic curve over a number field $K$ is finitely generated. The torsion subgroup $E(K)_{\text{tors}}$ is finite and isomorphic to $\mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ (denoted as $(m, n)$ ).
Definitions:
- $\Phi(d)$ : The set of all possible torsion groups for elliptic curves over any number field of degree $d$ .
- $\Phi_\infty(d)$ : The subset of $\Phi(d)$ consisting of groups that appear as torsion subgroups for infinitely many non-isomorphic (over $\mathbb{Q}$ ) elliptic curves over degree $d$ fields.
The Gap: While $\Phi_\infty(d)$ was known for degrees $d \le 6$ (determined by Derickx, Sutherland, Jeon, Kim, and others), the set $\Phi_\infty(7)$ (septic fields) was undetermined at the time of writing. The full set $\Phi(d)$ for $d > 4$ remains unknown, but determining the "infinite" subset $\Phi_\infty(d)$ is a critical step in understanding the arithmetic landscape of these curves.

2. Methodology

The author employs a combination of theoretical bounds, modular curve geometry, and computational algebraic geometry to determine $\Phi_\infty(7)$ .

A. Theoretical Framework

Cyclic vs. Product Groups:
- For a septic field (degree 7), the group structure $(m, n)$ is constrained. Since $E(K)$ can contain $(n, n)$ only if $\mathbb{Q}(\zeta_n) \subset K$ , and $[\mathbb{Q}(\zeta_n):\mathbb{Q}] = \phi(n)$ , it follows that for degree 7, $m$ must be 1 or 2. Thus, the search is limited to cyclic groups $(1, n)$ and groups of the form $(2, 2n)$ .
Gonality and Density Degree:
- The paper utilizes the concept of gonality ( $\text{gon}_K X$ ), the smallest degree of a morphism from a curve $X$ to $\mathbb{P}^1$ defined over $K$ .
- Proposition 2.1 (from Derickx-Sutherland) establishes a link between the minimum density degree $\delta(X/K)$ (the smallest degree $n$ such that $X$ has infinitely many points of degree $n$ ) and gonality: $\delta(X/K) \le \text{gon}_K X \le 2\delta(X/K)$ .
- Crucially, if the rank of the Jacobian $J(X)(\mathbb{Q})$ is 0, then $\delta(X/\mathbb{Q}) = \text{gon}_{\mathbb{Q}} X$ .
- Conclusion: To prove a group $G \in \Phi_\infty(7)$ , one must show that the corresponding modular curve $X$ has infinitely many points of degree 7. This is guaranteed if $\text{gon}_{\mathbb{Q}} X \le 7$ (and rank conditions are met). Conversely, if $\text{gon}_{\mathbb{Q}} X > 7$ , the group does not appear infinitely often.

B. Computational Strategy

The proof relies heavily on computing the gonality of modular curves $X_1(2, 2n)$ over finite fields to bound the gonality over $\mathbb{Q}$ .

Lower Bounds: The author uses the inequality $\text{gon}_{\mathbb{Q}} X \ge \text{gon}_{\mathbb{F}_p} X$ for primes $p$ of good reduction.
Techniques:
- Abramovich's Bound: Used to establish high gonality for large $n$ ( $n \ge 16$ ) via index calculations of congruence subgroups.
- Derickx-Terao [DT26] Algorithms: New techniques for computing class groups and gonalities of curves over finite fields were utilized to compute exact or lower-bound gonalities for specific cases ( $11 \le n \le 15$ ).
- Castelnuovo-Severi Inequality: Applied specifically for the case $n=15$ to avoid heavy computation by analyzing the geometry of maps between $X_1(2, 30)$ and $X_1(30)$ .

3. Key Contributions and Results

The main result is Theorem 1.1, which completely characterizes $\Phi_\infty(7)$ :
$\Phi_\infty(7) = \{(1, n) : 1 \le n \le 30, n \notin \{25, 29\}\} \cup \{(2, 2n) : 1 \le n \le 10\}$

The proof is broken down into three lemmas addressing different ranges of $n$ :

Cyclic Groups $(1, n)$ :
- Relying on Theorem 2.2 (from Derickx-Van Hoeij), the author cites that cyclic groups $(1, n)$ are in $\Phi_\infty(7)$ if and only if $1 \le n \le 30$ and $n \neq 25, 29$ . This part of the result was previously established.
Product Groups $(2, 2n)$ for $n \ge 16$ :
- Lemma 3.1 proves that for $n \ge 16$ , the gonality of the modular curve $X_1(2, 2n)$ over $\mathbb{Q}$ is strictly greater than 7.
- This is achieved by combining Abramovich's lower bound on gonality with the index of the congruence subgroup $\Gamma_1(2, 2n)$ . Since $\text{gon}_{\mathbb{Q}} X > 7$ , these groups cannot appear infinitely often over degree 7 fields.
Product Groups $(2, 2n)$ for $n \le 10$ :
- Lemma 3.2 confirms that these groups do appear infinitely often.
- For $n \le 6$ , this follows from genus considerations ( $g \le 1$ ) and Hilbert's Irreducibility Theorem.
- For $7 \le n \le 10$ , the result relies on unpublished code by Maarten Derickx which explicitly constructs modular units of degree 7, proving the existence of infinitely many degree 7 points.
The Critical Gap: $11 \le n \le 15$ :
- Lemma 3.3 is the novel contribution of this paper. It proves that $(2, 2n) \notin \Phi_\infty(7)$ for $n \in \{11, 12, 13, 14, 15\}$ .
- The author computes the gonality of $X_1(2, 2n)$ over small finite fields ( $\mathbb{F}_3, \mathbb{F}_7$ ).
- Results:
  - $n=11, 12$ : $\text{gon}_{\mathbb{F}_p} \ge 8$ .
  - $n=13, 14$ : $\text{gon}_{\mathbb{F}_p} = 12$ .
  - $n=15$ : $\text{gon}_{\mathbb{F}_7} \ge 8$ (verified via Castelnuovo-Severi inequality).
- Since the rank of the Jacobian $J_1(2, 2n)(\mathbb{Q})$ is 0 for these $n$ (Theorem 2.3), the condition $\text{gon}_{\mathbb{Q}} X > 7$ holds, excluding these groups from $\Phi_\infty(7)$ .

4. Significance

Completeness: This paper closes a major gap in the classification of torsion groups for elliptic curves over number fields. It provides the first complete description of $\Phi_\infty(7)$ .
Methodological Advancement: The paper demonstrates the power of combining classical algebraic geometry (gonality bounds, Castelnuovo-Severi) with modern computational number theory (Derickx-Terao algorithms for finite field gonalities).
Foundation for Future Work: While $\Phi(7)$ (the set of all possible torsion groups, including those appearing finitely often) remains unknown, determining $\Phi_\infty(7)$ is a prerequisite for tackling the full set. The methods used here, particularly the computational verification of gonalities for specific modular curves, set a precedent for analyzing higher-degree fields.
Verification of Conjectures: The results confirm that the "infinite" behavior of torsion groups stabilizes in a predictable pattern up to degree 7, with specific exclusions (like $n=25, 29$ for cyclic groups and $n \ge 11$ for product groups) driven by the geometric complexity (gonality) of the underlying modular curves.

In summary, Najman successfully determines that over septic fields, the only torsion groups appearing infinitely often are cyclic groups up to order 30 (excluding 25 and 29) and product groups of the form $(2, 2n)$ where $n \le 10$ .