New optimal function field towers over finite fields of quartic power

TL;DR

This paper constructs two new towers of Drinfeld modular curves over finite fields of quartic power, demonstrating their asymptotic optimality and extending the understanding of function field towers in algebraic geometry.

Contribution

It introduces novel types of Drinfeld modular curve towers associated with a specific domain and ideal, showing their asymptotic optimality over finite fields of degree four.

Findings

The towers are asymptotically optimal over _{q^4}.

The towers originate from a domain related to the projective line over _q.

The construction generalizes previous rank-two Drinfeld modular curve towers.

Abstract

We introduce two new types of towers of Drinfeld modular curves. These towers originate from a specific domain $\mathcal{A} $ and are analogous to the towers of rank-two Drinfeld modular curves over the polynomial ring. Specifically, the domain $\mathcal{A} $ corresponds to the projective line over the finite field $ \mathbb{F}_q $, equipped with an infinite place of degree two. We select an arbitrary non-zero principal $\mathcal{A} $-ideal $ I_{\eta} $ of degree two. Notably, the $ I_{\eta} $-reduction of the tower of minimal Drinfeld modular curves is asymptotically optimal over the finite field $ \mathbb{F}_{q^4} $.