Number of points on abelian and Jacobian varieties over finite fields

TL;DR

This paper establishes bounds and exact formulas for the number of points on abelian and Jacobian varieties over finite fields, enhancing understanding of their point distributions and extremal properties.

Contribution

It provides new bounds and explicit formulas for the number of points on abelian and Jacobian varieties, including specific results for Jacobian surfaces.

Findings

Upper and lower bounds for abelian varieties

Exact formulas for Jacobian surfaces

Bounds specific to Jacobian varieties

Abstract

We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.