On pseudopoints of algebraic curves

Abstract

Following Kraitchik and Lehmer, we say that a positive integer n ≡ 1 (mod 8) is an x-pseudosquare if it is a quadratic residue for each odd prime p ≤ x, yet it is not a square. We extend this definition to algebraic curves and say that n is an x-pseudopoint of a curve defined by f (U, V ) = 0 (where f ∈ Z[U, V ]) if for all sufficiently large primes p ≤ x the congruence f (n, m) ≡ 0 (mod p) is satisfied for some m. We use the Bombieri bound of exponential sums along a curve to estimate the smallest x-pseudopoint, which shows the limitations of the modular approach to searching for points on curves.