Distribution of rational points of an algebraic surface over finite fields

TL;DR

This paper investigates the distribution of point counts on algebraic surfaces over finite fields, linking character sums to Catalan numbers and hypergeometric functions, revealing their asymptotic behavior as parameters grow.

Contribution

It introduces a detailed analysis of the distribution of character sums associated with algebraic surfaces over finite fields, connecting them to Catalan numbers and hypergeometric functions.

Findings

Distribution of $A_p(\lambda)$ converges as $p$ grows.

Power moments relate to Catalan numbers.

Limiting distributions of hypergeometric functions are derived.

Abstract

The number of points on a certain one parameter family of algebraic surface over a finite field $\F_p$ can be expressed as $p^2+A_p(\lambda),$ where $A_p(\lambda)$ is a character sum and $\lambda$ is an element of the finite field $\F_p.$ In this paper, we study the distribution of the term $A_p(\lambda)$ as the surface varies over a large family of algebraic surfaces of fixed genus and growing $p.$ The power moments of $A_p$'s are weighted sums of Catalan numbers. As a consequence of these results, we obtain limiting distributions of certain families of hypergeometric functions over large finite fields.