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_{\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} $.