Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields

TL;DR

This paper introduces three geometric methods—theta divisor indecomposability, Galois descent, and Honda-Tate theory—to improve upper bounds on the number of rational points on algebraic curves over finite fields, surpassing existing bounds in various cases.

Contribution

It presents novel geometric techniques for tightening bounds on rational points, extending beyond prior bounds by leveraging advanced algebraic geometry and number theory tools.

Findings

Lowered bounds for small genus curves over specific finite fields.

Achieved isolated improvements for large genus curves in certain finite fields.

Demonstrated the effectiveness of geometric methods in bounding rational points.

Abstract

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus is small compared to q, or from Oesterle's optimization of the explicit formulae method if the genus is large. This paper presents three methods for improving these bounds. The arguments used are the indecomposability of the theta divisor of a curve, Galois descent, and Honda-Tate theory. Examples of improvements on the bounds include lowering them for a wide range of small genus when q=2^3, 2^5, 2^{13}, 3^3, 3^5, 5^3, 5^7, and when q=2^{2s}, s>1. For large genera, isolated improvements are obtained for q=3,8,9.