Lower bounds on the number of rational points of Jacobians over finite fields and application to algebraic function fields in towers

TL;DR

This paper provides effective bounds on the class number of algebraic function fields over finite fields, utilizing partial information on places, with applications to towers of function fields and their invariants.

Contribution

It introduces explicit bounds for class numbers of function fields based on partial data, aiding analysis of towers with positive Tsfasman-Vladut invariants.

Findings

Effective bounds depend on partial information about places.

Bounds are applicable to individual steps in towers, not just asymptotic cases.

Estimates provided for class numbers in specific tower configurations.

Abstract

We give effective bounds for the class number of any algebraic function field of genus $g$ defined over a finite field. These bounds depend on the possibly partial information on the number of places on each degree $\leq g$. Such bounds are especially useful for estimating the class number of function fields in towers of function fields over finite fields. We give examples in the case of asymptotically good towers. In particular we estimate the class number of function fields which are steps of towers having one or several positive Tsfasman-Vladut invariants. Note that the study is not done asymptotically, but for each individual step of the towers for which we determine precise parameters.