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Herivelto Borges

Publications and source records attributed to Herivelto Borges.

At least 19 recordsLinked to original sources
arXiv2026

Double Artin-Schreier extensions of rational function fields with many lifted automorphisms

In this paper we investigate algebraic function fields in positive characteristic mainly obtained as double Artin-Schreier extensions of rational function fields with a plane model. The goal is to extend to such extensions large automorphism groups of the rational function field. In this way, we construct some new families of ordinary function fields and determine their full automorphism groups. Such groups are large with respect to the genus, compared with the known upper bounds on the size of the automorphism group of an ordinary function field.

math.AG↗
arXiv2025

Minimal Value Set Polynomials

A well-known problem in the theory of polynomials over finite fields is the characterization of minimal value set polynomials (MVSPs) over the finite field $\mathbb{F}_q$, where $q = p^n$. These are the nonconstant polynomials $F \in \mathbb{F}_q[x]$ whose value set $V_F = \{F(a) : a \in \mathbb{F}_q\}$ has the smallest possible size, namely $\lceil \frac{q}{°(F)} \rceil$. In this paper, we describe the family $\mathcal{A}_q$ of all subsets $S \subseteq \mathbb{F}_q$ with $\# S>2$ that can be realized as the value set of an MVSP $F \in \mathbb{F}_q[x]$. Affine subspaces of $\mathbb{F}_q$ are a fundamental type of set in $\mathcal{A}_q$, and we provide the complete list of all MVSPs with such value sets. Building on this, we present a conjecture that characterizes all MVSPs $F \in \mathbb{F}_q[x]$ with $V_F=S$ for any $S \in \mathcal{A}_q$. The conjecture is confirmed by prior results for $q \in\left\{p, p^2, p^3\right\}$ or $\# S \geq p^{n / 2}$, and additional instances, including the cases for $q=p^4$ and $\# S>p^{n / 2-1}$, are proved. We further show that the conjecture leads to the complete characterization of the $\mathbb{F}_q$-Frobenius nonclassical curves of type $y^d=f(x)$, which we establish as a theorem for $q=p^4$.

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arXiv2024

Tate-Shafarevich results for quartic twists in characteristic $2$

The aim of this paper is to present elliptic curves defined over function fields of even characteristic having arbitrarily large Mordell-Weil rank. More precisely, we study elliptic curves arising as quartic twist of a supersingular elliptic curve defined over $\mathbb{F}_2$ using the function field of a maximal curve $C$ that admits an order 4 automorphism. For such elliptic curves we provide a rank formula for its Mordell-Weil group in terms of the genera of $C$ and of another curve covered by $C$.

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arXiv2023

On the number of elements with prescribed norm and trace

Let F_q be the finite field with cardinality q, where q is a prime power. Given a finite field extension F_q^n over F_q and a,b in (F_q)^{*}, we investigate in this article the number N_n(a,b) of elements in F_q^n whose norm equals a and trace equals b. Our approach to probe N_n(a,b) is to connect it with the number of rational points on certain Artin-Schreier curve. After establish an improvement of the Hasse-Weil bound for that Artin-Schreier curve, we improve the known estimates for N_n(a,b) when (roughly speaking) n \geq \sqrt{q}-1. Moreover, we use this approach to improve the bound given by Moisio and Wan for the number of rational points on the toric Calabi-Yau variety studied by Rojas-Leon and Wan in 2011. We finish the paper with explicit calculations of N_n(a,b) and an application to the number of irreducible monic polynomials in an arithmetic progression.

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arXiv2023

The p-rank of curves of Fermat type

Let $\mathbb{K}$ be an algebraically closed field of characteristic $p>0$. A pressing problem in the theory of algebraic curves is the determination of the $p$-rank of a (nonsingular, projective, irreducible) curve $\mathcal{X}$ over $\mathbb{K}$, This birational invariant affects arithmetic and geometric properties of $\mathcal{X}$, and its fundamental role in the study of the automorphism group $\operatorname{Aut}(\mathcal{X})$ has been noted by many authors in the past few decades. In this paper, we provide an extensive study of the $p$-rank of curves of Fermat type $y^m = x^n + 1$ over $\mathbb{K}=\bar{\mathbb{F}}_p$. We determine a combinatorial formula for this invariant in the general case and show how this leads to explicit formulas of the $p$-rank of several such curves. By way of illustration, we present explicit formulas for more than twenty subfamilies of such curves, where $m$ and $n$ are generally given in terms of $p$. We also show how the approach can be used to compute the $p$-rank of other types of curves.

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arXiv2022

An elementary abelian $p$-cover of the Hermitian curve with many automorphisms

The full automorphism group of a certain elementary abelian $p$-cover of the Hermitian curve in characteristic $p>0$ is determined. It is remarkable that the order of Sylow $p$-groups of the automorphism group is close to Nakajima's bound in terms of the $p$-rank. Weierstrass points, Galois points, Frobenius nonclassicality, and arc property are also investigated.

