arXiv subjects
Publications and source records attributed to João Paulo Guardieiro.
In this paper, we characterize all minimal value set binomials over $\mathbb{F}_q$, that is, binomials whose size of the set of images is the smallest possible. With this information, we also classify all quadrinomial curves with separated variables that are $\mathbb{F}_q$-Frobenius nonclassical for the morphism of lines.
Let $\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}_{q^3}$ of the family $$ f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\in \mathbb{F}_{q}[X]. $$ Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials.
We study the algebraic curve over $\mathbb{F}_{q^2}$ defined by $y^{q+1} = x^n(x^n+1)$, where $n$ is a positive integer coprime to the characteristic. We first prove (when $q$ is odd) that the nonsingular model of this curve is $\mathbb{F}_{q^2}$-maximal if and only if $n \mid (q+1)$. Writing $n = \frac{q+1}{m}$, we obtain a family of maximal curves parameterized by the divisors $m$ of $q+1$, which extends the previously studied case $m=3$ corresponding to maximal curves with the third largest possible genus. For this family, we determine the Weierstrass semigroups at several classes of rational points, including those lying above the branch points of the natural projection. These semigroups are described explicitly in terms of $q$ and $m$, and exhibit different behaviors depending on the arithmetic properties of $m$. Moreover, we determine the full automorphism group of the curve under a mild condition on the characteristic. Our results extend an earlier work on the case $m=3$ and provide new insight into the structure of this family of maximal curves.
Starting from the elliptic curve $E: y^2 = x^3 - x$ over $\mathbb{F}_9$, a curve $\mathcal{X}$ over $\mathbb{F}_{3^{2n}}$ and a cyclic cover of $\mathcal{X}$ of degree $m \in \{2,3,4,6\}$, we construct the corresponding $m$-twists over the function field $\mathbb{F}_{3^{2n}}(\mathcal{X})$. We also obtain the Mordell-Weil rank of these twists in terms of the Zeta functions of $\mathcal{X}$ and of suitable Kummer and Artin-Schreier extensions of it. Finally, we also describe the fibers of the elliptic fibration associated to such twists in the case $\mathcal{X} = \mathbb{P}^1$.
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 $C$.
Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the rational function $f_1(x)/f_2(x)$ belongs to a certain set which we define, we present a sufficient condition for the existence of a primitive element $α\in \mathbb{F}_{q^n}$, normal over $\mathbb{F}_q$, such that $f_1(α)/f_2(α)$ is also primitive.