The Distribution of Values of Short Hybrid Exponential Sums on Curves over Finite Fields

Let $p$ be a prime number, $X$ be an absolutely irreducible affine plane curve over $\mathbb{F}_p$, and $g,f\in\mathbb{F}_p(x,y)$. We study the distribution of the values of the hybrid exponential sums \begin{equation*} S_n = \sum_{\substack{P_i\in X, n<x(P_i)\leq n+H y(P_i)\in\mathcal{J}}}\chi(g(P_i))\psi(f(P_i)) \end{equation*} on $n\in\mathcal{I}$ for some short interval $\mathcal{I}$. We show that under some natural conditions the limiting distribution of the projections of the sum $S_n$, $n\in\mathcal{I}$ on any straight line through the origin is Gaussian as $p$ tends to infinity.

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Published 28 February 2011