Nazar Arakelian, Saeed Tafazolian, Fernando Torres
Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper ,we discuss the spectrum $\mathbf{M}(q^2)$ for the genera of maximal curves over finite fields of order $q^2$ with $7\leq q\leq 16$. In particular, by using a result in Kudo and Harashita(2016) paper, the set $\mathbf{M}(7^2)$ is completely determined.
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We determine the zeta functions of trinomial curves in terms of Gauss sums and Jacobi sums, and we obtain an explicit formula of the genus of a trinomial curve over a finite field, then we study the conditions for a trinomial curve to be a maximal curve over a finite field.
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.
In 2017 Skabelund constructed two new examples of maximal curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ as covers of the Suzuki and Ree curves, respectively. The resulting Skabelund curves are analogous to the Giulietti-Korchmáros cover of the Hermitian curve. In this paper a complete characterization of all Galois subcovers of the Skabelund curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ is given. Calculating the genera of the corresponding curves, we find new additions to the list of known genera of maximal curves over finite fields.
We show that the generalized Giulietti-Korchmáros curve is not a Galois subcover of the Hermitian curve over $\mathbb{F}_{q^{2n}}$. This answers a question raised by Garcia, Güneri and Stichtenoth.
In this note, we characterize certain maximal superspecial curves of genus $5$ over finite fields. Specifically, we prove that the desingularization $T_p$ of $x y z^3 + x^5 + y^5 = 0$ is a maximal superspecial trigonal curve of genus $5$ if and only if $p \equiv -1 \pmod{15}$ or $p \equiv 11 \pmod{15}$. Moreover, we give families of maximal curves of more general type, which include $T_p$.
Torsion of elliptic curves with rational $j$-invariant over the maximal elementary abelian 2-extension of $\mathbb{Q}$
本文针对定义在有理数域所有二次扩张合成域上、j不变量为非0或1728的有理数的椭圆曲线，完成了对其可能挠子群结构的分类。
For every $q=l^3$ with $l$ a prime power greater than 2, the GK curve $X$ is an $F_{q^2}$-maximal curve that is not $F_{q^2}$-covered by any $F_{q^2}$-maximal Deligne-Lusztig curve. Interestingly, $X$ has a very large $F_{q^2}$-automorphism group with respect to its genus. In this paper we compute the genera of a large variety of curves that are Galois-covered by the GK curve, thus providing several new values in the spectrum of genera of $F_{q^2}$-maximal curves.
Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an Artin-Schreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous to the product of these three elliptic curves. In this paper we exhibit explicit models for genus 3 curves with many involutions, and we compute explicit equations for the elliptic quotients. We then character…
In this paper we solve three open problems on maximal curves with Frobenius dimension 3. In particular, we prove the existence of a maximal curve with order sequence (0,1,3,q).
For every $q=n^3$ with $n$ a prime power greater than $2$, the GK-curve is an $\mathbb F_{q^2}$-maximal curve that is not $\mathbb F_{q^2}$-covered by the Hermitian curve. In this paper some Galois subcovers of the GK curve are investigated. We describe explicit equations for some families of quotients of the GK-curve. New values in the spectrum of genera of $\mathbb F_{q^2}$-maximal curves are obtained. Finally, infinitely many further examples of maximal curves that cannot be Galois covered by the Hermitian curve are provided.
We discuss sufficient conditions for a given curve to be covered by a maximal curve with the covering being unramified; it turns out that the given curve itself will be also maximal. We relate our main result to the question of whether or not a maximal curve is covered by the Hermitian curve. We also provide examples illustrating the results.
In this work we study the properties of maximal and minimal curves of genus 3 over finite fields with discriminant -19. We prove that any such curve can be given by an explicit equation of certain form. Using these equations we obtain a table of maximal and minimal curves over finite fields with discriminant -19 of cardinality up to 997. We also show that existence of a maximal curve implies that there is no minimal curve and vice versa.
We study geometrical properties of maximal curves having classical Weierstrass gaps.
A new family of maximal curves over a finite field is presented and some of their properties are investigated.
We construct new covers of the Suzuki and Ree curves which are maximal with respect to the Hasse-Weil bound over suitable finite fields. These covers are analogues of the Giulietti-Korchmáros curve, which covers the Hermitian curve and is maximal over a base field extension. We show that the maximality of these curves implies that of certain ray class field extensions of each of the Deligne-Lusztig curves. Moreover, we show that the Giulietti-Korchmáros curve is equal to the above-mentioned ray class field extension of the Hermitian curve.
