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Exact counts of elliptic curves of bounded height over $\mathbb F_q(t)$ in characteristics $2$ and $3$

This paper establishes exact formulas for the weighted and unweighted counts of elliptic curves of bounded height over $\mathbb{F}_q(t)$ in characteristics 2 and 3, revealing that while the leading term remains consistent across all characteristics, the lower-order terms exhibit distinct behaviors driven by nonsmooth loci and wild automorphisms unique to small characteristics.

Original authors: Jun-Yong Park

Published 2026-08-21

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CC BY 4.0

Original authors: Jun-Yong Park

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

In the vast landscape of mathematics, there is a branch dedicated to understanding the shapes and behaviors of equations that define curves. Among these, elliptic curves are particularly famous; they are smooth, looping shapes that, despite their simple appearance, hide deep and complex secrets about numbers. For centuries, mathematicians have studied these curves over familiar number systems like the real numbers or fractions. However, a different kind of arithmetic exists where the usual rules of counting behave strangely. In these systems, known as fields of characteristic two and three, the number one plus one equals zero, and the number two plus one equals zero. This creates a world where standard geometric intuition often fails, and equations can behave in ways that seem impossible elsewhere.

To make sense of these curves, mathematicians use a measuring stick called "height." Think of height not as physical distance, but as a measure of complexity. A curve with a low height is built from simple, small numbers, while a curve with a high height requires increasingly large and complicated numbers to describe it. By counting how many distinct curves exist below a certain height limit, researchers can map out the distribution of these mathematical objects. This counting is not just an exercise in enumeration; it reveals the fundamental structure of the number systems themselves. For a long time, mathematicians could count these curves accurately in most number systems, but the strange arithmetic of characteristics two and three remained a stubborn gap, where previous methods broke down and left the counts incomplete.

A recent study by Jun-Yong Park has finally closed this gap, providing the first exact counts of these elliptic curves in characteristics two and three. The work does not merely offer an estimate or a simulation; it delivers a precise, proven formula that works for every possible height limit. Park's approach involved looking closely at the equations that define these curves, specifically a standard format known as the generalized Weierstrass equation. In most number systems, mathematicians could safely assume that if a curve had a "singular" point—a place where the curve pinches or crosses itself—that point would be visible and rational within the system. However, Park discovered that in characteristics two and three, this assumption is false. There are specific equations where the singular point exists, but it is hidden, becoming visible only after a special, non-standard extension of the number system. These hidden points create a "nonsmooth locus" that previous counting methods missed entirely.

To solve this, Park had to account for these hidden singularities, which he calls quasi-elliptic-type strata. These are regions in the space of equations where the geometry is subtly broken in a way that only reveals itself through a specific kind of mathematical extension. By carefully identifying and counting these hidden cases, he was able to correct the total count. Furthermore, the study had to grapple with the symmetries of the curves. In these specific characteristics, some curves possess extra symmetries—ways to rotate or flip the shape that leave it looking exactly the same—that do not exist in other number systems. These extra symmetries are often non-commutative, meaning the order in which you apply them matters, and they can be "wild," behaving in ways that are highly sensitive to the underlying arithmetic. Park developed a method to weigh these curves correctly, ensuring that curves with many symmetries were not overcounted, while also tracking how these symmetries change when the curve is twisted or modified.

The result is a set of exact formulas that tell us precisely how many elliptic curves exist below any given height bound. The most striking finding is that while the leading term of the count—the dominant number that grows the fastest as the height increases—is the same across all characteristics, the smaller, lower-order terms differ significantly. In characteristics two and three, the count includes additional terms that appear nowhere else. For instance, in characteristic two, there is a specific term that grows at a rate proportional to the height raised to the power of five-twelfths, and another that grows at the power of one-quarter, but only when the size of the number system is an even power of two. These terms are not random; they are direct mathematical reflections of the hidden singular points and the complex, non-commutative symmetries unique to these fields.

Perhaps most surprisingly, some of these lower-order terms are negative. In the context of counting, a negative term does not mean there are fewer curves than zero; rather, it represents a necessary subtraction to correct for overcounting that occurred when the researchers first looked at the raw equations. It is a correction factor that removes the "ghosts" of non-minimal models and hidden singularities that do not correspond to unique, valid curves. The study confirms that when you strip away the complexity of these special characteristics, the underlying growth rate of the number of curves remains consistent with the rest of mathematics, but the path to that growth is paved with unique, characteristic-specific corrections.

This work completes a long-standing project to enumerate elliptic curves over function fields in every possible characteristic. By resolving the issues of hidden singularities and wild symmetries, Park has provided a definitive map for this corner of number theory. The formulas are exact and proven, offering a clear view of how these curves are distributed. The findings show that while the broad strokes of the mathematical landscape are uniform, the fine details are deeply influenced by the specific arithmetic rules of the world in which the curves live. For anyone interested in the structure of numbers, this study reveals that even in the most exotic and counterintuitive arithmetic systems, order and precision can be found, provided one knows exactly where to look for the hidden points and how to weigh the complex symmetries.

Technical Summary: Exact Counts of Elliptic Curves of Bounded Height over $\mathbb{F}_q(t)$ in Characteristics 2 and 3

Problem Statement
This paper addresses the enumeration of $K$ -isomorphism classes of elliptic curves over the function field $K = \mathbb{F}_q(t)$ , where $q = p^r$ and $p \in \{2, 3\}$ . The goal is to determine the exact weighted and unweighted counts of these curves with bounded Faltings height (equivalently, bounded minimal discriminant degree). While the case for characteristic $p > 3$ was previously resolved by Bejleri, Park, and Satriano [BPS22], the characteristics 2 and 3 present distinct arithmetic and geometric obstructions that prevent the direct application of those methods. Specifically, the standard rational-singular-section parametrization fails to exhaust the nonsmooth locus, and the automorphism groups of curves with $j=0$ exhibit non-abelian structures containing "wild" elements (orders divisible by $p$ ) that interact non-trivially with twisting.

