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

Maosheng Xiong[1]

  • [1] Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay, Kowloon P. R. China

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 3, page 755-769
  • ISSN: 1246-7405

Abstract

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Let l ≥ 2 be a positive integer, 𝔽 q a finite field of cardinality q with q ≡ 1 ( mod l ) . 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 ( X ) , where F is drawn at random uniformly from the set of all monic l -th power-free polynomials F ∈ 𝔽 q [ X ] of degree d as d → ∞ . 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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