Maximal curves of genus 5 over finite fields
研究有限域上亏格5且判别式为-19的极大曲线的存在性，利用自同构群证明当q≡2,3,4 mod 5时不存在，并给出q≡1 mod 5时的模型及q≡0 mod 5时仅可能q=5^7。
For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all $q$ large enough, the existence of a curve of genus $g$ over ${\mathbb F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally,…
In this article we construct for any prime power $q$ and odd $n \ge 5$, a new $\mathbb{F}_{q^{2n}}$-maximal curve $\mathcal X_n$. Like the Garcia--G\" uneri--Stichtenoth maximal curves, our curves generalize the Giulietti--Korchmáros maximal curve, though in a different way. We compute the full automorphism group of $\mathcal X_n$, yielding that it has precisely $q(q^2-1)(q^n+1)$ automorphisms. Further, we show that unless $q=2$, the curve $\mathcal{X}_n$ is not a Galois subcover of the Hermitian curve. Finally, we find new values of the genus spectrum of $\mathbb{F}_{q^{2n}}$-maximal curves, …
In this paper, we compute the fundamental group of the complement of linear torus curves of maximal contact and we show that it is isomorphic that of generic linear torus curves.
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。
Some linear codes associated to maximal algebraic curves via Feng-Rao construction are investigated. In several case, these codes have better minimum distance with respect to the previously known linear codes with same length and dimension.
In this paper we compute the automorphism group of the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$ introduced in Tafazolian et al. in 2016 as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.
In this note, we give an overview of a new technique for studying Brill--Noether curves in projective space via degeneration. In particular, we give a roadmap to the proof of the Maximal Rank Conjecture.
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism maximal curves known when r \equiv 3 (mod 4). Here, a maximal curve with genus g_2 and a non-singular plane model is characterized as a Fermat curve of degree (r+1)/2.
In this work we present explicit examples of maximal and minimal curves over finite fields in odd characteristic. The curves are of Artin-Schreier type and the construction is closely related to quadratic forms from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$.
The maximal number of automorphisms of a Mumford curve over a non-archimedean valued field of positive characteristic is known. In this note, the unique family of curves that attains this bound in genus not equal to 5,6,7 or 8 is explicitly determined, as is its automorphism group.
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。
We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus $3$ in c…
The purpose of this paper is to extend the symmetry of maximals of the ring of a germ of reducible plane curve proved by Delgado to a relation between the relative maximals of a fractional ideal and the absolute maximals of its dual for any admissible ring. In particular, it includes the case of germs of reduced reducible curve of any codimension. We then apply this symmetry to characterize the elements in the set of values of a fractional ideal from some of its projections and the irreducible absolute maximals of the dual ideal.
The classification of maximal plane curves of degree $3$ over $\mathbb{F}_4$ will be given, which complements Hirschfeld-Storme-Thas-Voloch's theorem on a characterization of Hermitian curves in $\mathbb{P}^2$. This complementary part should be understood as the classification of Sziklai's example of maximal plane curves of degree $q-1$ over $\mathbb{F}_q$. Although two maximal plane curves of degree $3$ over $\mathbb{F}_4$ up to projective equivalence over $\mathbb{F}_4$ appear, they are birationally equivalent over $\mathbb{F}_4$ each other.
CM theory, maximal hyperelliptic curves, and Chebyshev polynomials
本文研究由 Chebyshev 多项式定义的超椭圆曲线，结合 Frobenius 斜率、复乘 Abel 子簇与 Jacobian 的 $2$-下降技术，给出并排除若干 $\FF_{q^2}$ 上的极大性情形。
Maximal singular curves over finite fields from elliptic and hyperelliptic curves
本文构造有限域上达到Aubry-Perret界的极大奇异曲线族，利用椭圆/超椭圆曲线的线性投影和Gorenstein曲线理论，并确定所得曲线的gonality。
We study families of n-gonal curves with maximal variation of moduli, which have a rational section. Certain numerical results on the degree of the modular map are obtained for such families of hyperelliptic and trigonal curves. In the last case we use the description of the relative Picard group of the universal family of trigonal curves.
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.
In 2014 A. Degtyarev, I. Itenberg and the author gave a description, up to fiberwise equivariant deformations, of maximally inflected real trigonal curves of type~I (over a base $ B $ of an arbitrary genus) in terms of the combinatorics of sufficiently simple graphs and for $ B=\mathbb{P}^1 $ obtained a complete classification of such curves. In this paper, the mentioned results are extended to maximally inflected real trigonal curves of type~II over $ B=\mathbb{P}^1 $. As usual, the results for such curves are extended to real Jacobian elliptic surfaces with all singular fibers real.
In this paper we determine the generalized Weierstrass semigroup $ \widehat{H}(P_{\infty}, P_1, \ldots , P_{m})$, and consequently the Weierstrass semigroup $H(P_{\infty}, P_1, \ldots , P_{m})$, at $m+1$ points on the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$. These curves has been introduced in Tafazolian et al. as new examples of curves which cannot be covered by the Hermitian curve.
