The distribution of zeros of an irreducible curve over a finite field

Let f(x, y) be a polynomial defined over Z in two variables of total degree d greater than or equal to 2, and let Vp = {(x, y) is an element of C-p: f(x, y) = 0 (mod p)} for each prime p, where C-p = {(x, y) is an element of Z(2): 0 less than or equal to x < p and 0 less than or equal to y < p}. In this paper, we show that if f(x, y) is absolutely irreducible module p for all sufficiently large p, then we have the following distribution formula for the zeros of f(x, y) module p, [GRAPHICS] where delta is any given real number with 0 less than or equal to delta less than or equal to 1, and [GRAPHICS] Here \\theta\\ denotes the nearest distance from theta to integer, and beta is a real number with 0 less than or equal to beta less than or equal to 1/2. (C) 1996 Academic Press, Inc.