An upper bound for the genus of a curve without points of small degree

TL;DR

This paper establishes an upper bound on the genus of algebraic curves over finite fields that lack points of small degree, providing a key limitation related to the distribution of rational points.

Contribution

It proves the existence of curves with genus bounded by a constant times the field size power, without points of degree less than a given threshold, for any prime p.

Findings

Existence of curves with genus ≤ C_p q^n without small degree points

Bound on genus independent of specific curve constructions

Applicable for all primes p and finite fields of size q

Abstract

In this paper I prove that for any prime $p$ there is a constant $C_p>0$ such that for any $n>0$ and for any $p$-power $q$ there is a smooth, projective, absolutely irreducible curve over $\mathbb{F}_q$ of genus $g\leq C_p q^n$ without points of degree smaller than $n$.