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Kodaira-Neron statistics for rational elliptic curves with $j$ -invariant 0 and 1728

This paper counts rational elliptic curves with $j$ -invariants 0 and 1728 by height, deriving asymptotic statistics for their Kodaira-Néron types at the primes of bad reduction (3 and 2, respectively) while also analyzing these distributions under fixed torsion subgroups and isogeny-torsion graphs.

Original authors: John Cullinan, Sebastian Sargenti

Published 2026-05-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: John Cullinan, Sebastian Sargenti

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 the world of mathematics as a vast, infinite library. Inside this library, there are millions of books, but instead of stories, they contain complex equations called Elliptic Curves. These aren't just any equations; they are the building blocks of modern cryptography and number theory.

This paper, written by John Cullinan and Sebastian Sargenti, is like a census taker entering that library. Their goal? To count how many of these "books" (curves) have a specific, rare feature: a "signature" called a j-invariant of either 0 or 1728.

Here is a simple breakdown of what they did, using everyday analogies:

1. The "Bad Weather" Report (Reduction Types)

Imagine every elliptic curve is a house. Most houses are sturdy, but some have structural issues when the weather gets bad (specifically, when you look at them through the "lens" of a prime number like 2 or 3).

The Problem: When a curve has "bad reduction" at a prime, it's like the house developing a crack or a leak. Mathematicians need to know exactly how bad the damage is.
The Kodaira-Néron Type: This is the official "damage report." It doesn't just say "broken"; it gives a specific code (like II, III, IV, or I*). Think of these codes as different levels of structural failure:
- Type II or III: A minor crack in the foundation.
- Type IV: A slightly bigger issue.
- Type I or II:** A major structural collapse.

The authors noticed that for curves with the signature 1728, the "bad weather" always happens at the number 2. For curves with the signature 0, it always happens at the number 3.

2. The Great Counting Game

The authors wanted to answer a big question: "If we look at all possible curves with these signatures, how often does each specific 'damage report' (Kodaira-Néron type) appear?"

To do this, they had to organize the library. They couldn't just count randomly; they had to sort the books by "size" (called Height).

The Analogy: Imagine sorting books by their thickness. They counted how many thin books there are, how many medium ones, and how many thick ones, up to a certain limit. As they looked at thicker and thicker books (larger numbers), they looked for a pattern.

3. The Two Main Families

The paper splits the investigation into two distinct groups, like two different neighborhoods in a city:

Neighborhood A: The "1728" District

The Vibe: Every house here has a specific architectural style.
The Twist: Some houses belong to small, rare families (called isogeny-torsion graphs). Most houses, however, belong to a massive, dominant family called L2(2).
The Discovery: The authors found that the rare families are so small compared to the big family that they barely affect the overall statistics.
The Result: They calculated the exact percentage of houses that have a "Type II" crack, a "Type III" crack, etc.
- Example: They found that "Type III" damage is the most common, occurring about 53% of the time (8/15), while "Type II" happens about 27% of the time (4/15).

Neighborhood B: The "0" District

The Vibe: This neighborhood is more diverse. The houses are sorted by their "furniture" (the torsion subgroup).
The Groups:
1. Empty House: No furniture (Trivial torsion).
2. House with a Chair: One specific piece of furniture (Z/2Z).
3. House with a Table: A different piece of furniture (Z/3Z).
The Discovery: The authors realized that the "damage report" depends heavily on which furniture the house has.
- If the house is empty, you get a wide variety of damage types (from II to IV*).
- If the house has a "Chair," you mostly see specific types (III and III*).
- If the house has a "Table," you see a different set of types (II, III, IV, IV*).
The Result: They provided a precise recipe (mathematical formulas) to predict exactly how many houses of each furniture type will have each specific damage code as the library grows infinitely large.

4. How They Did It (The Detective Work)

To get these numbers, the authors didn't just guess. They used a mathematical tool called Tate's Algorithm.

The Metaphor: Imagine a robot inspector that walks up to every house, checks the foundation, measures the cracks, and assigns a code.
The authors programmed this robot to check millions of theoretical houses.
They then used a technique called Sieve Methods. Think of this like sifting sand through a net. They filtered out the houses that didn't fit their criteria (like houses that weren't "minimal" or "simplest" versions) to get a clean count of the valid ones.

5. The Bottom Line

The paper is essentially a statistical map of a very specific part of the mathematical universe.

For the "1728" curves: They proved that one specific family of curves (L2(2)) completely dominates the statistics, and they gave the exact odds of seeing every type of "bad reduction."
For the "0" curves: They showed that the "furniture" (torsion) changes the odds entirely, and they provided the exact odds for each furniture type.

They even ran computer simulations (using a tool called Pari/GP) to check their math, and the real-world counts matched their theoretical predictions almost perfectly.

