Towers of Function Fields over Non-prime Finite Fields

TL;DR

This paper constructs recursive towers of function fields over non-prime finite fields, significantly improving lower bounds for Ihara's quantity by relating explicit equations to Drinfeld modular varieties.

Contribution

It introduces new recursive towers over non-prime finite fields and connects their explicit equations to Drinfeld modular varieties, advancing lower bounds for Ihara's quantity.

Findings

Improved lower bounds for Ihara's quantity $A(\, ext{ell})$ for certain finite fields.

Construction of recursive towers with many rational places.

Explicit equations related to Drinfeld modular varieties.

Abstract

Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara's quantity $A(\ell)$, for $\ell = p^n$ with $p$ prime and $n>3$ odd. We relate the explicit equations to Drinfeld modular varieties.