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Local statistics and average rank of genus $g$ hyperelliptic curves with a Weierstrass point

This paper establishes probability formulas for various reduction types of genus $g$ hyperelliptic curves with a Weierstrass point over number fields and, under standard conjectures, derives an explicit upper bound for their average analytic rank.

Original authors: Keunyoung Jeong, Junyeong Park

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Keunyoung Jeong, Junyeong 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

Imagine a vast, cosmic garden filled with millions of different kinds of plants called hyperelliptic curves. These aren't your average garden flowers; they are complex, multi-dimensional shapes defined by specific mathematical recipes. Some of these plants have a special "Weierstrass point," which you can think of as a unique, glowing seed that anchors the whole structure.

Mathematicians Keunyoung Jeong and Junyeong Park decided to take a census of this garden. They wanted to answer two big questions: First, if you pick one of these plants at random, what are the odds it will look "perfect" (smooth and healthy) when you zoom in on a specific spot? Second, on average, how "twisted" or complex are these plants?

The Health Check: Good vs. Bad Reduction

To understand the first question, imagine you are a botanist checking the soil quality at a specific location in the garden. In the world of these curves, the "soil" is a prime number (let's call it $p$ ). The researchers found that if the soil is rich enough (specifically, if the prime number is larger than $2g + 1$ , where $g$ is the "genus" or complexity of the curve), you can predict the plant's health with surprising precision.

They discovered that most of these plants are incredibly robust. If you pick a random curve with a Weierstrass point, there is a very high probability—specifically, a chance that looks like $1 - \frac{1}{q}$ (where $q$ is the size of the soil's residue field)—that the plant will be in "good reduction." This means it stays smooth and doesn't break apart when you look at it through the lens of that prime number.

However, sometimes the soil causes the plant to develop a "knot" or a "singularity." The paper categorizes these knots into three types, much like how a knot in a rope can be a simple loop, a tight twist, or a messy tangle:

Abelian Rank ( $a$ ): The plant stays mostly intact, just losing a little bit of its "toric" or "unipotent" flexibility.
Toric Rank ( $t$ ): The plant develops a specific kind of loop (like a multiplicative reduction in simpler curves).
Unipotent Rank ( $u$ ): The plant gets a messy tangle (like an additive reduction).

The authors calculated the exact odds for these different outcomes. For example, the chance of a plant having a "toric" knot (a specific type of loop) is roughly $\frac{q-1}{q+1} \times \frac{1}{q^{2g}}$ . They even worked out the odds for more complex scenarios, like a plant having a "tacnode" (a very specific type of double-point knot), which happens with a probability of $\frac{q-1}{q^4(1-q^{-20})}$ for genus 2 curves.

What they ruled out: The paper is very careful to say that these probabilities are for curves with a Weierstrass point. They explicitly note that if you try to apply their formulas for "additive reduction" (the messy tangle) in the simplest case (genus 1, which is just an elliptic curve), the math breaks down because the "reduction map" lands in a weird, exceptional spot that doesn't behave like the others. So, their general formula doesn't work for that one specific, tiny case without adjustment.

The Twistiness: Average Analytic Rank

The second part of their adventure is about measuring the "twistiness" of these plants. In math, this is called the analytic rank. Think of it as a score that tells you how many independent loops or cycles exist within the plant's structure. A higher score means a more complex, twisted plant.

The authors wanted to know: "If we look at the entire garden of these curves, what is the average twistiness score?"

To get this answer, they had to make some big assumptions. They assumed two famous, unproven (but widely believed) mathematical ideas: the Hasse–Weil conjecture and the Generalized Riemann Hypothesis. These are like assuming the laws of physics hold true in a parallel universe to make the math work.

Under these assumptions, they proved a strict upper limit. They found that the average analytic rank for these curves over a number field $K$ (which has a degree $D$ ) is bounded above by:

$\frac{1}{2} + \frac{3g(2g+1)D}{2}$

This is a concrete, proven ceiling. It means that no matter how many curves you pick, the average twistiness cannot exceed this number.

How sure are they?
The paper is very confident in the upper bound. They didn't just guess; they derived an explicit formula. However, they acknowledge that this is an upper limit, not the exact average. They mention that, based on the "Katz–Sarnak philosophy" (a guiding principle in the field), they expect the true average to be exactly $\frac{1}{2}$ . But their paper only proves it is less than or equal to the formula above.

They also point out that while other mathematicians (Bhargava and Gross) found a tighter bound of $\frac{3}{2}$ for curves over the rational numbers ( $\mathbb{Q}$ ), Jeong and Park's result is special because it works for any number field, not just the rational ones.

The Bottom Line

Jeong and Park have mapped out the statistical landscape of these complex curves. They showed us that:

Most are healthy: The vast majority of these curves stay smooth when viewed through the lens of large prime numbers.
Knots are predictable: If they do develop knots, we know exactly how likely each type of knot is.
Twistiness has a limit: Assuming standard mathematical hypotheses, the average complexity of these curves is capped at a specific number that depends on the curve's genus and the field it lives in.

They haven't solved the mystery of the exact average (which they suspect is 0.5), but they have built a very sturdy fence around it, proving it can't get too wild.

