Functiones et Approximatio Commentarii Mathematici

Poisson type phenomena for points on hyperelliptic curves modulo $p$

Kit-Ho Mak and Alexandru Zaharescu

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Abstract

Let $p$ be a large prime, and let $C$ be a hyperelliptic curve over $\mathbb{F}_p$. We study the distribution of the $x$-coordinates in short intervals when the $y$-coordinates lie in a prescribed interval, and the distribution of the distance between consecutive $x$-coordinates with the same property. Next, let $g(P,P_0)$ be a rational function of two points on $C$. We study the distribution of the above distances with an extra condition that $g(P_i,P_{i+1})$ lies in a prescribed interval, for any consecutive points $P_i,P_{i+1}$.

Article information

Source
Funct. Approx. Comment. Math., Volume 47, Number 1 (2012), 65-78.

Dates
First available in Project Euclid: 25 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.facm/1348578277

Digital Object Identifier
doi:10.7169/facm/2012.47.1.5

Mathematical Reviews number (MathSciNet)
MR2987111

Zentralblatt MATH identifier
1320.11056

Subjects
Primary: 11G20: Curves over finite and local fields [See also 14H25]
Secondary: 11T99: None of the above, but in this section

Keywords
Poisson distribution hyperelliptic curves

Citation

Mak, Kit-Ho; Zaharescu, Alexandru. Poisson type phenomena for points on hyperelliptic curves modulo $p$. Funct. Approx. Comment. Math. 47 (2012), no. 1, 65--78. doi:10.7169/facm/2012.47.1.5. https://projecteuclid.org/euclid.facm/1348578277


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