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

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 $S_H(x;C)={\sum }_{P\in C,x<x(P)\le x+H}\chi (g(P))\psi (f(P))$ 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.

21 pages