# The fluctuations in the number of points on a family of curves over a finite field

• [1] Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay, Kowloon P. R. China
• Volume: 22, Issue: 3, page 755-769
• ISSN: 1246-7405

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## Abstract

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Let $l\ge 2$ be a positive integer, ${𝔽}_{q}$ a finite field of cardinality $q$ with $q\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}l\right)$. In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of ${𝔽}_{q}$-points on the curve ${ℂ}_{F}$ given by the affine model ${ℂ}_{F}:{Y}^{l}=F\left(X\right)$, where $F$ is drawn at random uniformly from the set of all monic $l$-th power-free polynomials $F\in {𝔽}_{q}\left[X\right]$ of degree $d$ as $d\to \infty$. The method also enables us to study the fluctuations in the number of ${𝔽}_{q}$-points on the same family of curves arising from the set of monic irreducible polynomials.

## How to cite

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Xiong, Maosheng. "The fluctuations in the number of points on a family of curves over a finite field." Journal de Théorie des Nombres de Bordeaux 22.3 (2010): 755-769. <http://eudml.org/doc/116433>.

@article{Xiong2010,
abstract = {Let $l \ge 2$ be a positive integer, $\{\mathbb\{F\}\}_q$ a finite field of cardinality $q$ with $q \equiv 1 \hspace\{4.44443pt\}(\@mod \; l)$. In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of $\{\mathbb\{F\}\}_q$-points on the curve $\mathbb\{C\}_F$ given by the affine model $\mathbb\{C\}_F: Y^l=F(X)$, where $F$ is drawn at random uniformly from the set of all monic $l$-th power-free polynomials $F \in \{\mathbb\{F\}\}_q[X]$ of degree $d$ as $d \rightarrow \infty$. The method also enables us to study the fluctuations in the number of $\{\mathbb\{F\}\}_q$-points on the same family of curves arising from the set of monic irreducible polynomials.},
affiliation = {Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay, Kowloon P. R. China},
author = {Xiong, Maosheng},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {curves over finite fields; irreducible polynomials; Dirichlet character},
language = {eng},
number = {3},
pages = {755-769},
publisher = {Université Bordeaux 1},
title = {The fluctuations in the number of points on a family of curves over a finite field},
url = {http://eudml.org/doc/116433},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Xiong, Maosheng
TI - The fluctuations in the number of points on a family of curves over a finite field
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 3
SP - 755
EP - 769
AB - Let $l \ge 2$ be a positive integer, ${\mathbb{F}}_q$ a finite field of cardinality $q$ with $q \equiv 1 \hspace{4.44443pt}(\@mod \; l)$. In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of ${\mathbb{F}}_q$-points on the curve $\mathbb{C}_F$ given by the affine model $\mathbb{C}_F: Y^l=F(X)$, where $F$ is drawn at random uniformly from the set of all monic $l$-th power-free polynomials $F \in {\mathbb{F}}_q[X]$ of degree $d$ as $d \rightarrow \infty$. The method also enables us to study the fluctuations in the number of ${\mathbb{F}}_q$-points on the same family of curves arising from the set of monic irreducible polynomials.
LA - eng
KW - curves over finite fields; irreducible polynomials; Dirichlet character
UR - http://eudml.org/doc/116433
ER -

## References

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1. J. Bergström, Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves. Preprint, http://arxiv.org/abs/math/0611813v1, 2006. Zbl1211.14030MR2538614
2. P. Billingsley, Probability and Measure. Third ed., Wiley Ser. Probab. Math. Stat., John Wiley & Sons Inc., Ney Youk, 1995, A Wiley-Interscience Publication. MR1324786
3. A. Bucur, C. David, B. Feigon, M. Lalín, Statistics for traces of cyclic trigonal curves over finite fields. International Mathematics Research Notices (2010), 932–967. Zbl1201.11063MR2595014
4. A. Bucur, C. David, B. Feigon, M. Lalín, Biased statistics for traces of cyclic $p$-fold covers over finite fields. To appear in Proceedings of Women in Numbers, Fields Institute Communications. Zbl1258.11071
5. P. Diaconis, M. Shahshahani, On the eigenvalues of random matrices. Studies in Applied Probability, J. Appl. Probab. 31A (1994), 49–62. Zbl0807.15015MR1274717
6. P. Kurlberg, Z. Rudnick, The fluctuations in the number of points on a hyperelliptic curve over a finite field. J. Number Theory Vol. 129 3 (2009), 580–587. Zbl1221.11141MR2488590
7. N. M. Katz, P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy. Amer. Math. Soc. Colloq. Publ., vol. 45, American Mathematical Socitey, Providence, RI, 1999. Zbl0958.11004MR1659828
8. N. M. Katz, P. Sarnak, Zeroes of zeta functions and symmetry. Bull. Am. Math. Soc. 36 (1999), 1–26. Zbl0921.11047MR1640151
9. L. A. Knizhnerman, V. Z. Sokolinskii, Some estimates for rational trigonometric sums and sums of Legendre symbols. Uspekhi Mat. Nauk 34 (3 (207))(1979), 199–200. Zbl0444.10030MR542248
10. L. A. Knizhnerman, V. Z. Sokolinskii, Trigonometric sums and sums of Legendre symbols with large and small absolute values. Investigations in Number Theory, Saratov. Gos. Univ., Saratov, 1987, 76–89. Zbl0654.10036MR1032540
11. M. Larsen, The normal distribution as a limit of generalized sato-tate measures. Preprint.
12. M. Rosen, Number theory in function fields. Graduate Texts in Mathematics, 210, Springer-Verlag, New York, 2002. Zbl1043.11079MR1876657
13. A. Weil, Sur les Courbes Algébriques et les Variétés qui s’en Déduisent. Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, 1948. iv+85 pp. Zbl0036.16001MR27151

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