Conference abstracts
Session S07 - Finite Fields
July 29, 16:30 ~ 16:55
A problem of Beelen, Garcia and Stichtenoth on an Artin-Schreier tower
Horacio Navarro
IMAL, Argentina - hnavarro@santafe-conicet.gov.ar
A \textit{tower of function fields} over $\mathbb{F}_q$ is a sequence of algebraic function fields $\mathcal{F}=\{F_i\}_{i=0}^{\infty}$ such that for all $i\geq 0$ $ F_i\subsetneq F_{i+1}$, $F_{i+1}/F_i$ is a separable finite extension, $\mathbb{F}_q$ is algebraically closed in $F_i$ and there exists $F_j$ with genus greater than one.\\
A tower $\mathcal{F}$ is called \textit{asymptotically good} if $\gamma(\mathcal{F})<\infty$ and $\nu(\mathcal{F})>0$ where $$\gamma(\mathcal{F}):=\lim_{i\rightarrow \infty}g(F_i)/[F_i:F_0] \text{ \quad and \quad} \nu(\mathcal{F}):=\lim_{i\rightarrow \infty } N(F_i)/[F_i:F_0],$$ $g(F_i)$ is the genus of $F_i$ and $N(F_i)$ is the number of rational places of $F_i$. Otherwise, $\mathcal{F}$ is called \textit{asymptotically bad}. \\
In $2006$ Beleen, Garcia and Stichtenoth proved that any recursive tower of function fields over $\mathbb{F}_2$ defined by $g(Y)=f(X)$ with $g(T), f(T) \in \mathbb{F}_2(T)$ and $\deg f=\deg g=2$ is defined by the Artin-Schreier equation \begin{equation}\label{e1} Y^2+Y=\frac{1}{(1/X)^2+(1/X)+b}+c, \end{equation} with $b,c \in \mathbb{F}_2$. They checked that all the posible cases were already considered in previous works, except when $b=c=1$. In fact, they left as an open problem to determine whether or not this tower is asymptotically good over $\mathbb{F}_{2^s}$ for some positive integer $s$.\\
In this talk we will show that the recursive tower defined by equation $(1)$ with $b=c=1$ is asymptotically bad over $\mathbb{F}_{2^s}$ when $s$ is odd and where the main difficulty arises in the study of this tower when $s$ is even.
Joint work with Ricardo Toledano (Universidad Nacional del Litoral-IMAL) and María Chara (Universidad Nacional del Litoral-IMAL).