The distribution of values of short hybrid exponential sums on curves over finite fields II

TL;DR

This paper proves that the distribution of short hybrid exponential sums on algebraic curves over finite fields tends to a Gaussian distribution under certain conditions, extending previous results in the field.

Contribution

It generalizes earlier work by establishing Gaussian distribution for exponential sums on arbitrary curves over finite fields.

Findings

Distribution of sums is Gaussian under natural conditions

Results apply to all absolutely irreducible affine plane curves

Extends previous specific cases to general curves

Abstract

Let $p$ be a prime number, $C$ be any absolutely irreducible affine plane curve over $\mathbb{F}_p$, and $g,f\in\mathbb{F}_p(x,y)$ be rational functions. We continue the study of the distribution of the values of short hybrid exponential sums of the form $S_{H}(x;C) = \sum_{P\in C, x<x(P)\leq x+H}\chi(g(P))\psi(f(P))$ on $x\in\mathcal{I}$ for some short interval $\mathcal{I}$. We show that under some natural conditions, the limiting distribution of the sum $S_{H}(x;C)$ is Gaussian for all curve $C$. This largely generalizes a previous result of the author and Zaharescu.