The distribution of rational points on a curve defined modulo q (Q1071813)

Let f(x,y) be a polynomial over the integers of degree at least 2, which is absolutely irreducible (mod p) for all sufficiently large primes. The author considers the number of points of the curve f(x,y)\(\equiv 0\) (mod p) in boxes, i.e. x and y being restricted to some subintervals of [0,p]. The main results of the paper are of conditional nature, namely consequences of some natural hypotheses on incomplete exponential sums. As applications of special cases, the author gives conditional asymptotic results on the sum functions of the arithmetic functions d(n) and r(n) over an arithmetic progression.