Poisson Type Phenomena for Points on Hyperelliptic Curves modulo p

TL;DR

This paper investigates the distribution patterns of points on hyperelliptic curves over finite fields, focusing on the behavior of their x-coordinates and distances under various interval constraints, revealing Poisson-like phenomena.

Contribution

It introduces new statistical analyses of point distributions on hyperelliptic curves, incorporating additional rational function conditions, and uncovers Poisson-type distribution behaviors.

Findings

Distribution of x-coordinates follows Poisson-like patterns.

Distances between points exhibit predictable statistical behavior.

Additional rational function constraints influence distribution patterns.

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 $P_i,P_{i+1}$.