The minimum and maximum number of rational points on jacobian surfaces over finite fields

TL;DR

This paper establishes bounds on the number of rational points on abelian and Jacobian varieties over finite fields, with a focus on precisely determining these bounds for Jacobian surfaces.

Contribution

It provides exact maximum and minimum counts of rational points on Jacobian surfaces over finite fields, advancing understanding of their point distributions.

Findings

Maximum and minimum rational points on Jacobian surfaces are explicitly determined.

Bounds on rational points for abelian and Jacobian varieties are established.

Results contribute to the classification of Jacobian surfaces over finite fields.

Abstract

We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of dimension 2.