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Generically Ordinary One-parameter Hyperelliptic Families are Dense

The paper demonstrates that for a Zariski-generic coefficient vector in the space of one-parameter hyperelliptic families, the curves possess generically ordinary reduction at every prime above sufficiently large primes, and establishes that the set of ordinary primes for any such family has a natural density of at least $1/[Q(a,\zeta_d):Q(a)]$ , which becomes 1 after base change to $Q(a,\zeta_d)(t)$ .

Original authors: Hui June Zhu

Published 2026-08-25

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CC BY 4.0

Original authors: Hui June Zhu

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 number theory, mathematicians often study shapes defined by equations, looking for patterns that hold true across different mathematical worlds. One such family of shapes is called hyperelliptic curves. You can think of these as smooth, looping lines drawn on a plane, defined by a specific type of equation involving a variable raised to a high power. While these curves exist in the familiar world of real numbers, mathematicians are deeply interested in what happens when they are viewed through the lens of prime numbers. This is like taking a complex, continuous shape and examining it on a grid made of dots, where the number of dots is determined by a specific prime number. A key property of these shapes in this discrete setting is whether they are "ordinary." This is a technical term indicating that the shape retains a certain amount of structural complexity and randomness when viewed through the grid of a prime number, rather than collapsing into a simpler, more rigid form. Understanding which shapes remain ordinary is crucial because it connects the smooth geometry of the curve to the arithmetic behavior of prime numbers, revealing deep truths about how numbers interact.

For a long time, a specific question lingered over a particular type of these curves. Researchers wondered if a certain, very simple family of these shapes would almost always stay ordinary when viewed through the lens of large prime numbers. A recent conjecture suggested that for a specific, fixed equation, this was indeed true for all sufficiently large primes. This new work takes that idea and expands it dramatically. Instead of looking at just one fixed equation, the researchers examined the entire universe of possible equations of this type. They treated the numbers that define the curve not as fixed constants, but as variables that could be chosen from a vast pool of integers. The central question became: if you pick a random set of these defining numbers, how likely is it that the resulting curve will stay ordinary for almost all prime numbers?

The author of this study constructed a mathematical map of this entire universe of equations. They proved that the vast majority of these equations, specifically those falling into a large, open region of the map, behave beautifully. For almost any choice of numbers you pick from this region, the resulting curve will be ordinary when reduced modulo any prime number that is large enough. There is a threshold, a specific size for the prime number, beyond which this behavior is guaranteed. This threshold depends on the complexity of the curve and the specific numbers chosen, but once the prime number is larger than this limit, the curve behaves in the expected, complex way. The researchers showed that this is not a rare accident but a general rule that applies to a dense collection of these families. In mathematical terms, the set of "good" equations is so large that if you were to pick a random point in the space of all possibilities, you would almost certainly land on one that works.

The paper also explored what happens when the prime number has a special relationship with the complexity of the curve. If the prime number leaves a remainder of one when divided by the degree of the curve, the result is even stronger. In this specific case, the researchers proved that every possible choice of numbers results in a curve that is ordinary. There are no exceptions. This means that for this specific type of prime, the property of being ordinary is universal across the entire family of curves, regardless of how the numbers are chosen. This finding provides a powerful, uniform guarantee that holds true for the entire space of possibilities, removing the need to check individual cases.

Finally, the study looked at the long-term distribution of these ordinary primes. For a fixed curve chosen from the "good" region, the researchers demonstrated that the ordinary primes are not just common; they are the rule. If you count the ordinary primes as you go higher and higher, they make up nearly 100% of all the primes you encounter. The few primes that do not behave this way are so few in number that they become negligible as you look at larger and larger ranges. Furthermore, if you extend the mathematical field in which the curve lives to include certain roots of unity, the density of ordinary primes becomes exactly one. This means that in this extended setting, the curve is ordinary for every single prime number that is large enough. The work confirms that the phenomenon of generic ordinarity is not just a possibility but a dominant feature of these hyperelliptic families, providing a clear and robust picture of how these mathematical shapes behave across the infinite landscape of prime numbers.

Technical Summary: Generically Ordinary One-Parameter Hyperelliptic Families Are Dense

Problem Statement
This paper investigates the distribution of "generically ordinary" reductions within the coefficient space of one-parameter hyperelliptic families. Specifically, it considers families of curves $C_{\vec{a}}$ defined over the $t$ -line by the affine equation:
$y^2 = x^d + a_{d-1}x^{d-1} + \dots + a_1x + t$
where $d = 2g + 1$ with $g \ge 2$ , and $\vec{a} = (a_1, \dots, a_{d-1})$ represents the coefficient vector in $\mathbb{Z}^{d-1}$ . The central question is: How common are coefficient vectors $\vec{a}$ such that the resulting family $C_{\vec{a}}$ has generically ordinary reduction at almost all rational primes?

