On maximal curves having classical Weierstrass gaps

The Weierstrass subgroup of a curve has maximal rank

We show that the Weierstrass points of the generic curve of genus $g$ over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian.

关于函数域$\mathbb{F}_q(x)$的任意Kummer扩张中的广义Weierstrass半群

On generalized Weierstrass Semigroups in arbitrary Kummer extensions of $\mathbb{F}_q(x)$

本文研究函数域$\mathbb{F}_q(x)$任意Kummer扩张中广义Weierstrass半群的结构、性质及极大元，给出极大元集合的明确描述并应用于特定极大曲线，推广统一了早期成果，为半群与函数域关系提供新视角。

关于最大费马函数域的魏尔斯特拉斯半群

On Weierstrass semigroups of maximal Fermat function fields

本文研究了最大费马函数域在不同位置处的魏尔斯特拉斯半群，揭示了其复杂性和丰富的结构特性。

线性化函数域中的广义Weierstrass半群

On generalized Weierstrass semigroups in linearized function fields

利用差异概念，研究线性化函数域中由n个一次全分歧点构成的n元组Q的广义Weierstrass半群，刻画并显式确定绝对最大元素集和相对最大元素集，推广了现有结果。

具有第三大亏格的极大曲线的Weierstrass半群与自同构群

Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus

本文确定具有第三大亏格的极大曲线$\mathcal{X}_3$在所有点处的Weierstrass半群与完整自同构群，揭示其Weierstrass点异常丰富，并证明除小$q$外其自同构群完全继承自Hermitian曲线，为唯一性公开问题提供关键线索。

Weierstrass semigroup at $m+1$ rational points in maximal curves which cannot be covered by the Hermitian curve

We determine the Weierstrass semigroup $H(P_\infty,P_1,\ldots,P_m)$ at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced by Tafazolian, Teherán-Herrera, and Torres. Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain AG codes with better relative parameters than comparable one-point AG codes arising from these curves.

The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines

For any smooth Hurwitz curve $\mathcal{H}_n: \, XY^n+YZ^n+X^nZ=0$ over the finite field $\mathbb{F}_{p}$, an explict description of its Weierstrass points for the morphism of lines is presented. As a consequence, the full automorphism group ${\rm Aut}(\mathcal{H}_n)$, as well as the genera of all Galois subcovers of $\mathcal{H}_n$, with $n\neq 3, p^r$, are computed. Finally, a question by F. Torres on plane non nonsingular maximal curves is answered.

第三大可能亏格的极大函数域在 $q \equiv 1 \pmod 3$ 时的 Weierstrass 半群与自同构群

Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 1 \pmod 3$

针对 q≡1 mod 3 时具有第三大亏格的极大函数域 Y_3，本文显式确定其所有位的 Weierstrass 半群与完整自同构群，发现其 Weierstrass 位结构异常丰富，但自同构群除小 q 外均继承自 Hermitian 函数域。

AG codes from the second generalization of the GK maximal curve

The second generalized GK maximal curves $\mathcal{GK}_{2,n}$ are maximal curves over finite fields with $q^{2n}$ elements, where $q$ is a prime power and $n \geq 3$ an odd integer, constructed by Beelen and Montanucci. In this paper we determine the structure of the Weierstrass semigroup $H(P)$ where $P$ is an arbitrary $\mathbb{F}_{q^2}$-rational point of $\mathcal{GK}_{2,n}$. We show that these points are Weierstrass points and the Frobenius dimension of $\mathcal{GK}_{2,n}$ is computed. A new proof of the fact that the first and the second generalized GK curves are not isomorphic for any $…

Automorphisms of Curves and Weierstrass semigroups for Harbater-Katz-Gabber covers

We study $p$-group Galois covers $X \rightarrow \mathbb{P}^1$ with only one fully ramified point. These covers are important because of the Katz-Gabber compactification of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero $p$--rank: we focus on maximal curves and curves that admit a big action. Moreover the Galois module structure of polydifferentials is…

Weierstrass Gap Sequence at Total Inflection Points of Nodal Plane Curves

Let $Γ$ be a plane curve of degree $d$ with $δ$ ordinary nodes and no other singularities. If $P$ is a smooth point on $Γ$ then the Weierstrass gap sequence at $P$ is considered as that at the corresponding point on the normalization of $Γ$. A smooth point $P\inΓ$ is called a total inflection point if $i(Γ,T;P)=d$ where $T$ is the tangent line to $Γ$ at $P$. There are many possible Weierstrass gap sequences at total inflection points. Our main results are: Among them (1) There exists a pair $(P,Γ)$ such that the gap sequence at $P$ is the minimal (in the sense of weight). (2) There exists a pa…

具有第三大可能亏格的极大函数域的 Weierstrass 半群与自同构群，$q \equiv 0 \pmod 3$

Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 0 \pmod 3$

本文针对 $q\equiv0\pmod3$，显式确定第三大亏格的 $\mathbb{F}_{q^2}$-极大函数域 $Z_3$ 的各位 Weierstrass 半群与完整自同构群。

