Families of curves over any finite field with a class number greater than the Lachaud - Martin-Deschamps bounds

TL;DR

This paper constructs explicit families of algebraic function fields over finite fields with class numbers exceeding known bounds, demonstrating the existence of dense towers with exceptional properties.

Contribution

It introduces new explicit constructions of asymptotically exact sequences of algebraic function fields with class numbers surpassing classical bounds over any finite field.

Findings

Families have class numbers greater than Lachaud-Martin-Deschamps bounds

Constructs dense towers over any finite field, including non-square q

Provides explicit examples of such function field sequences

Abstract

We study and explicitly construct some families of asymptotically exact sequences of algebraic function fields. It turns out that these families have an asymptotical class number widely greater than the general Lachaud - Martin-Deschamps bounds. We emphasize that we obtain asymptotically exact sequences of algebraic function fields over any finite field $\F_q$, in particular when $q$ is not a square and that these sequences are dense towers.