Technical Summary: Local Statistics and Average Rank of Genus $g$ Hyperelliptic Curves with a Weierstrass Point

Problem Statement
This paper investigates the statistical distribution of reduction types for families of hyperelliptic curves of genus $g \geq 2$ over a number field $K$ , specifically those possessing a Weierstrass point. The primary objectives are twofold: first, to determine the probability that such a curve has specific local reduction properties (characterized by abelian, toric, and unipotent ranks) at a prime $p$ of residue characteristic $> 2g + 1$ ; and second, to derive an explicit upper bound for the average analytic rank of this family, assuming the Hasse–Weil conjecture and the Generalized Riemann Hypothesis (GRH).

Methodology
The authors employ a moduli-theoretic approach combined with analytic number theory techniques.

Moduli Space Identification: The study is grounded in the isomorphism between the weighted projective space $\mathbb{P}(w_g) := \mathbb{P}(4, 6, \dots, 4g+2)$ and the compactified moduli stack of genus $g$ hyperelliptic curves with a Weierstrass point. Curves are represented by equations of the form $y^2 = x^{2g+1} + a_2 x^{2g-1} + \dots + a_{2g+1}$ .
Local Statistics via Reduction Maps: The authors define a reduction map $\psi_p$ $ψ_{p}$ from the $p$ $p$ -adic points of the moduli stack to the set of points over the residue field $\kappa(p) \cong \mathbb{F}_q$ $κ (p) ≅ F_{q}$ . The reduction type of a curve is determined by the multiplicity partition of the roots of the polynomial on the right-hand side of the defining equation.
- The abelian rank ( $a$ ), toric rank ( $t$ ), and unipotent rank ( $u$ ) of the Jacobian are explicitly linked to the singularity types of the special fiber, which correspond to partitions of $2g+1$ . For instance, a partition $(2, 1, \dots, 1)$ corresponds to $(a, t, u) = (g-1, 1, 0)$ .
Counting and Height Functions: Using a weighted height function $H_{w_g, K}$ on $\mathbb{P}(w_g)(K)$ , the authors count the number of curves with height $\leq X$ satisfying specific local conditions. They utilize results from Cho–Jeong–Park and others regarding the distribution of points in weighted projective spaces over finite fields, employing Burnside's Lemma and generating functions to count squarefree polynomials with specific root multiplicities.
Average Rank Estimation: To bound the average analytic rank, the authors utilize the explicit formula for $L$ -functions. They compute the first and second moments of the trace of Frobenius ( $a_p$ and $a_p^2$ ) over the family. By averaging the explicit formula over the family and applying the GRH, they relate the average rank to the conductor and the distribution of local reduction types.

Key Contributions and Results

Local Reduction Probabilities (Theorem 1.1): The paper provides explicit probability formulas for the occurrence of specific rank tuples $(a, t, u)$ at a prime $p$ with residue characteristic $> 2g+1$ .
- The probability of good reduction (abelian rank $g$ , toric/unipotent rank 0) is approximately $1 - 1/q$ .
- Explicit formulas are derived for cases with toric rank 1, unipotent rank 1, and toric rank 2.
- For genus $g=2$ , a complete classification of reduction types is provided, including probabilities for cases like $(0, 1, 1)$ and the occurrence of tacnodes.
- These results generalize known local density results for elliptic curves (genus $g=1$ ) to higher genera.
Average Analytic Rank Bound (Theorem 1.2): Assuming the Hasse–Weil conjecture and GRH, the authors establish an upper bound for the average analytic rank of genus $g$ hyperelliptic curves with a Weierstrass point over a number field $K$ of degree $D$ :
$\text{Average Rank} \leq \frac{1}{2} + \frac{3g(2g+1)D}{2}$
- This bound recovers the known result for elliptic curves ( $g=1$ ) by Phillips.
- The derivation relies on showing that the contribution of the first moment of the trace of Frobenius vanishes (Proposition 3.1, 3.2) and estimating the second moment (Proposition 3.3).

Significance and Claims
The paper positions itself as a higher-genus analogue to local-density results for elliptic curves established by Cho–Jeong, Shankar–Shankar–Wang, and Phillips.

Generalization: The work extends the understanding of local statistics from elliptic curves to hyperelliptic curves of arbitrary genus $g \geq 2$ with a Weierstrass point, utilizing the specific geometry of the weighted projective moduli space.
Rank Bounds: While Bhargava and Gross previously established a bound of $3/2$ for the algebraic rank of Jacobians of such curves over $\mathbb{Q}$ , this paper provides a bound for the analytic rank over arbitrary number fields. The authors note that their bound is weaker than the Bhargava–Gross result for $g=1$ over $\mathbb{Q}$ but applies more broadly to arbitrary number fields and addresses the analytic rank directly, which is of independent interest in the context of the Birch–Swinnerton-Dyer conjecture.
Philosophical Context: The authors acknowledge the Katz–Sarnak philosophy, which predicts the average analytic rank to be $1/2$ . Their derived bound is consistent with this expectation, though it includes a term dependent on the genus and the degree of the number field.

The paper does not claim to prove the Hasse–Weil conjecture or GRH but rather uses them as hypotheses to derive the rank bound, following the standard methodology in the field for such conditional results.

The Health Check: Good vs. Bad Reduction

The Twistiness: Average Analytic Rank

The Bottom Line

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