Resumen de Frobenius indices of certain curves over finite fields

Keisuke Toki

• We consider an algebraic curve $X\subset\mathbb{P}^N(N\geq 3)$ defined over a finite field of characteristic $p>0$ which possesses an order-sequence in the sense of [6]. Let $N$, an odd prime number $p$ and an integer $I(1\leq I\leq N)$ be arbitrarily given. Then we shall give an example of a curve as above whose $q'-$\textit{Frobenius index} in the sense of [1] equals $I$, which is a complete intersection in $\mathbb{P}^N$ of $N - I$ Fermat equations and $I - 1$ Artin-Schreier equations over a finite field $\mathbb{F}_{q'}$ with $q'$ elements, where $q'$ is some power of $p$ (see the Theorem of Section 1). In the case of $N = 3$ and $I = 1$, our example is the same one as Example 3 in [1] or [2].