The distribution of points on curves over finite fields in some small rectangles
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)].
scientific article; zbMATH DE number 6336199
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