Poisson Type Phenomena for Points on Hyperelliptic Curves modulo p

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 $P_i,P_{i+1}$.

13 pages. To appear in Funct. Approx. Comment. Math