Methodology
The author employs a refined version of de Jong's weighted-counting framework, adapted to handle the specific pathologies of small characteristics. The methodology proceeds through several key stages:

Correction of the Nonsmooth Locus:
Standard counting arguments (e.g., [dJ02]) assume that a geometric singular point of a Weierstrass equation extends to a $K$ -rational section. Over the imperfect field $K = \mathbb{F}_q(t)$ , this fails in characteristics 2 and 3. The paper identifies "quasi-elliptic-type strata" where the unique geometric singular point is defined only over a nontrivial purely inseparable extension of $K$ . The normalization of the closure of such points maps purely inseparably to $\mathbb{P}^1$ rather than providing a section. Proposition 2.4 quantifies the additional equations in these strata, which are missed by the rational-singular count.
Marked Inertia and Automorphism Groups:
To transition from weighted counts (where a curve $E$ is weighted by $1/|\text{Aut}_K(E)|$ ) to unweighted counts, the paper utilizes a groupoid-theoretic approach. It counts "marked pairs" $(E, \alpha)$ where $\alpha$ is an automorphism distinct from the identity and inversion $[-1]$ .
- Characteristic 3: The analysis focuses on the $j=0$ locus, where the geometric automorphism group has order 12. The paper classifies fixed pairs under the residual coordinate group, identifying automorphisms of orders 3, 4, and 6.
- Characteristic 2: Similarly, the $j=0$ locus has a geometric automorphism group of order 24. The paper classifies fixed pairs, distinguishing between automorphisms of orders 3, 4, and 6. Crucially, the descent of these automorphisms to $K$ depends on the parity of $r$ (specifically, whether $\mathbb{F}_q$ contains specific roots of unity).
Minimality Defect and Intrinsic Divisors:
The paper relates the count of coefficient-degree- $d$ equations to the exact Faltings height $n$ via the "minimality defect" $e = d - n$ . Unlike the $p > 3$ case, where divisibility conditions on coefficients can be imposed invariantly, the residual coordinate translations in characteristics 2 and 3 destroy visible coefficient divisibility. Consequently, the paper formulates the defect intrinsically on the resolved elliptic surface using de Jong's blowup divisor $D$ . This allows for a convolution formula relating the ambient marked counts to the exact height counts.
Generating Series and Inversion:
The relationship between the ambient counts (coefficient degree) and the exact height counts is expressed via generating series involving the symmetric powers of $\mathbb{P}^1$ . The author inverts this convolution to derive explicit formulas for the cumulative counts up to a height bound $B$ .

Key Results
The paper establishes exact formulas for the weighted count $N^w(K, B)$ and the unweighted count $N(K, B)$ for $B = q^{12M}$ .

Leading Term: In both characteristics, the leading term of the unweighted count is of order $B^{5/6}$ with the coefficient $2 \frac{q^9-1}{q^8-q^7}$ . This is exactly twice the leading term of the weighted count, consistent with the generic case where $|\text{Aut}_K(E)| = 2$ .
Lower-Order Terms: The lower-order terms differ significantly from the $p > 3$ $p > 3$ case and depend on $p$ $p$ and the parity of $r$ $r$ (via the indicator $\varepsilon_r$ $ε_{r}$ , which is 1 if $r$ $r$ is even and 0 if odd).
- Characteristic 3: The unweighted count includes terms of order $B^{1/2}$ , $B^{1/3}$ , and $B^{1/6}$ . The $B^{1/3}$ term involves $\varepsilon_r$ , reflecting the descent of order-4 automorphisms.
- Characteristic 2: The unweighted count includes terms of order $B^{1/2}$ , $B^{5/12}$ , $B^{1/3}$ , $B^{1/4}$ , and $B^{1/6}$ . Notably, the $B^{5/12}$ term (arising from order-4 automorphisms) and the $B^{1/4}$ term (arising from order-3/6 automorphisms when $r$ is even) are unique to characteristic 2.
Negative Coefficients: Several lower-order coefficients are negative. These arise from the subtraction of non-minimal models (removing the minimality defect) and the correction for the nonsmooth locus.
Height Zero: At $B=1$ , the formulas recover the known classification of $j=0$ twists over $\mathbb{F}_q$ (Kronberg–Soomro–Top [KST17]), confirming the consistency of the method.

Significance and Claims
The paper claims to complete the exact enumeration of elliptic curves of bounded height over $\mathbb{F}_q(t)$ in every characteristic. By combining the results of this work with the $p > 3$ formulas from [BPS22], the author provides a unified, characteristic-independent framework for the leading asymptotic behavior, while explicitly detailing the characteristic-specific lower-order corrections.

The significance lies in resolving two specific "small-characteristic phenomena":

Nonsmooth Locus Corrections: The paper demonstrates that the rational-singular-section argument is insufficient in characteristics 2 and 3 due to purely inseparable singular points, requiring a correction term that alters the lower-order asymptotics.
Wild Automorphisms and Twisting: The paper rigorously handles the non-abelian, wild automorphism groups at $j=0$ . It shows that the passage from weighted to unweighted counts requires a "marked-inertia" calculation involving conjugacy classes and centralizer weights, where the descent of automorphisms is sensitive to the arithmetic of the constant field (specifically the parity of $r$ ).

The results show that while the leading asymptotic behavior is universal, the lower-order terms are highly sensitive to the characteristic and the arithmetic of the base field, reflecting the complex interplay between the geometry of the moduli space and the arithmetic of the function field.

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