Frobenius order-sequences of curves. (English) Zbl 0824.14019

Garcia, Arnaldo; Homma, Masaaki

Frey, Gerhard (ed.) et al., Algebra and number theory. Proceedings of a conference held at the Institute of Experimental Mathematics, University of Essen, Germany, December 2-4, 1992. Berlin: de Gruyter. 27-41 (1994).

Let \(X \subseteq \mathbb{P}^ N\) be a nondegenerate, absolutely irreducible algebraic curve over \(\mathbb{F}_ q\). In a previous paper, K.-O. Stöhr and J. F. Voloch [Proc. Lond. Math. Soc., III. Ser. 52, 1- 19 (1986; Zbl 0593.14020)] introduced the notion of “\(q\)-Frobenius order-sequence of \(X\),” and gave a geometric proof of Weil’s basic theorem on rational points of curves over \(\mathbb{F}_ q\). The order sequence of \(X\) is the set of possible intersection multiplicities of \(X\) with the hyperplanes of \(\mathbb{P}^ N\) at general points. It is basic that the \(q\)-Frobenius sequence differs from the order sequence by the deletion of one order. The paper being reviewed studies this phenomenon and establishes some geometric results, e.g. on the inseparability degree of Gauss maps.
For the entire collection see [Zbl 0793.00015].

Reviewer: D.Goss (Columbus/Ohio)

MSC:

14H25	Arithmetic ground fields for curves
14G15	Finite ground fields in algebraic geometry
14G05	Rational points
11G20	Curves over finite and local fields
14H45	Special algebraic curves and curves of low genus

Keywords:

rational points of curves over finite fields; Frobenius sequence; Weil number