The number of rational points of hyperelliptic curves over subsets of finite fields

We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a prime field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and non-residues in the image of such subsets over uniformly random hyperelliptic curves of given degrees. We find a constant probability of such a high difference and show the existence of sets with an exceptionally large discrepancy.