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

TL;DR

This paper establishes concentration inequalities for the number of rational points on hyperelliptic curves over finite field subsets, revealing significant discrepancies between quadratic residues and non-residues with high probability.

Contribution

It introduces new concentration inequalities and demonstrates the existence of subsets with exceptionally large quadratic residue discrepancies on hyperelliptic curves.

Findings

Constant probability of large quadratic residue discrepancy

Existence of sets with exceptionally high discrepancy

Quantitative bounds on rational point distributions

Abstract

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 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.