Maximal curves with respect to quadratic extensions over finite fields
本文详细研究了一个规范界，该界关联了曲线在有限域及其二次扩张上的有理点个数，通过方差连接给出了最优改进，并刻画了达到该界的Hallouin-Perret-极大曲线及其与Ihara界曲线的联系，得到了低亏格曲线列表及涉及Suzuki曲线的极大性结果，最后确定了亏格2的Hallouin-Perret-极大曲线雅可比对应的多项式。
Some new results on plane F_{q^2}-maximal curves are stated and proved. It is known that the degree d of such curves is upper bounded by q+1 and that d=q+1 if and only if the curve is F_{q^2}-isomorphic to the Hermitian. We show that d\le q+1 can be improved to d\le (q+2)/2 apart from the case d=q+1 or q\le 5. This upper bound turns out to be sharp for q odd. We also study the maximality of Hurwitz curves of degree n+1. We show that they are F_{q^2}-maximal if and only if (q+1) divides (n^2-n+1). Such a criterion is extended to a wider family of curves.
We prove the following statement, which has been conjectured since 1985: There exists a constant $K$ such that for all natural numbers $d,g$ with $g\le Kd^{3/2}$ there exists an irreducible component of the Hilbert scheme of $\mathbb{P}^3$ whose general element is a smooth, connected curve of degree $d$ and genus $g$ of maximal rank.
Maximality of moduli spaces of vector bundles on curves
研究证明非奇异实射影曲线上的半稳定向量丛模空间的最大性取决于基曲线的最大性，并引入了更强的希格尔-表达性性质。
Let $\mathbb{F}$ be the finite field of order $q^2$, $q=p^h$ with $p$ prime. It is commonly atribute to J.P. Serre the fact that any curve $\mathbb{F}$-covered by the Hermitian curve $\mathcal{H}_{q+1}:\, y^{q+1}=x^q+x$ is also $\mathbb{F}$-maximal. Nevertheless, the converse is not true as the Giulietti-Korchmáros example shows provided that $q>8$ and $h\equiv 0\pmod{3}$. In this paper, we show that if an $\mathbb{F}$-maximal curve $\mathcal{X}$ of genus $g\geq 2$ where $q=p$ is such that $|Aut(\mathcal{X})|>84(g-1)$ then $\mathcal{X}$ is Galois-covered by $\mathcal{H}_{p+1}$. Also, we show t…
Maximal Mumford Curves from Planar Graphs
本文研究极大Mumford曲线的构造，通过在典范曲线的Hilbert概型中利用平面图曲线的一阶形变，构建了具有全维数的MM曲线族。
Let $\mathbb{F}_{q^2}$ be the finite field with $q^2$ elements. Most of the known $\mathbb{F}_{q^2}$-maximal curves arise as quotient curves of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_{q}$. After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of $\mathcal{H}_q$, but their complete determination is a challenging open problem. In this paper we determine completely the spectrum of genera of quotients of $\mathcal{H}_q$ for any $q\equiv1\pmod4$.
We investigate the Jacobian decomposition of some algebraic curves over finite fields with genus $4$, $5$ and $10$. As a corollary, explicit equations for curves that are either maximal or minimal over the finite field with $p^2$ elements are obtained for infinitely many $p$'s. Lists of small $p$'s for which maximality holds are provided. In some cases we describe the automorphism group of the curve.
Maximality and configuration spaces on real algebraic curves
本文证明实代数曲线C的n点无标号构型空间为极大（Smith-Thom意义）当且仅当C极大或亏格为零且n为偶数。
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…
The ordinary algebraic curves of maximal rank are also the arithmetically Cohen-Maccaulay curves of minimal rank. We give sufficient conditions for such curves to exist as well as examples, generalizing results of [GHL] in the dimension three.
Fix positive integers d;m such that $(m^2+4m+6)/6 \leq d < (m^2+4m+6)/3$ (the so-called Range A for space curves). Let G(d;m) be the maximal genus of a smooth and connected curve, of degree d, $C \subset P^3$ such that $h^0(I_C(m-1)) = 0$. Here we prove that $G(d,m) = 1+(m-1)d -\binom{m+2}{3}$ if $m\ge 13.8\cdot 10^5$.
We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs $(q,d)$ such that the hyperelliptic curve $\cC$ over a finite field $\FF_{q^2}$ corresponding to the equation $y^2 = φ_{d}(x)$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. Here $φ_{d}(x)$ denotes the Chebyshev polynomial of degree $d$. The same question is studied for the curves corresponding to $y^2=(x\pm 2) φ_{d}(x)$, and also for $y^2=(x^2-4)φ_d(x)$.
