arXiv subjects

Maria Montanucci

Publications and source records attributed to Maria Montanucci.

At least 55 records · Page 3Linked to original sources

arXiv2018

An $\mathbb{F}_{p^2}$-maximal Wiman's sextic and its automorphisms

In 1895 Wiman introduced a Riemann surface $\mathcal{W}$ of genus $6$ over the complex field $\mathbb{C}$ defined by the homogeneous equation $\mathcal{W}:X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2=0$, and showed that its full automorphism group is isomorphic to the symmetric group $S_5$. The curve $\mathcal{W}$ was previously studied as a curve defined over a finite field $\mathbb{F}_{p^2}$ where $p$ is a prime, and necessary and sufficient conditions for its maximality over $\mathbb{F}_{p^2}$ were obtained. In this paper we first show that the result of Wiman concerning the automorphism group of $\mathcal{W}$ holds also over an algebraically closed field $\mathbb{K}$ of positive characteristic $p$, provided that $p \geq 7$. For $p=2,3$ the polynomial $X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2$ is not irreducible over $\mathbb{K}$, while for $p=5$ the curve $\mathcal{W}$ is rational and $Aut(\mathcal{W}) \cong PGL(2,\mathbb{K})$. We also show that the $\mathbb{F}_{19^2}$-maximal Wiman's sextic $\mathcal{W}$ is not Galois covered by the Hermitian curve $\mathcal{H}_{19}$ over $\mathbb{F}_{19^2}$.

math.AG↗

arXiv2018

Quotients of the Hermitian curve from subgroups of ${\rm PGU}(3,q)$ without fixed points or triangles

In this paper we deal with the problem of classifying the genera of quotient curves $\mathcal{H}_q/G$, where $\mathcal{H}_q$ is the $\mathbb{F}_{q^2}$-maximal Hermitian curve and $G$ is an automorphism group of $\mathcal{H}_q$. The groups $G$ considered in the literature fix either a point or a triangle in the plane ${\rm PG}(2,q^6)$. In this paper, we give a complete list of genera of quotients $\mathcal{H}_q/G$, when $G \leq {\rm Aut}(\mathcal{H}_q) \cong {\rm PGU}(3,q)$ does not leave invariant any point or triangle in the plane. As a result, the classification of subgroups $G$ of ${\rm PGU}(3,q)$ satisfying this property is given up to isomorphism.

math.AG↗

arXiv2018

Pure gaps on curves with many rational places

We consider the algebraic curve defined by $y^m = f(x)$ where $m \geq 2$ and $f(x)$ is a rational function over $\mathbb{F}_q$. We extend the concept of pure gap to {\bf c}-gap and obtain a criterion to decide when an $s$-tuple is a {\bf c}-gap at $s$ rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places.

math.CO↗

arXiv2018

$\mathbb{F}_{p^2}$-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

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 that the hypothesis on the order of $Aut(\mathcal{X})$ is sharp, since there exists an $\mathbb{F}$-maximal curve $\mathcal{X}$ for $q=71$ of genus $g=7$ with $|Aut(\mathcal{X})|=84(7-1)$ which is not Galois-covered by the Hermitian curve $\mathcal{H}_{72}$.

math.AG↗

arXiv2017

Linear codes from Denniston maximal arcs

In this paper we construct functional codes from Denniston maximal arcs. For $q=2^{4n+2}$ we obtain linear codes with parameters $[(\sqrt{q}-1)(q+1),5,d]_q$ where $\lim_{q \to +\infty} d=(\sqrt{q}-1)q-3\sqrt{q}$. We also find for $q=16,32$ a number of linear codes which appear to have larger minimum distance with respect to the known codes with same length and dimension.

math.CO↗

arXiv2017

AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

We investigate several types of linear codes constructed from two families $\tilde{\mathcal S}_q$ and $\tilde{\mathcal R}_q$ of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such curves are provided, and the Weierstrass semigroup $H(P)$ at an $\mathbb{F}_{q}$-rational point $P$ is shown to be symmetric.

