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The Smallest Invariant Factor of Elliptic Curves, and Coincidences

This paper investigates the conditions under which the density constant for primes where the smallest invariant factor of an elliptic curve's group of points equals a specific integer is positive, utilizing adelic Galois representations to establish a group-theoretic criterion and identifying that this constant vanishes precisely when coincidences occur among the curve's abelian division fields.

Original authors: Alexander Milner, Jack Shotton

Published 2026-04-24

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexander Milner, Jack Shotton

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, infinite collection of locks, one for every prime number (2, 3, 5, 7, 11, etc.). Each lock is attached to a specific elliptic curve—a special kind of mathematical shape that looks a bit like a squashed donut.

When you try to "unlock" one of these locks using a specific prime number $p$ , the lock doesn't just click open or stay shut. Instead, it opens into a small group of keys. Mathematicians call this group $E(\mathbb{F}_p)$ .

The "Smallest Invariant Factor" (The Keychain)

Usually, this group of keys can be described as two smaller groups stuck together, like a chain with two links.

Link A: The smallest link.
Link B: The bigger link (which is always a multiple of Link A).

The paper focuses entirely on Link A. Let's call the size of this smallest link $d_{E,p}$ .

If $d_{E,p} = 1$ , the lock opens into a single, simple chain. The group is "cyclic" (simple).
If $d_{E,p} = 2$ , the smallest link has size 2.
If $d_{E,p} = 5$ , the smallest link has size 5.

The Big Question: For a specific number $j$ (like 1, 2, or 5), does the smallest link ever equal $j$ ? And if it does, does it happen infinitely many times as we try different prime locks?

The "Density" Calculator ( $C_{E,j}$ )

The authors look at a formula created by a mathematician named Cojocaru. This formula, $C_{E,j}$ , acts like a probability meter.

If the meter reads positive (greater than zero), it means: "Yes! There are infinitely many prime locks where the smallest link is exactly size $j$ ."
If the meter reads zero, it means: "No. After a certain point, you will never find a prime lock where the smallest link is size $j$ ."

The paper's main goal is to figure out when the meter reads zero.

The "Coincidence" Mystery

The authors discovered that the meter usually reads positive. It only reads zero when something weird happens in the background of the math universe. They call these weird events "Coincidences."

Imagine the elliptic curve has a set of "division fields." Think of these as secret rooms or nested boxes inside the curve's structure.

Room $Q(E[j])$ contains information about the $j$ -th division.
Room $Q(E[jp])$ contains information about the $j \times p$ -th division.

Usually, Room $A$ is smaller than Room $B$ . But sometimes, Room A and Room B are actually the exact same room. They overlap perfectly.

The Discovery:
The paper proves that the "probability meter" ( $C_{E,j}$ ) drops to zero if and only if these two secret rooms are identical.

If $Q(E[j]) = Q(E[jp])$ (the rooms are the same), then the smallest link $d_{E,p}$ can never be $j$ . The universe forbids it.
If the rooms are different, the meter stays positive, and the pattern repeats forever.

The "Entanglement" Analogy

Think of the elliptic curve as a complex knot.

Division fields are like the different ways you can untangle the knot.
A Coincidence is when two different untangling methods accidentally result in the exact same shape.
The authors found that for most curves, this only happens with very specific "twists" (multiplying by 2 or 3). They suspect that for these curves, you can never get a coincidence by multiplying by 5, 7, or any larger prime. It's as if the knot is too rigid to allow those specific overlaps.

Why Does This Matter?

Fixing a Mistake: The paper corrects a small error in a famous 2004 paper. The old paper said the meter was always positive for $j=1$ (unless the curve had a specific flaw). The new paper proves this is true but explains why the old proof had a tiny gap.
Predicting Patterns: By understanding these "coincidences," mathematicians can predict exactly which numbers will appear as the smallest link size for any given curve.
The "Serre" Curve: Most elliptic curves are "well-behaved" (called Serre curves). For these, the meter is always positive. The weird zero-reading cases are the rare, special exceptions where the curve's structure has these hidden "coincidences."

Summary in One Sentence

This paper explains that the frequency of a specific pattern in elliptic curves drops to zero only when the curve's internal "secret rooms" (division fields) accidentally overlap perfectly, a phenomenon that mostly happens when multiplying by 2 or 3.

1. Problem Statement

The paper investigates the distribution of the smallest invariant factor of the group of points on an elliptic curve $E$ over $\mathbb{Q}$ reduced modulo a prime $p$ .

For a prime $p$ of good reduction, the group $E(\mathbb{F}_p)$ is isomorphic to $\mathbb{Z}/d_{E,p}\mathbb{Z} \times \mathbb{Z}/e_{E,p}\mathbb{Z}$ with $d_{E,p} \mid e_{E,p}$ .
The integer $d_{E,p}$ is the smallest invariant factor. If $d_{E,p}=1$ , the group is cyclic.
Cojocaru's Theorem (2004) establishes that, under the Generalized Riemann Hypothesis (GRH), the density of primes $p$ such that $d_{E,p} = j$ is given by a constant $C_{E,j}$ :
$C_{E,j} = \sum_{k=1}^{\infty} \frac{\mu(k)}{[\mathbb{Q}(E[jk]) : \mathbb{Q}]}$
The Core Question: While the formula exists, it is not immediately clear when $C_{E,j} > 0$ . If $C_{E,j} = 0$ , there are only finitely many primes $p$ with $d_{E,p} = j$ . The authors aim to determine necessary and sufficient conditions for $C_{E,j} > 0$ for elliptic curves without Complex Multiplication (non-CM).

2. Methodology

The authors employ a combination of group-theoretic analysis of Galois representations and field-theoretic arguments regarding division fields.

