Lower bounds on the maximal number of rational points on curves over finite fields

TL;DR

This paper establishes new lower bounds on the maximum number of rational points on algebraic curves over finite fields, using theoretical and explicit construction methods, with implications for algebraic geometry and coding theory.

Contribution

It introduces improved lower bounds for rational points on curves over finite fields, combining Katz-Sarnak theory, trace sums, and tower constructions for the first time.

Findings

Existence of curves with at least 1+q+(2g-ε)√q rational points for large q.

A lower bound of 1+q+1.71√q for g≥3 and odd q≥11.

Explicit tower constructions yield bounds of 1+q+4√q-32 for all g≥2 and all q.

Abstract

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all $q$ large enough, the existence of a curve of genus $g$ over ${\mathbb F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result, with a bound of the form $1+q+4 \sqrt{q} -32$ valid for all $g\ge 2$ and for all $q$.