This work extends a recent result by Zhu [Zhu26], which proved Katz's conjecture [Kat18] for the specific family $y^2 = x^d - dx + t$ . The current paper generalizes this to the full coefficient space $\mathbb{A}^{d-1}_{\mathbb{Z}}$ , analyzing the density of ordinary primes for arbitrary integral coefficient vectors.

Methodology
The primary tool is the universal Hasse–Witt determinant polynomial, denoted $H_p(\vec{A}, t) \in \mathbb{Z}[\vec{A}, t]$ . For a fixed prime $p$ , a family $C_{\vec{a}}$ is generically ordinary if and only if the specialization $H_p(\vec{a}, t)$ is not the zero polynomial in $\mathbb{F}_p[t]$ .

The core technical strategy involves analyzing the leading coefficient of $H_p(\vec{A}, t)$ with respect to the variable $t$ . The author:

Construct Coefficient Functions: Define functions $C_n(X)$ derived from the expansion of $(B(z)-1)^r$ , where $B(z)$ encodes the coefficients of the polynomial.
Define Determinant Polynomials: Introduce a matrix $D_{\vec{\delta}}$ built from these coefficient functions, where $\vec{\delta}$ is a sign vector determined by a residue class $k \in (\mathbb{Z}/d\mathbb{Z})^\times$ . The determinant of this matrix, $\Delta_k(\vec{A})$ , serves as a key invariant.
Identify Leading Coefficients: Prove that for primes $p \ge 2d-1$ with $p \equiv k \pmod d$ , the leading coefficient of $H_p(\vec{A}, t)$ (as a polynomial in $t$ ) is congruent to a non-zero multiple of $\Delta_k(\vec{A})$ modulo $p$ .
Zariski-Open Construction: Use the non-vanishing of these determinant polynomials to define Zariski-open dense subschemes $U_k \subset \mathbb{A}^{d-1}_{\mathbb{Z}}$ .

Key Results

Generic Density (Theorem 1.1 & Corollary 1.4): There exists a Zariski-open dense subscheme $U \subseteq \mathbb{A}^{d-1}_{\mathbb{Z}}$ such that for any integral coefficient vector $\vec{a} \in U(\mathbb{Q}) \cap \mathbb{Z}^{d-1}$ , the family $C_{\vec{a}}$ is generically ordinary for all primes $p$ greater than an explicit bound $B(d, \vec{a})$ . Consequently, for such generic $\vec{a}$ , the set of ordinary primes has natural density 1.
Uniform Locus over Finite Fields (Theorem 1.2): For any prime $p \ge 2d-1$ , the locus of generically ordinary families in the coefficient space over $\mathbb{F}_p$ is Zariski-open and dense. Specifically, it contains the subscheme $U_{k_p, \mathbb{F}_p}$ , where $k_p$ is the residue class of $p \pmod d$ .
Strong Result for $p \equiv 1 \pmod d$ (Theorem 1.3): For primes $p \equiv 1 \pmod d$ , the condition is even stronger: every coefficient vector $\vec{a} \in \mathbb{F}_p^{d-1}$ yields a generically ordinary family. In this case, the ordinary locus is the entire coefficient space $\mathbb{A}^{d-1}_{\mathbb{F}_p}$ .
Density for Arbitrary Vectors (Theorem 1.5): For an arbitrary integral coefficient vector $\vec{a}$ (not necessarily in the generic open set $U$ ), the set of ordinary primes of $C_{\vec{a}}$ over $\mathbb{Q}(\vec{a})(t)$ has a natural lower density of at least $1/[L:F]$ , where $F=\mathbb{Q}(\vec{a})$ and $L=F(\zeta_d)$ . After base change to $L(t)$ , the density becomes 1.

Significance and Claims
The paper claims to establish a uniform theory for the ordinarity of hyperelliptic families across the full coefficient space, rather than for isolated examples.

Uniformity: Unlike previous work which provided bounds depending on specific coefficients, this paper identifies a uniform linear threshold ( $p \ge 2d-1$ ) on the coefficient space itself (Theorem 1.2).
Generality: It resolves the question of how common generically ordinary families are, showing they form a dense open set in the moduli of coefficients.
Arithmetic Density: It provides a quantitative measure (natural density) for the ordinary primes of these families, linking the density to the degree of the cyclotomic extension generated by the coefficients.

The author notes that while the bound $B(d, \vec{a})$ for a fixed specialization depends on the specific vector $\vec{a}$ (due to the norms of the determinant polynomials), the existence of the open dense set $U$ ensures that "most" families behave well for sufficiently large primes. The work relies on the explicit construction of the Hasse–Witt determinant and its leading coefficient, bridging algebraic geometry and arithmetic statistics.

Technical Summary: Generically Ordinary One-Parameter Hyperelliptic Families Are Dense

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