Adelic Galois Representations: They analyze the image $H$ of the adelic Galois representation $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\widehat{\mathbb{Z}})$ . By Serre's Open Image Theorem, $H$ is an open subgroup of $\text{GL}_2(\widehat{\mathbb{Z}})$ .
Level Reduction: They define the adelic level $m_E$ as the smallest integer such that the kernel of reduction modulo $m_E$ is contained in $H$ . This allows them to reduce the infinite sum defining $C_{E,j}$ to a finite calculation involving the structure of $H \pmod{m_E}$ .
Inclusion-Exclusion Principle: They translate the condition $C_{E,j} = 0$ into a group-theoretic condition involving the union of subgroups. Specifically, $C_{E,j} = 0$ if and only if the subgroup $H \cap \Gamma_j$ (where $\Gamma_j$ is the kernel of reduction mod $j$ ) is covered by the union of subgroups $H \cap \Gamma_{jp}$ for primes $p$ .
Division Field Coincidences: They investigate "coincidences" where division fields are equal, i.e., $\mathbb{Q}(E[j]) = \mathbb{Q}(E[jp])$ . They prove that such coincidences are the primary obstruction to the positivity of $C_{E,j}$ .

3. Key Contributions and Results

A. Positivity Criteria

The authors provide a refined criterion for when $C_{E,j} > 0$ , improving upon previous results by Kim (2015).

Theorem 3.2 (Main Positivity Criterion): Let $j \mid m_E$ $j ∣ m_{E}$ . Let $R$ $R$ be the product of $j$ $j$ and the odd prime factors of $m_E/j$ $m_{E} / j$ . If:
1. $\mathbb{Q}(E[j]) \cap \mathbb{Q}(\zeta_R) = \mathbb{Q}(\zeta_j)$ (a condition on the intersection of division fields and cyclotomic fields), and
2. $\mathbb{Q}(E[2^{a+1}]) \neq \mathbb{Q}(E[2^a])$ where $a = v_2(j)$ ,
  Then $C_{E,j} > 0$ .
Corollary 3.8: If $\gcd(j, 2A(E)) = 1$ and $\mathbb{Q}(E[2]) \neq \mathbb{Q}$ , then $C_{E,j} > 0$ . Here $A(E)$ is the product of primes where the $p$ -adic representation is not surjective. This generalizes Kim's result by removing the condition that $j$ must be coprime to the conductor $N_E$ .
Serre Curves: They prove that for "Serre curves" (where the image of the adelic representation has index 2 in $\text{GL}_2(\widehat{\mathbb{Z}})$ ), $C_{E,j} > 0$ for all $j$ .

B. Coincidences and Vanishing Constants

The paper characterizes when $C_{E,j} = 0$ .

Conjecture 2.6: For a non-CM elliptic curve, $C_{E,j} = 0$ if and only if there exists a prime $p$ such that $\mathbb{Q}(E[j]) = \mathbb{Q}(E[jp])$ .
Conjecture 4.2: Such coincidences only occur for $p=2$ or $p=3$ . They conjecture that for $p \geq 5$ , no non-CM elliptic curves over $\mathbb{Q}$ exhibit a $p$ -coincidence.
Verification: They verify these conjectures for all curves in the LMFDB (L-Functions and Modular Forms Database) with conductor $\leq 500,000$ .

C. Construction of Coincidences

The authors document infinite families of curves exhibiting coincidences, primarily arising from abelian division fields:

Abelian 2-torsion: If $\mathbb{Q}(E[2])$ is an abelian extension of odd conductor $m$ , then $\mathbb{Q}(E[m]) = \mathbb{Q}(E[2m])$ .
Abelian 3-torsion: If $\mathbb{Q}(E[3]) = \mathbb{Q}(\zeta_3)$ and $3$ divides the square-free part of the discriminant, specific coincidences arise (e.g., $\mathbb{Q}(E[14]) = \mathbb{Q}(E[42])$ ).
Examples: They provide explicit parametric families of curves (using modular curves $X_H$ ) that generate these coincidences, such as curves with $\mathbb{Q}(E[2]) = \mathbb{Q}(i)$ leading to $\mathbb{Q}(E[2j]) = \mathbb{Q}(E[4j])$ .

D. Correction of Previous Literature

Correction of [CM04]: The authors identify and repair a small error in the proof of Theorem 1.2 (that $C_{E,1} > 0$ if $\mathbb{Q}(E[2]) \neq \mathbb{Q}$ ) found in the paper by Cojocaru and Murty. The original proof incorrectly assumed linear disjointness of certain fields; the authors provide a corrected argument using a more careful analysis of the inclusion-exclusion sum.

4. Significance

Theoretical Advancement: The paper moves beyond the existence of the density constant to a structural understanding of when it is non-zero. It links the arithmetic of the smallest invariant factor directly to the Galois-theoretic structure of division fields.
Resolution of Open Questions: It strengthens Kim's result by relaxing coprimality conditions and provides a comprehensive framework for analyzing "entanglements" (coincidences) of division fields.
Empirical Validation: By verifying conjectures against the LMFDB, the paper provides strong evidence for the classification of division field coincidences, suggesting a deep rigidity in the Galois images of elliptic curves over $\mathbb{Q}$ .
Methodological Utility: The group-theoretic approach using the adelic image and inclusion-exclusion principles offers a robust toolkit for studying similar distribution problems in arithmetic geometry.

In summary, Milner and Shotton establish that the vanishing of the density of primes with a specific smallest invariant factor is almost exclusively caused by "coincidences" in the division fields of the curve, specifically $\mathbb{Q}(E[j]) = \mathbb{Q}(E[jp])$ for small primes $p$ , and they provide explicit criteria and families to detect these phenomena.