On the Maximality, Weierstrass Semigroups, and Automorphism Group of the Curve $Y^{q+1} = X^n(X^n + 1)$

Technical Summary: Maximality, Weierstrass Semigroups, and Automorphism Group of the Curve $y^{q+1} = x^n(x^n + 1)$

Problem Statement
This paper investigates the family of algebraic curves $X_{n,q}$ defined over the finite field $\mathbb{F}_{q^2}$ by the affine equation:
$$y^{q+1} = x^n(x^n + 1)$$
where $n$ is a positive integer coprime to the characteristic $p$. The primary objectives are to determine precise conditions under which the nonsingular model of this curve is $\mathbb{F}_{q^2}$-maximal, to compute the Weierstrass semigroups at various classes of rational points, and to determine the full automorphism group of the curve. The study aims to extend previous results on specific cases (notably $m=3$) to a broader family parameterized by divisors of $q+1$.

Methodology
The authors employ a combination of algebraic geometry and arithmetic techniques:

1. Maximality Analysis: The authors utilize the property that maximality is preserved under non-constant rational morphisms. They construct a covering map from the classical Hermitian curve $H: u^{q+1} + v^{q+1} + 1 = 0$ to $X_{n,q}$ and analyze the quotient structure. They also relate the curve to hyperelliptic curves of the form $y^2 = x^n + 1$ to derive necessary and sufficient conditions for maximality.
2. Divisor and Automorphism Theory: The paper utilizes Kummer extension theory to analyze the ramification structure of the projection $X_{n,q} \to \mathbb{P}^1$. Explicit divisor formulas for functions $x$ and $y$ are derived to determine the genus and the structure of the automorphism group.
3. Weierstrass Semigroups: The authors determine the Weierstrass semigroups $H(P)$ at rational points by constructing explicit rational functions with prescribed pole orders. This involves:
   • Analyzing local parameters at branch points (places over $0, \infty$, and roots of $x^{(q+1)/m} + 1 = 0$).
   • Using expansions in terms of local parameters to identify gaps and non-gaps in the semigroup.
   • Distinguishing between points in the set $O$ (branch points) and points outside $O$ by analyzing the rank of specific matrices formed by coefficients of monomial expansions.
4. Automorphism Group Determination: The authors identify the automorphism group by analyzing the orbits of rational points. They leverage the fact that points in the same orbit must share the same Weierstrass semigroup. By proving that the semigroups at points in $O$ differ from those outside $O$ (under mild characteristic assumptions), they restrict the possible automorphisms to a specific subgroup generated by diagonal maps and specific involutions.

Key Contributions and Results

• Characterization of Maximality:
  The paper proves that the nonsingular model of $X_{n,q}$ is $\mathbb{F}_{q^2}$-maximal if and only if $n$ divides $q+1$. Writing $q+1 = nm$, the authors define a family of maximal curves $X_{m,q}$ parameterized by the divisors $m$ of $q+1$. This generalizes the known case $m=3$.

• Weierstrass Semigroups:
  The authors provide explicit descriptions of the Weierstrass semigroups at several classes of rational points:

  • Points in $O$ (Branch Points): The semigroups at places over $0$, $\infty$, and the roots of $x^{(q+1)/m} + 1 = 0$ are determined explicitly in terms of $q$ and $m$. The structure depends on the parity of $m$. For instance, at places over $\infty$, the semigroup is generated by $\{q+1-m, q+1-\lfloor m/2 \rfloor, \dots, q+1\}$ (for even $m$) or $\{q+1-m, q+1-m+2, \dots, q, q+1\}$ (for odd $m$).
  • Points Outside $O$: The paper demonstrates that for most rational points $P(a,b)$ outside the set $O$, the Weierstrass semigroup differs from those in $O$. Specifically, for the case $m=4$, the authors derive generators involving $q, q+1, q-1$ and specific arithmetic progressions depending on a parameter $\alpha$.
  • Orbit Distinction: A significant result is that the set $O$ forms a single orbit under the automorphism group if and only if $m = (q+1)/2$ and $m$ is even (implying $q \equiv 3 \pmod 4$). In other cases, the set $O$ splits into distinct orbits based on the Weierstrass semigroups.

• Automorphism Group:
  Under the mild assumption that the characteristic $p$ does not divide $\binom{m/2 + (m \pmod 2)}{1}$ (a condition satisfied for most $m$), the full automorphism group $\text{Aut}(X_{m,q})$ is determined:

  • General Case ($m \neq (q+1)/2$ or $m$ odd): The group is isomorphic to a semidirect product of order $2(q+1)^2/m$, generated by diagonal automorphisms and an involution $\theta_2(x,y) = (y^m/x, y)$.
  • Special Case ($m = (q+1)/2$ and $m$ even): The group is larger, of order $8(q+1)$, generated by the previous group and an additional automorphism $\theta_4$ involving fourth roots of unity, provided $q \equiv 3 \pmod 4$.

Significance
The paper claims to provide a complete classification of the maximal curves of the form $y^{q+1} = x^n(x^n + 1)$. By extending the study from the specific case $m=3$ to the general family parameterized by divisors of $q+1$, the work offers new explicit examples of maximal curves with varying genera.

The explicit determination of Weierstrass semigroups is highlighted as having direct implications for geometric coding theory. Specifically, the precise knowledge of these semigroups and the number of rational points is essential for constructing Algebraic Geometry (AG) codes (such as Goppa codes) with optimal parameters, including maximized minimum distance and high dimension. The results provide the necessary theoretical and computational tools for these applications.

Furthermore, the paper contributes to the understanding of the structure of maximal curves by clarifying the relationship between the Hermitian curve and its quotients, and by characterizing the automorphism groups of these curves, which is a fundamental problem in the arithmetic geometry of curves over finite fields.