Everywhere ramified towers of global function fields

TL;DR

This paper constructs and analyzes towers of global function fields with specific ramification and growth properties, addressing open questions about their structure and limits in algebraic geometry and number theory.

Contribution

It introduces new towers of function fields with controlled ramification and growth, answering previously open questions posed by Stichtenoth.

Findings

Constructed towers with ramification at all places and finite genus ratio limit.

Developed towers with positive limit of degree-one places per extension.

Provided examples that resolve open problems in the theory of function fields.

Abstract

We consider a tower of function fields F_0 < F_1 < ... over a finite field such that every place of every F_i ramified in the tower and the sequence genus(F_i)/[F_i:F_0] has a finite limit. We also construct a tower in which every place ramifies and the sequence N_i/[F_i:F_0] has a positive limit, where N_i is the number of degree-one places of F_i. These towers answer questions posed by Stichtenoth.