Mathematics Research Institute


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

Alonso Sepúlveda Castellanos (Universidade Federal de Uberlândia, Brasil)

Fecha: 02/06/2020 16:00
Lugar: Webex (número de reunión 326 251 668)


Let $\mathcal{X}$ be a non-singular, projective, irreducible, algebraic curve of genus $g \geq 1$ over a finite field $\mathbb{F}_{q}$. Fix $m$ distinct rational points $P_1,\ldots,P_m$ on $\mathcal{X}$. The set $H(P_{1}, \ldots , P_{m}) = \{(a_{1}, \ldots, a_{m}) \in \mathbb{N}_{0}^ {m} \mbox{ ; } \exists f \in \mathbb{F}_q(\mathcal{X}) \mbox{ with } (f)_{\infty} = \sum_{i=1}^ {m} a_{i}P_{i} \}$ is called the Weierstrass Semigroup in the points $P_1,\ldots,P_m$. This semigroup is very important to calculate the parameters of algebraic geometry codes (AG codes) over $\mathcal{X}$. In general, is very complicate determinate this semigroup and various efforts have been possible for certain types of curves. In 2018, joint with G. Tizziotti, we determinate the generator set $\Gamma(P_1,\ldots,P_m)$ of $H(P_1,\ldots,P_m)$ for curves $\mathcal{X}$ with affine plane model $f(y)=g(x)$, using the concept of discrepancy on two rational points $P,Q$ over $\mathcal{X}$, introduced by Duursma and Park. With certain conditions, we will show how we can calculate the set $\Gamma(P_1,\ldots,P_m)$ for two types of maximal curves which cannot covered by the Hermitian curve. The first family the curves that we present is the example given by Giulietti and Korchmáros: For $q=n^3$, with $n \geq 2$ a prime power, the $GK$ curve over $\mathbb{F}_{q^2}$ is the curve in $\mathbb{P}^{3}(\overline{\mathbb{F}}_{q^2})$ with equations $Z^{n^2-n+1} = Y \sum_{i=0}^{n} (-1)^{i+1} X^{i(n-1)}$, $X^{n} + X = Y^{n+1}$. The second family was introduced in 2016, by Tafazolian, Teherán and Torres: For $a,b,s\geq 1,n\geq 3$ integers such that $n$ is odd. Let $q=p^a$ a power of a prime, $b$ is a divisor of $a$, $s$ is a divisor of $(q^n+1)/(q+1)$ and $c\in \mathbb{F}_{q^2}$ with $c^{q-1}=-1$. We define the curve $\mathcal{X}_{a,b,n,s}$ over $\mathbb{F}_{q^{2n}}$ with equations $cy^{q+1}=t(x):=\sum_{i=0}^{a/b-1}x^{p^{ib}}$ and $z^{M}=y^{q^2}-y$ where $M=(q^n+1)/(s(q+1))$.

El seminario tendrá lugar en Webex:
Número de reunión: 326 251 668

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