In short: The authors took a chaotic, infinite collection of mathematical curves, organized them by their "bad spots," and gave us a precise probability chart for what kind of "damage" we are most likely to find in each group.

Technical Summary: Kodaira-Néron Statistics for Rational Elliptic Curves with $j$ -Invariant 0 and 1728

Problem Statement
The paper addresses the statistical distribution of Kodaira-Néron types for rational elliptic curves over $\mathbb{Q}$ with specific $j$ -invariants: $j=0$ and $j=1728$ . While the Kodaira-Néron type $typ_p(E)$ provides a fine measure of bad reduction at a prime $p$ (incorporating component group size and potential reduction type), previous studies have largely focused on families with fixed torsion subgroups. This work focuses on families where every member shares a common prime of bad reduction: $p=3$ for $j=0$ and $p=2$ for $j=1728$ . The central question is to determine the exact asymptotic proportions with which each Kodaira-Néron type occurs within these families, ordered by height.

Methodology
The authors employ a combination of arithmetic geometry and analytic number theory:

Minimal Models and Tate's Algorithm: The authors first establish necessary and sufficient conditions for an elliptic curve with $j=0$ or $j=1728$ to be minimal at the relevant primes ( $p=2$ or $p=3$ ). They utilize Tate's algorithm to derive precise arithmetic criteria (involving valuations and congruences of the coefficients $A$ and $B$ in the short Weierstrass form) that determine the Kodaira-Néron type.
Parameterization by Torsion and Isogeny: The families are refined by fixing the torsion subgroup $E(\mathbb{Q})_{tors}$ and, for $j=1728$ , the isogeny-torsion graph. The curves are parameterized by integers $t$ (e.g., $y^2 = x^3 + t^3$ for $j=0, Z/2Z$ torsion) subject to specific square-free, cube-free, or fourth-power-free conditions to ensure minimality.
Sieve Methods and Asymptotics: Using standard sieve methods, the authors count the number of integers $t$ satisfying the minimality and reduction type conditions up to a height bound $X$ . They leverage known proportions of $k$ -th power-free integers and adjust local Euler factors to account for specific congruence conditions (e.g., parity or modulo 9 constraints).

Key Contributions and Results

Case $j=1728$ :
- The authors distinguish between curves belonging to isogeny-torsion graphs of type $T_4^3$ and $L_2(2)$ . They prove that curves in the $L_2(2)$ class dominate the asymptotic count ( $O(X^{1/3})$ ) compared to $T_4^3$ ( $O(X^{1/6})$ ).
- Theorem 1.5 provides explicit asymptotic formulas for the number of curves $N_T^{1728}(X)$ with a specific Kodaira-Néron type $T$ at $p=2$ , separated by graph type.
- Corollary 1.7 establishes that the overall proportions for $j=1728$ are determined by the $L_2(2)$ class. The types occurring are $\{II, III, III^*, I_2^*, I_3^*\}$ with specific constants $c_T$ (e.g., $8/15$ for type $III$, $4/15$ for type $II$).
- Numerical verification using Pari/GP up to height $10^{18}$ confirms the theoretical asymptotics.
Case $j=0$ :
- The study is refined by the torsion subgroup $E(\mathbb{Q})_{tors} \in \{\{ \infty \}, \mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/3\mathbb{Z}\}$ .
- Theorem 1.8 ( $Z/2Z$ torsion): Shows types $III$ and $III^*$ occur at $p=3$ with proportions $3/4$ and $1/4$ , respectively.
- Theorem 1.9 ( $Z/3Z$ torsion): Shows types $II, III, IV, IV^*$ occur with proportions $6/13, 3/13, 3/13, 1/13$ .
- Theorem 1.10 (Trivial torsion): Shows types $II, III, IV, II^*, III^*, IV^*$ occur with a complex set of proportions derived from congruences modulo 36.
- The authors provide explicit asymptotic formulas for each case, involving constants derived from $\zeta(k)$ and specific arithmetic factors (e.g., $62/63$ for the trivial torsion case).
Tables and Classification: The paper includes detailed tables (Tables 2–9) mapping the arithmetic conditions of the curve parameters to specific Kodaira-Néron types for various torsion and isogeny configurations.

Significance and Claims
The paper claims to determine the "exact proportion" with which each Kodaira-Néron type occurs within these specific families of rational elliptic curves. By holding the torsion subgroup and isogeny-torsion graph constant, the authors refine previous statistical studies to provide a more granular understanding of bad reduction behavior. The work serves as a repository of explicit asymptotic counts and arithmetic criteria, verified by computational examples, contributing to the broader understanding of the distribution of arithmetic invariants in families of elliptic curves. The authors note that their results for $j=1728$ with non-trivial torsion were partially known via prior work (cited as [1]), but they provide complete details for the cases not previously covered, particularly regarding the trivial torsion case for $j=0$ and the distinction between isogeny-torsion graphs for $j=1728$ .