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arXiv2021

Weierstrass pure gaps on curves with three distinguished points

Let $\mathbb{K}$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $\mathbb{K}$, containing three points, $P_1,P_2,$ and $P_3$, such that $nP_1+P_2$, $nP_2+P_3$, and $nP_3+P_1$ are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on $\{P_1,P_2,P_3\}$, as well as an explicit description of all pure gaps at any subset of $\{P_1,P_2,P_3\}$. When $\mathbb{K}=\overline{\mathbb{F}}_q$, this class of curves, which includes the Hermitian curve, is used to construct algebraic geometry codes having minimum distance better than the Goppa bound.

math.NT↗
arXiv2019

On the Zeta function and the automorphism group of the generalized Suzuki curve

For $p$ an odd prime number, $q_{0}=p^{t}$, and $q=p^{2t-1}$, let $\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}$ be the nonsingular model of $$ Y^{q}-Y=X^{q_{0}}(X^{q}-X). $$ In the present work, the number of $\mathbb{F}_{q^{n}}$-rational points and the full automorphism group of $\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}$ are determined. In addition, the L-polynomial of this curve is provided, and the number of $\mathbb{F}_{q^{n}}$-rational points on the Jacobian $J_{\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}}$ is used to construct étale covers of $\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}$, some with many rational points.

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arXiv2019

On some generalized Fermat curves and chords of an affinely regular polygon inscribed in a hyperbola

Let $\mathcal{G}$ be the projective plane curve defined over $\mathbb{F}_q$ given by $$aX^nY^n-X^nZ^n-Y^nZ^n+bZ^{2n}=0,$$ where $ab\notin\{0,1\}$, and for each $s\in\{2,\ldots,n-1\}$, let $\mathcal{D}_s^{P_1,P_2}$ be the base-point-free linear series cut out on $\mathcal{G}$ by the linear system of all curves of degree $s$ passing through the singular points $P_1=(1:0:0)$ and $P_2=(0:1:0)$ of $\mathcal{G}$. The present work determines an upper bound for the number $N_q(\mathcal{G})$ of $\mathbb{F}_q$-rational points on the nonsingular model of $\mathcal{G}$ in cases where $\mathcal{D}_s^{P_1,P_2}$ is $\mathbb{F}_q$-Frobenius classical. As a consequence, when $\mathbb{F}_q$ is a prime field, the bound obtained for $N_q(\mathcal{G})$ improves in several cases the known bounds for the number $n_P$ of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point $P$ distinct from its vertices.

math.AG↗
arXiv2018

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.

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arXiv2018

Galois points for double-Frobenius nonclassical curves

We determine the distribution of Galois points for plane curves over a finite field of $q$ elements, which are Frobenius nonclassical for different powers of $q$. This family is an important class of plane curves with many remarkable properties. It contains the Dickson-Guralnick-Zieve curve, which has been recently studied by Giulietti, Korchmaros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.

math.AG↗
arXiv2018

Subcovers and codes on the $X_{n,r}$ curves

In this work, subcovers $\mathcal{X}_{n,r}^s$ of the curve $\mathcal{X}_{n,r}$ are constructed, the Weierstrass semigroup $H(P_\infty)$ at the point $P_\infty \in \mathcal{X}_{n,r}^s$ is determined and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and $108$ improvements on MinT tables are obtained.

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arXiv2016

Bounds for the number of points on curves over finite fields

Let $\mathcal{X}$ be a projective irreducible nonsingular algebraic curve defined over a finite field $\mathbb{F}_q$. This paper presents a variation of the Störh-Voloch theory and sets new bounds to the number of $\mathbb{F}_{q^r}$-rational points on $\mathcal{X}$. In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil's, Störh-Voloch's and Ihara's.

math.AG↗
arXiv2015

Weierstrass points on Kummer extensions

For Kummer extensions $y^m=f(x)$, we discuss conditions for an integer be a Weierstrass gap at a place $P$. In the case of totally ramified places, the conditions will be necessary and sufficient. As a consequence, we extend independent results of several authors.

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arXiv2015

Points on singular Frobenius nonclassical curves

In 1990, Hefez and Voloch proved that the number of $F_q$-rational points on a nonsingular plane $q$-Frobenius nonclassical curve of degree $d$ is $N = d(q-d+2)$. We address these curves in the singular setting. In particular, we prove that $d(q-d + 2)$ is a lower bound on the number of $F_q$-rational points on such curves of degree $d$.

math.AG↗
arXiv2015

Frobenius nonclassicality of Fermat curves with respect to cubics

For Fermat curves $\mathcal{F}:aX^n+bY^n=Z^n$ defined over $\mathbb{F}_q$, we establish necessary and sufficient conditions for $\mathcal{F}$ to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane cubics. In the $\mathbb{F}_q$-Frobenius nonclassical cases, we determine explicit formulas for the number $N_q(\mathcal{F})$ of $\mathbb{F}_q$-rational points on $\mathcal{F}$. For the remaining Fermat curves, nice upper bounds for $N_q(\mathcal{F})$ are immediately given by the Stöhr-Voloch Theory.

math.AG↗
arXiv2014

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

For each integer $s\geq 1$, we present a family of curves that are $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case $s = 2$, we give necessary and sufficient conditions for such curves to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of conics. In the $\mathbb{F}_q$-Frobenius nonclassical cases, we determine the exact number of $\mathbb{F}_q$-rational points. In the remaining cases, an upper bound for the number of $\mathbb{F}_q$-rational points will follow from Stöhr-Voloch theory.

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