On the Invariants of Towers of Function Fields over Finite Fields

TL;DR

This paper investigates the invariants of towers of function fields over finite fields, providing methods to construct towers with specific invariant properties and demonstrating the existence of towers attaining bounds for various invariants.

Contribution

It introduces explicit extension methods for constructing towers with prescribed invariants and proves the existence of towers reaching the Drinfeld-Vladut bound for any order r.

Findings

Constructed towers with finitely many positive invariants.

Established towers over square fields with at least one positive invariant.

Proved existence of recursive towers attaining the Drinfeld-Vladut bound.

Abstract

We consider a tower of function fields F=(F_n)_{n\geq 0} over a finite field F_q and a finite extension E/F_0 such that the sequence \mathcal{E):=(EF_n)_{n\goq 0} is a tower over the field F_q. Then we deal with the following: What can we say about the invariants of \mathcal{E}; i.e., the asymptotic number of places of degree r for any r\geq 1 in \mathcal{E}, if those of F are known? We give a method based on explicit extensions for constructing towers of function fields over F_q with finitely many prescribed invariants being positive, and towers of function fields over F_q, for q a square, with at least one positive invariant and certain prescribed invariants being zero. We show the existence of recursive towers attaining the Drinfeld-Vladut bound of order r, for any r\geq 1 with q^r a square. Moreover, we give some examples of recursive towers with all but one invariants equal to zero.