Even points on an algebraic curve
type: Item
instance of: scholarly article
title: Even points on an algebraic curve
zbMATH Open document ID: 1468.11145
author: Przemysław Koprowski
author: Alfred Czogała
published in: Finite Fields and their Applications
publication date: 2021-02-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1904.12177
review text: Let \(\mathbb{F}_q\) be a finite field of odd characteristic and \(K \) an algebraic function field of one variable over \(\mathbb{F}_q\). We denote by \(X\) the set of classes of valuations of \(K\). For a point \(p \in X\), we denote by \([p]\) its class in the Picard group Pic\(X\) of \(X\). The point \(p\) is called {\em even} if its class in the Picard group is 2-divisible. In this paper, it is proved that the graph of even points of \(K\) is always connected and its diameter is precisely 2. Furthermore, it is shown that the set of even points have positive density, and the quotient group Pic\(X/2\mathrm{ Pic }X \) is naturally isomorphic to the group of square classes of K that have even valuations everywhere on X. Moreover, an analogue of global square theorem that concerns points 2-divisible in the Picard group is presented.
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11R65
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 05C40
zbMATH DE Number: 7313150
zbMATH Keywords: global function fields
zbMATH Keywords: curves over finite fields
zbMATH Keywords: global square theorem
zbMATH Keywords: Picard groups
zbMATH Keywords: connected graphs
zbMATH Keywords: graph's diameter
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
OpenAlex ID: W3115205815
arXiv ID: 1904.12177
cites work: The birational type of the moduli space of even spin curves
cites work: On the Picard group: Torsion and the kernel induced by a faithfully flat map
cites work: The Picard group of the moduli space of curves with level structures
cites work: A descent map for curves with totally degenerate semi-stable reduction
cites work: Wild and even points in global function fields
cites work: Wild sets in global function fields
cites work: Q4317713
cites work: Q2770573
cites work: Q2776773
cites work: Basic number theory.
cites work: Algebraic function fields and codes
cites work: Introduction to quadratic forms
cites work: The Haase norm theorem mod squares
DOI: 10.1016/J.FFA.2020.101792
description: scientific article