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Maximal and minimal curves of the form $y^3=x^{(q^2+1)/2}+x$

This paper establishes that the curve $y^3=x^{(q^2+1)/2}+x$ over $\mathbb{F}_{q^6}$ (where $q=p^r$ and $p\equiv -1\pmod 3$ ) is maximal or minimal depending on the parity of $r$ , by deriving an exact point-count formula using birational Kummer models and Jacobi sums.

Original authors: Guilherme Dias, Saeed Tafazolian

Published 2026-08-24

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CC BY 4.0

Original authors: Guilherme Dias, Saeed Tafazolian

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 counting the solutions to equations when the numbers involved are not the familiar infinite set we use for everyday arithmetic, but rather a finite, self-contained universe of numbers. These finite fields act like closed loops where numbers eventually wrap around, creating a rigid structure that is surprisingly rich with hidden patterns. Within this world, mathematicians study geometric shapes defined by equations, known as curves. A central question in this field is to determine exactly how many points, or solutions, such a curve possesses within a specific finite field. There is a famous theoretical limit, a boundary set by nature itself, that dictates the maximum and minimum number of points a curve of a certain complexity can have. Curves that hit this maximum limit are called maximal, while those that hit the minimum are minimal. Finding these extreme curves is like discovering the most efficient or most constrained structures possible in this mathematical universe, and they have deep connections to cryptography and coding theory, where efficiency and error correction are paramount.

For decades, researchers have been cataloging these extreme curves, looking for families of equations that consistently produce them. While some patterns were known, there remained a puzzling, isolated example that did not fit the existing rules. A specific curve defined by a cubic equation involving a power of thirteen was known to be maximal over a particular large field, yet the standard mathematical tools used to explain why it worked failed because the numbers involved did not meet the usual criteria. It was a solitary case that stood alone, lacking a clear explanation or a connection to a broader family of similar shapes. This gap in understanding left mathematicians wondering if this was a unique fluke or the first glimpse of a much larger, hidden structure.

A team of researchers has now solved this mystery by proving that this isolated example is actually just the first member of an infinite family of curves that behave in a remarkably predictable way. They focused on a specific type of equation where a variable cubed equals a more complex expression involving another variable raised to a high power plus the variable itself. By carefully analyzing the structure of these equations over a specific type of finite field, they derived a precise formula that counts the exact number of solutions for any member of this family. Their work reveals that the behavior of these curves is governed by a simple switch: whether a specific parameter, denoted as 'r' (which defines the size of the field), is an odd or even number. If 'r' is odd, the curve reaches the theoretical maximum number of points, making it a maximal curve. If 'r' is even, it reaches the theoretical minimum, becoming a minimal curve. This discovery transforms a previously confusing outlier into a clear, systematic pattern.

The researchers achieved this by translating the original, complicated equation into a different, simpler form that is easier to analyze, a technique that allows them to see the underlying symmetry of the shape. They then used a sophisticated counting method involving special sums, which are like weighted tallies of how the numbers interact within the field. A crucial step in their proof involved showing that these complex sums could be reduced to simpler, well-understood quantities. They demonstrated that the sign of the final count, which determines whether the curve is maximal or minimal, is directly tied to the behavior of a much simpler, classic shape known as the Fermat cubic. This connection allowed them to determine the exact number of points without needing to count them one by one, a feat that would be impossible given the enormous size of the numbers involved.

The significance of this work extends beyond just solving a single puzzle. It provides a uniform explanation for a phenomenon that was previously thought to be an exception. The specific curve that puzzled mathematicians, which involves the number thirteen, is now understood as the first case of this new infinite family, occurring when the prime number five is used in a specific configuration where the resulting exponent happens to be thirteen. The researchers proved that this family works for every possible variation of the parameters, provided a specific condition about the prime numbers involved is met. They also clarified that this family exists outside the scope of previous classifications, meaning it represents a genuinely new territory in the landscape of maximal curves. While the genus, or the topological complexity, of these curves was known to be possible in other contexts, this paper provides the first explicit, infinite family of equations that generates them, complete with a formula that works for every case.

The paper also addresses what happens when the underlying conditions are changed. It explicitly notes that if the prime numbers involved do not meet a specific requirement regarding their remainder when divided by three, the elegant symmetry that allows for this precise counting breaks down. In those cases, the researchers could not force the same conclusion, leaving that scenario as an open question for future study. However, for the cases they did solve, the results are definitive. They have established a clear, proven rule that dictates the exact number of points on these curves, confirming that the transition between maximum and minimum efficiency is controlled entirely by the parity of the parameter 'r'. This work not only resolves a specific historical anomaly but also opens the door to a deeper understanding of how these finite geometric shapes can be constructed and manipulated, offering a new toolkit for mathematicians exploring the boundaries of what is possible in the arithmetic of finite fields.