曲线 $Y^{q+1} = X^n(X^n + 1)$ 的最大性、Weierstrass 半群和自同构群

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

研究有限域 $\mathbb{F}_{q^2}$ 上曲线 $y^{q+1} = x^n(x^n+1)$，证明当 $q$ 为奇数时其非奇异模型是 $\mathbb{F}_{q^2}$-最大的当且仅当 $n \mid (q+1)$，并确定了该族最大曲线的 Weierstrass 半群和自同构群。

On maximal curves in characteristic two

The genus g of an F_{q^2}-maximal curve satisfies g=g_1:=q(q-1)/2 or g\le g_2:= [(q-1)^2/4]. Previously, such curves with g=g_1 or g=g_2, q odd, have been characterized up to isomorphism. Here it is shown that an F_{q^2}-maximal curve with genus g_2, q even, is F_{q^2}-isomorphic to the nonsingular model of the plane curve \sum_{i=1}^{t}y^{q/2^i}=x^{q+1}, q=2^t, provided that q/2 is a Weierstrass non-gap at some point of the curve.

Two-point AG codes on the GK maximal curves

We determine de Weierstrass semigroup of a pair of certain rational points on the GK-curves. We use this semigroup to obtain two-point AG codes with better parameters than comparable one-point AG codes arising from these curves. These parameters are new records in the MinT's tables.

Abdón-Bezerra-Quoos型Serre极大曲线的a数

The a-number of the Serre-type maximal curves of Abdón-Bezerra-Quoos

该研究确定了由Abdón等人引入的Serre型极大曲线X(q,n)的a数，利用Weierstrass半群与Cartier算子将a数计算转化为格点计数并得到闭式结果，是对Serre曲线的推广研究。

Two-point AG codes from the Beelen-Montanucci maximal curve

In this paper we investigate two-point algebraic-geometry codes (AG codes) coming from the Beelen-Montanucci (BM) maximal curve. We study properties of certain two-point Weierstrass semigroups of the curve and use them for determining a lower bound on the minimum distance of such codes. AG codes with better parameters with respect to comparable two-point codes from the Garcia-Güneri-Stichtenoth (GGS) curve are discovered.

来自一条Skabelund极大曲线的两点代数几何码

Two-point AG codes from one of the Skabelund maximal curves

本文研究由Suzuki曲线的循环覆盖得到的Skabelund极大曲线上的两点代数几何码，利用广义阶界和两点Weierstrass半群估计其最小距离。

Skabelund曲线的最大子覆盖：通过亏格和自同构群的唯一性

Maximal Subcovers of the Skabelund Curve: Uniqueness via Genus and Automorphism Groups

通过亏格和自同构群的组合，Skabelund曲线的中间覆盖在同构意义下被唯一确定。

关于无穷远点处仅有一个点的 Kummer 扩展

On Kummer extensions with one place at infinity

本文显式描述了 Kummer 扩展曲线在唯一无穷远点处的 Weierstrass 半群，确定了其 Frobenius 数与重数，讨论了对称性条件，并刻画了极大 Castle 曲线。

On the genus of a maximal curve

Previous results on genera g of F_{q^2}-maximal curves are improved: (1) Either g\leq (q^2-q+4)/6, or g=\lfloor(q-1)^2/4\rfloor, or g=q(q-1)/2; (2) The hypothesis on the existence of a particular Weierstrass point in \cite{at} is proved; (3) For q\equiv 1\pmod{3}, q\ge 13, no F_{q^2}-maximal curve of genus (q-1)(q-2)/3 exists; (4) For q\equiv 2\pmod{3}, q\ge 11, the non-singular F_{q^2}-model of the plane curve of equation y^q+y=x^{(q+1)/3} is the unique F_{q^2}-maximal curve of genus g=(q-1)(q-2)/6; (5) Assume \dim(\cD_\cX)=5, and char(\fq)\geq 5. For q\equiv 1\pmod{4}, q\geq 17, the Fermat c…

关于亏格为 $\mathfrak{g}=\frac{1}{8}(q-1)^2$ 的 Hermitian 曲线的 Galois 覆盖

On a Galois cover of the Hermitian curve of genus $\mathfrak{g}=\frac{1}{8}(q-1)^2$

本文研究亏格为 $\frac{1}{8}(q-1)^2$、$q\equiv1\pmod4$ 的 Hermitian 曲线的 4 阶循环 Galois 覆盖，分析其 Frobenius 嵌入、Weierstrass 半群与自同构群。

On maximal curves

We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F_{q^2} whose number of F_{q^2}-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a rational point, we prove that maximal curves are F_{q^2}-isomorphic to y^q + y = x^m, for some $m \in Z^+$. As a consequence we show that a maximal curve of genus g=(q-1)^2/4 is F_{q^2}-isomorphic to the curve y^q + y = x^{(q+1)/2}.

On curves over finite fields with many rational points

We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of $\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are $\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.

Explicit two-cover descent for genus 2 curves

Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a number field, we construct a family of genus $5$ curves that realize descent by maximal unramified abelian two-covers of $C$, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus $2$ quintic curves over $\mathbb{Q}$ of Mordell-Weil rank $2$ or $3$ whos…