New optimal function field towers over finite fields of quartic power
Chuangqiang Hu, Xiuwu Zhu
TL;DR
The paper constructs two explicit towers of Drinfeld modular curves from a domain $\\mathcal{A}$ associated with $\\mathbb{P}^1_{\\mathbb{F}_q}$ with an infinite degree-two place, and analyzes their $I_\infty$-reduction by a degree-two ideal $I_\eta$ to obtain asymptotically optimal towers over $\\mathbb{F}_{q^4}$. It provides detailed normalized and minimal models, derives recursive isogeny relations linking $k$-th twists, and describes primitive torsion via explicit polynomials, enabling concrete function-field Towers. The genus and supersingular-point analysis for the $I_\eta$-reduced curves yields exact counts and shows that the towers meet Ihara’s bound, establishing asymptotic optimality. The results extend the landscape of explicit Drinfeld modular towers and provide tools for constructing optimal algebraic-geometric codes from these 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_η $ of degree two. Notably, the $ I_η $-reduction of the tower of minimal Drinfeld modular curves is asymptotically optimal over the finite field $ \mathbb{F}_{q^4} $.