On the Number of Rational Points of Jacobians over Finite Fields

TL;DR

This paper establishes improved bounds for class numbers of algebraic curves over finite fields using methods from the asymptotic theory of global fields, offering concrete applications of these theoretical tools.

Contribution

It provides new, tighter bounds for class numbers of algebraic curves over finite fields utilizing explicit asymptotic methods from global field theory.

Findings

Bounds are sharper than previous combinatorial estimates.

Application of global field asymptotic theory to algebraic curve invariants.

Demonstrates effectiveness of global field zeta-function techniques.

Abstract

In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in the proof are essentially those from the explicit asymptotic theory of global fields. We thus provide a concrete application of effective results from the asymptotic theory of global fields and their zeta-functions.