We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and nonresidues 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.
"The number of rational points of hyperelliptic curves over subsets of finite fields." Involve 12 (5) 755 - 765, 2019. https://doi.org/10.2140/involve.2019.12.755