Technical Summary: Maximal and Minimal Curves of the Form $y^3 = x^{(q^2+1)/2} + x$

Problem Statement
The paper addresses the classification of smooth projective curves over finite fields that achieve the Hasse–Weil bound, known as maximal or minimal curves. Specifically, it investigates the family of superelliptic curves defined by the equation:
$C: y^3 = x^{(q^2+1)/2} + x$
over the field $\mathbb{F}_{q^6}$ , where $q = p^r$ , $p \geq 5$ is a prime satisfying $p \equiv -1 \pmod 3$ , and $r \geq 1$ .

While previous work (Tafazolian and Torres) provided maximality criteria for curves of the form $y^n = x^m + x$ over $\mathbb{F}_{p^2}$ when the field size's square root is prime, this condition fails for specific known examples. Notably, the curve $y^3 = x^{13} + x$ is maximal over $\mathbb{F}_{5^6}$ , yet $5^3 = 125$ is not prime. The authors aim to provide a uniform explanation for such phenomena, determining the exact number of rational points for the general family and characterizing its behavior based on the parity of the exponent $r$ .

Methodology
The proof strategy relies on transforming the curve into a birational Kummer model and utilizing character sum techniques, specifically Jacobi and Gauss sums.

Birational Transformation: The authors define $d = (q^2-1)/2$ and $m = (q^2+1)/2$ . They establish a birational equivalence between the original curve $C$ and the model:
$Y: v^{3d} = t(1-t)^d$
This transformation allows the problem to be analyzed through the lens of Kummer extensions.
Point Counting via Character Sums: The number of rational points on the model $Y$ is expressed as a sum involving Jacobi sums $J(\chi^j, \chi^{dj})$ , where $\chi$ is a multiplicative character of exact order $3d$ .
- The authors analyze the ramification of the extension to determine the contribution of points at $t=0, 1, \infty$ .
- They utilize the congruence $q^4 \equiv d+1 \pmod{3d}$ and the Frobenius invariance of Gauss sums to reduce the relevant Jacobi sums to cubic Gauss sums. Specifically, they show that $J(\chi^j, \chi^{dj}) = g(\chi^{dj})$ .
Sign Determination: The sign of the resulting Gauss sums is determined by analyzing the behavior of the Fermat cubic curve $U^3 + V^3 = W^3$ over $\mathbb{F}_{q^6}$ . By applying known criteria for the maximality/minimality of Fermat curves, the authors deduce the sign of the cubic Gauss sum $g(\lambda)$ (where $\lambda$ is a character of order 3) as $(-1)^{r+1}q^3$ .

Key Results
The paper establishes the following main theorem:

Genus: The genus of the curve $C$ is exactly $g = (q^2 - 1)/2$ .
Point Count: The exact number of $\mathbb{F}_{q^6}$ -rational points is given by:
$\#C(\mathbb{F}_{q^6}) = q^6 + 1 + (-1)^{r+1}(q^2 - 1)q^3$
Maximality/Minimality:
- If $r$ is odd, the term $(-1)^{r+1}$ is positive, and the curve attains the upper Hasse–Weil bound, making it maximal.
- If $r$ is even, the term is negative, and the curve attains the lower bound, making it minimal.

A specific corollary confirms that the curve $y^3 = x^{13} + x$ (corresponding to $p=5, r=1$ ) has genus 12 and is maximal over $\mathbb{F}_{5^6}$ , with exactly 18,626 rational points.

Significance and Claims
The authors position their contribution as providing a uniform, explicit explanation for previously isolated examples of maximal curves.

Explicit Family: The paper presents an infinite family of superelliptic curves with an exact point-count formula valid for all $r \geq 1$ .
Parity-Dependent Transition: The work highlights a distinct parity-dependent transition between maximality and minimality controlled solely by the exponent $r$ in the field definition $q=p^r$ .
Contextualization: The authors note that while the genus value $(q^2-1)/2$ is not new (it appears in the spectrum of quotients of Hermitian curves), the novelty lies in the explicit construction of this specific superelliptic family and the proof of its point count.
Limitations: The results are strictly contingent on the condition $p \equiv -1 \pmod 3$ . The authors explicitly state that the case $p \equiv 1 \pmod 3$ is not addressed, as the Frobenius invariance argument used to equate the Gauss sums fails in that scenario.
Open Question: The paper concludes by noting that while the genus matches that of certain Hermitian quotients, it remains an open question whether the curve $C$ is covered by the Hermitian curve $H_{q^3}$ for odd $r$ .

Technical Summary: Maximal and Minimal Curves of the Form y3=x(q2+1)/2+xy^3 = x^{(q^2+1)/2} + xy3=x(q2+1)/2+x

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Technical Summary: Maximal and Minimal Curves of the Form $y^3 = x^{(q^2+1)/2} + x$