cs.IT↗

arXiv2017

Weierstrass semigroups on the Giulietti-Korchmáros curve

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti-Korchmáros curve $\mathcal{X}$. We show that as the point varies, exactly three possibilities arise: One for the $\mathbb{F}_{q^2}$-rational points (already known in the literature), one for the $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational points, and one for all remaining points. As a result, we prove a conjecture concerning the structure of $H(P)$ in case $P$ is an $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational point. As a corollary we also obtain that the set of Weierstrass points of $\mathcal{X}$ is exactly its set of $\mathbb{F}_{q^6}$-rational points.

math.AG↗

arXiv2017

On plane curves given by separated polynomials and their automorphisms

Let $\mathcal{C}$ be a plane curve defined over the algebraic closure $K$ of a prime finite field $\mathbb{F}_p$ by a separated polynomial, that is $\mathcal{C}: A(y)=B(x)$, where $A(y)$ is an additive polynomial of degree $p^n$ and the degree $m$ of $B(X)$ is coprime with $p$. Plane curves given by separated polynomials are well-known and studied in the literature. However just few informations are known on their automorphism groups. In this paper we compute the full automorphism group of $\mathcal{C}$ when $m \not\equiv 1 \pmod {p^n}$ and $B(X)$ has just one root in $K$, that is $B(X)=b_m(X+b_{m-1}/mb_m)^m$ for some $b_m,b_{m-1} \in K$. Moreover, some sufficient conditions for the automorphism group of $\mathcal{C}$ to imply that $B(X)=b_m(X+b_{m-1}/mb_m)^m$ are provided. As a byproduct, the full automorphism group of the Norm-Trace curve $\mathcal{C}: x^{(q^r-1)/(q-1)}=y^{q^{r-1}}+y^{q^{r-2}}+\ldots+y$ is computed. Finally, these results are used to construct multi point AG codes with many automorphisms.

math.AG↗

arXiv2017

AG codes and AG quantum codes from the GGS curve

In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all $\mathbb F_{q^2}$-rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes.

math.CO↗

arXiv2017

Generalized Artin-Mumford curves over finite fields

Let $\mathbb{F}_q$ be the finite field of order $q=p^h$ with $p>2$ prime and $h>1$, and let $\mathbb{F}_{\bar{q}}$ be a subfield of $\mathbb{F}_q$. From any two $\bar{q}$-linearized polynomials $L_1,L_2 \in \overline{\mathbb{F}}_q[T]$ of degree $q$, we construct an ordinary curve $\mathcal{X}_{(L_1,L_2)}$ of genus $(q-1)^2$ which is a generalized Artin-Schreier cover of the projective line $\mathbb{P}^1$. The automorphism group of $\mathcal{X}_{(L_1,L_2)}$ over the algebraic closure $\overline{\mathbb{F}}_q$ of $\mathbb{F}_q$ contains a semidirect product $Σ\rtimes Γ$ of an elementary abelian $p$-group $Σ$ of order $q^2$ by a cyclic group $Γ$ of order $\bar{q}-1$. We show that for $L_1 \neq L_2$, $Σ\rtimes Γ$ is the full automorphism group ${\rm Aut}(\mathcal{X}_{(L_1,L_2)})$ over $\overline{\mathbb{F}}_q$; for $L_1=L_2$ there exists an extra involution and ${\rm Aut}(\mathcal{X}_{(L_1,L_1)})=Σ\rtimes Δ$ with a dihedral group $Δ$ of order $2(\bar{q}-1)$ containing $Γ$. Two different choices of the pair $\{L_1,L_2\}$ may produce birationally isomorphic curves, even for $L_1=L_2$. We prove that any curve of genus $(q-1)^2$ whose $\overline{\mathbb{F}}_q$-automorphism group contains an elementary abelian subgroup of order $q^2$ is birationally equivalent to $\mathcal{X}_{(L_1, L_2)}$ for some separable $\bar{q}$-linearized polynomials $L_1,L_2$ of degree $q$. We produce an analogous characterization in the special case $L_1=L_2$. This extends a result on the Artin-Mumford curves, due to Arakelian and Korchmáros.

