Lower bounds on the class number of algebraic function fields defined over any finite field

TL;DR

This paper establishes lower bounds on the class number of algebraic function fields over finite fields, relating it to the number of places of specific degrees, and provides examples of towers with large class numbers.

Contribution

It introduces new lower bounds and asymptotic estimates for class numbers based on the distribution of places of certain degrees in algebraic function fields.

Findings

Lower bounds on the number of effective divisors of degree ≤ g-1.

Asymptotic estimates for the class number.

Examples of algebraic function field towers with large class numbers.

Abstract

We give lower bounds on the number of effective divisors of degree $\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and asymptotics for the class number, depending mainly on the number of places of a certain degree. We give examples of towers of algebraic function fields having a large class number.