New optimal function field towers over finite fields of quartic power

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 $I_{\hat{I}}\cdot$ of degree two. Notably, the $I_{\hat{I}}\cdot$-reduction of the tower of minimal Drinfeld modular curves is asymptotically optimal over the finite field ${\mathbb{F}}_{q^4}$.