math.AG↗

arXiv2017

On the spectrum of genera of quotients of the Hermitian curve

We investigate the genera of quotient curves $\mathcal H_q/G$ of the $\mathbb F_{q^2}$-maximal Hermitian curve $\mathcal H_q$, where $G$ is contained in the maximal subgroup $\mathcal M_q\leq{\rm Aut}(\mathcal H_q)$ fixing a pole-polar pair $(P,\ell)$ with respect to the unitary polarity associated with $\mathcal H_q$. To this aim, a geometric and group-theoretical description of $\mathcal M_q$ is given. The genera of some other quotients $\mathcal H_q/G$ with $G\not\leq\mathcal M_q$ are also computed. Thus we obtain new values in the spectrum of genera of $\mathbb F_{q^2}$-maximal curves. A plane model for $\mathcal H_q/G$ is obtained when $G$ is cyclic of order $p\cdot d$, with $d$ a divisor of $q+1$.

math.AG↗

arXiv2017

On some Galois covers of the Suzuki and Ree curves

We determine the full automorphism group of two recently constructed families $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ of maximal curves over finite fields. These curves are covers of the Suzuki and Ree curves, and are analogous to the Giulietti-Korchmáros cover of the Hermitian curve. We also show that $\tilde{\mathcal{S}}_q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}_{q^4}$, and $\tilde{\mathcal{R}}_q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}_{q^6}$. Finally, we compute the genera of many Galois subcovers of $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$; this provides new genera for maximal curves.

math.AG↗

arXiv2017

Transcendency Degree One Function Fields Over a Finite Field with Many Automorphisms

Let $\mathbb{K}$ be the algebraic closure of a finite field $\mathbb{F}_q$ of odd characteristic $p$. For a positive integer $m$ prime to $p$, let $F=\mathbb{K}(x,y)$ be the transcendency degree $1$ function field defined by $y^q+y=x^m+x^{-m}$. Let $t=x^{m(q-1)}$ and $H=\mathbb{K}(t)$. The extension $F|H$ is a non-Galois extension. Let $K$ be the Galois closure of $F$ with respect to $H$. By a result of Stichtenoth, $K$ has genus $g(K)=(qm-1)(q-1)$, $p$-rank (Hasse-Witt invariant) $γ(K)=(q-1)^2$ and a $\mathbb{K}$-automorphism group of order at least $2q^2m(q-1)$. In this paper we prove that this subgroup is the full $\mathbb{K}$-automorphism group of $K$; more precisely $Aut_{\mathbb {K}}(K)=Q\rtimes D$ where $Q$ is an elementary abelian $p$-group of order $q^2$ and $D$ has a index $2$ cyclic subgroup of order $m(q-1)$. In particular, $\sqrt{m}|Aut_{\mathbb{K}}(K)|> g(K)^{3/2}$, and if $K$ is ordinary (i.e. $g(K)=γ(K)$) then $|Aut_{\mathbb{K}}(K)|>g^{3/2}$. On the other hand, if $G$ is a solvable subgroup of the $\mathbb{K}$-automorphism group of an ordinary, transcendency degree $1$ function field $L$ of genus $g(L)\geq 2$ defined over $\mathbb{K}$, then by a result due to Korchmáros and Montanucci, $|Aut_{\mathbb{K}}(K)|\le 34 (g(L)+1)^{3/2}<68\sqrt{2}g(L)^{3/2}$. This shows that $K$ hits this bound up to the constant $68\sqrt{2}$. Since $Aut_{\mathbb{K}}(K)$ has several subgroups, the fixed subfield $F^N$ of such a subgroup $N$ may happen to have many automorphisms provided that the normalizer of $N$ in $Aut_{\mathbb{K}}(K)$ is large enough. This possibility is worked out for subgroups of $Q$.

