Frobenius indices of certain curves over finite fields
type: Item
instance of: scholarly article
title: Frobenius indices of certain curves over finite fields
zbMATH Open document ID: 0952.14015
author: Keisuke Toki
published in: Collectanea Mathematica
publication date: 2001-01-15T00:00:00Z
full work available at URL: https://eudml.org/doc/42679
review text: The author is concerned with an algebraic curve \(X \subset \mathbb{P}^N\) \((N\geq 3)\) over a finite field of characteristic \(p>0\) lying in an \(N\)-dimensional projective space \(\mathbb{P}^N\) which possesses an order-sequence in the sense of \textit{A. Garcia} and \textit{M. Homma} [in: Proc. Conf. Inst. Experimental Math., Essen 1992, 27-41 (1994; Zbl 0824.14019)] or of \textit{K.-O. Stöhr} and \textit{J. F. Voloch} [Proc. Lond. Math. Soc., III. Ser. 52, 1-19 (1986; Zbl 0593.14020)]. Considering such a curve \(X \subset \mathbb{P}^N\) over a finite field \(\mathbb{F}_{q'}\) with \(q'\) elements (\(q'\) is a power of \(p\)), it possesses the \(q'\)-Frobenius order-sequence in the sense of the papers cited above, too. Let \(x_0: x_1: x_2:\cdots: x_N\) be the coordinate functions, \(k(X)\) be the function-field of \(X \subset \mathbb{P}^N\), and \(\{D^{(r)}\); \(0\leq r\in \mathbb{Z}\}\) be the system of Hasse-Schmidt derivatives on \(k(X)\). Put \({\mathfrak x}= (x_0, x_1, x_2,\dots, x_N)\). Then each of both sequences which are denoted by \(0= \varepsilon_0< \varepsilon_1< \varepsilon_2<\dots< \varepsilon_N\) (order-sequence) and \(0= \nu_0< \nu_1<\dots< \nu_N\) (\(q'\)-Frobenius order-sequence), means the minimal sequence consisting of integers, in the lexicographic order, such that the \(N+1\) row-vectors \(D^{(\varepsilon_i)} \cdot {\mathfrak x}\) \((0\leq i\leq N)\) are linearly independent over \(k(X)\), the \(N+1\) row-vectors \({\mathfrak x}^{q'}\), \(D^{(\nu_i)} \cdot{\mathfrak x}\) \((0\leq i\leq N-1)\) are so, respectively. By proposition 2.1 in the paper by \textit{K.-O. Stöhr} and \textit{J. P. Voloch} cited above, for the relationship between both sequences, it is known that there exists an integer \(I\) depending on \(q'\) \((1\leq I\leq N)\) such that \(\nu_i= \varepsilon_i\) (whenever \(i<I\)), \(\varepsilon_{i+1}\) (whenever \(i\geq I\)). This value \(I\) is named Frobenius index. The author denotes by \(\iota(q';X)\) this value for the curve…
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 1454844
zbMATH Keywords: curves over finite fields
zbMATH Keywords: order-sequence
zbMATH Keywords: Frobenius index
MaRDI profile type: Publication
Recommended article: Fermat curves over finite fields
Recommended article: On the indices of curves over local fields
Recommended article: Frobenius fields for elliptic curves
Recommended article: Q3972365
Recommended article: Q5706770
Recommended article: Indices of hyperelliptic curves over \(p\)-adic fields
Recommended article: Jacobians of curves over finite fields
Recommended article: The index of certain hyperelliptic curves over \(p\)-adic fields
Recommended article: Q4673259
Recommended article: Q4314365
description: scientific article; zbMATH DE number 1454844