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}.
Let $C$ be a smooth projective curve and $W$ a symplectic bundle over $C$. Let $LQ_e (W)$ be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves $E \subset W$ of degree $e$. We give a closed formula for intersection numbers on $LQ_e (W)$. As a special case, for $g \ge 2$, we compute the number of Lagrangian subbundles of maximal degree of a general stable symplectic bundle, when this is finite. This is a symplectic analogue of Holla's enumeration of maximal subbundles in [13].
We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a prime degree Galois covering. We also compute the genus of maximal curves obtained by the quotient of \mathcal H by several automorphisms groups. Finally we discuss the value for the third largest genus that a maximal curve can have.
A curve attaining the Hasse-Weil bound is called a maximal curve. Usually classical error-correcting codes obtained from a maximal curve have good parameters. However, the quantum stabilizer codes obtained from such classical error-correcting codes via Euclidean or Hermitian self-orthogonality do not always possess good parameters. In this paper, the Hermitian self-orthogonality of algebraic geometry codes obtained from two maximal curves is investigated. It turns out that the stabilizer quantum codes produced from such Hermitian self-orthogonal classical codes have good parameters.
Let be a general curve of genus g embedded via a general linear series of degree d in P^r. The well-known Maximal Rank Conjecture asserts that the restriction maps H^0(O_{P^r}(m)) \to H^0(O_C(m) are of maximal rank; if known, this conjecture would determine the Hilbert function of C. In this paper, we prove an analogous statement for the hyperplane sections of unions general curves. More specifically, if H is a general hyperplane, we show that H^0(O_H(m)) \to H^0(O_{(C_1 \cup C_2 \cup \cdots \cup C_n) \cap H}(m)) is of maximal rank, except for some counterexamples when m = 2. As explained in a…
We prove that if $X$ is a complex projective K3 surface and $g>0$, then there exist infinitely many families of curves of geometric genus $g$ on $X$ with maximal, i.e., $g$-dimensional, variation in moduli. In particular every K3 surface contains a curve of geometric genus 1 which moves in a non-isotrivial family. This implies a conjecture of Huybrechts on constant cycle curves and gives an algebro-geometric proof of a theorem of Kobayashi that a K3 surface has no global symmetric differential forms.
We prove that curves in a non-primitive, base point free, ample linear system on a K3 surface have maximal variation. The result is deduced from general restriction theorems applied to the tangent bundle. We also show how to use specialisation to spectral curves to deduce information about the variation of curves contained in a K3 surface more directly. The situation for primitive linear systems is not clear at the moment. However, the maximal variation holds in genus two and can, in many cases, be deduced from a recent result of van Geemen and Voisin confirming a conjecture due to Matsushita.
A Mumford curve of genus g=5,6,7 or 8 over a non-archimedean field of characteristic p (such that if p=0, the residue field characteristic exceeds 5) has at most 12(g-1) automorphisms. In this paper, all curves that attain this bound and their automorphism groups (called of Lame type) are explicitly determined.
For each prime power $\ell$ the plane curve $\mathcal X_\ell$ with equation $Y^{\ell^2-\ell+1}=X^{\ell^2}-X$ is maximal over $\mathbb{F}_{\ell^6}$. Garcia and Stichtenoth in 2006 proved that $\mathcal X_3$ is not Galois covered by the Hermitian curve and raised the same question for $\mathcal X_\ell$ with $\ell>3$; in this paper we show that $\mathcal X_\ell$ is not Galois covered by the Hermitian curve for any $\ell>3$. Analogously, Duursma and Mak proved that the generalized GK curve $\mathcal C_{\ell^n}$ over $\mathbb{F}_{\ell^{2n}}$ is not a quotient of the Hermitian curve for $\ell>2$ and…
We show that a F_{q^2}-maximal curve of genus q(q-3)/6 in characteristic three is either a non-reflexive space curve of degree q+1, or it is uniquely determined up to F_{q^2}-isomorphism by a plane model of Artin-Schreier type
In 2017, D. Skabelund constructed a maximal curve over $\mathbb{F}_{q^4}$ as a cyclic cover of the Suzuki curve. In this paper we explicitly determine the structure of the Weierstrass semigroup at any point $P$ of the Skabelund curve. We show that its Weierstrass points are precisely the $\mathbb{F}_{q^4}$-rational points. Also we show that among the Weierstrass points, two types of Weierstrass semigroup occur: one for the $\mathbb{F}_q$-rational points, one for the remaining $\mathbb{F}_{q^4}$-rational points. For each of these two types its Apéry set is computed as well as a set of generator…
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