The distribution of points on curves over finite fields in some small rectangles
type: Item
instance of: scholarly article
title: The distribution of points on curves over finite fields in some small rectangles
zbMATH Open document ID: 1307.11077
author: Kit Ho Mak
published in: Monatshefte für Mathematik
publication date: 2014-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.5272
review text: Let \(p\) be a prime and let \(C\) be an absolutely irreducible affine plane curve of degree \(d>1\) defined over the field \(k\) of order \(p\). We identify the plane \(k^2\) with the integer coordinates in \([0,p-1]^2\). For \(\mathcal B\) a square in \([0,p-1]^2\), let \(N_{\mathcal B}(C)\) denote the number of \(k\)-rational points of \(C\) in \(\mathcal B\). If \(\mathcal B=[0,p-1]^2\), we just write \(N(C)\). The subject of the paper under review is related to the distribution of the points of \(C\) in the plane: it is true that \[ N_{\mathcal B}(C)\sim N(C)\text{vol}(\mathcal B)/p^2? \tag{1} \] The classical Weil's bound on the number of rational points together with Bombieri's estimate on exponential sums over \(k\) imply the relation \[ N_{(\mathcal B)}(C)=N(C)\text{vol}(\mathcal B)/p^2+O(d^2\sqrt{p} \log^2p). \tag{2} \] We say that \(\mathcal B\) is large if \(\operatorname{vol}(\mathcal B)\gg p^{3/2}\log^{2+\varepsilon}p\), otherwise \(\mathcal B\) is small. In the large case, (2) implies (1). The author deals with the small case. There are several authors who worked out this case by using results on average (see e.g., [the author and \textit{A. Zaharescu}, ``The distribution of rational points and polynomial maps on an affine variety over a finite field on average'', Preprint, \url{arXiv:1301.1359}]). There are also results whenever the curve \(C\) is the modular hyperbola being this of Gaussian distribution type [\textit{S. M. Gonek} et al., Acta Arith. 102, No. 4, 315--322 (2002; Zbl 0994.11034)]. Here the author applies the method of patterns of points on curves to show that on certain curves the distribution of \(N_{\mathcal B}(C)\) is also Gaussian. The key ingredient concerning the pattern approach generalizes a previous work of \textit{C. I. Cobeli} et al. [J. Number Theory 101, No. 2, 209--222 (2003; Zbl 1086.11045)].
reviewed by: Fernando Torres
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11K36
Mathematics Subject Classification ID: 11T99
zbMATH DE Number: 6336199
zbMATH Keywords: almost all
zbMATH Keywords: patterns
zbMATH Keywords: curves
zbMATH Keywords: rational points
zbMATH Keywords: uniform distribution
MaRDI profile type: Publication
OpenAlex ID: W1986818231
arXiv ID: 1304.5272
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cites work: Q5794285
DOI: 10.1007/S00605-013-0572-4
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description: scientific article; zbMATH DE number 6336199