math.AG↗

arXiv2016

The Geometry of the Artin-Schreier-Mumford Curves over an Algebraically Closed Field

For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c\neq 0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|<1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\sqrt{g}(\sqrt{g}+1)^2$, far away from the Hurwitz bound $84(g-1)$ valid in zero characteristic. In this paper we deal with the case where $\mathbb{K}$ is an algebraically closed field of characteristic $p$. We prove that the group $Aut(\mathcal{X})$ of all automorphisms of $\mathcal{X}$ fixing $\mathbb{K}$ elementwise has order $2q^2(q-1)$ and it is the semidirect product $Q\rtimes D_{q-1}$ where $Q$ is an elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini and Madan. Furthermore, we show that $ASM(q)$ has a nonsingular model $\mathcal{Y}$ in the three-dimensional projective space $PG(3,\mathbb{K})$ which is neither classical nor Frobenius classical over the finite field $\mathbb{F}_{q^2}$.

math.AG↗

arXiv2016

Ordinary algebraic curves with many automorphisms in positive characteristic

Let $\mathcal{X}$ be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus $\mathcal{g}(\mathcal{X}) \ge 2$ defined over an algebraically closed field $\mathbb{K}$ of odd characteristic $p$. Let $Aut(\mathcal{X})$ be the group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$ element-wise. For any solvable subgroup $G$ of $Aut(\mathcal{X})$ we prove that $|G|\leq 34 (\mathcal{g}(\mathcal{X})+1)^{3/2}$. There are known curves attaining this bound up to the constant $34$. For $p$ odd, our result improves the classical Nakajima bound $|G|\leq 84(\mathcal{g}(\mathcal{X})-1)\mathcal{g}(\mathcal{X})$, and, for solvable groups $G$, the Gunby-Smith-Yuan bound $|G|\leq 6(\mathcal{g}(\mathcal{X})^2+12\sqrt{21}\mathcal{g}(\mathcal{X})^{3/2})$ where $\mathcal{g}(\mathcal{X})>cp^2$ for some positive constant $c$.

math.AG↗

arXiv2016

Multi-Point AG Codes on the GK Maximal Curves

In this paper we investigate multi-point Algebraic-Geometric codes associated to the GK maximal curve, starting from a divisor which is invariant under a large automorphism group of the curve. We construct families of codes with large automorphism groups.

math.CO↗

arXiv2016

Some Ree and Suzuki curves are not Galois covered by the Hermitian curve

The Deligne-Lusztig curves associated to the algebraic groups of type $^2A_2$, $^2B_2$, and $^2G_2$ are classical examples of maximal curves over finite fields. The Hermitian curve $\mathcal H_q$ is maximal over $\mathbb F_{q^2}$, for any prime power $q$, the Suzuki curve $\mathcal S_q$ is maximal over $\mathbb F_{q^4}$, for $q=2^{2h+1}$, $h\geq1$ and the Ree curve $\mathcal R_q$ is maximal over $\mathbb F_{q^6}$, for $q=3^{2h+1}$, $h\geq0$. In this paper we show that $\mathcal S_8$ is not Galois covered by $\mathcal H_{64}$. We also give a proof for an unpublished result due to Rains and Zieve stating that $\mathcal R_3$ is not Galois covered by $\mathcal H_{27}$. Furthermore, we determine the spectrum of genera of Galois subcovers of $\mathcal H_{27}$, and we point out that some Galois subcovers of $\mathcal R_3$ are not Galois subcovers of $\mathcal H_{27}$.

math.AG↗

arXiv2016

The $a$-numbers of Fermat and Hurwitz curves

For an algebraic curve $\mathcal{X}$ defined over an algebraically closed field of characteristic $p >0$, the $a$-number $a(\mathcal{X})$ is the dimension of the space of exact holomorphic differentials on $\mathcal{X}$. We compute the $a$-number for an infinite families of Fermat and Hurwitz curves. Our results apply to Hermitian curves giving a new proof for a previous result of Gross.

math.NT↗