Drinfeld Modular Curves Subordinate to Conjugacy Classes of Nilpotent Upper-Triangular Matrices

TL;DR

This paper introduces a new class of normalized Drinfeld modular curves linked to conjugacy classes of nilpotent upper-triangular matrices, revealing a tree structure and establishing their geometric irreducibility and function field characteristics.

Contribution

It establishes a novel connection between nilpotent matrix conjugacy classes and Drinfeld modular curves, generalizing existing tower structures and analyzing their geometric properties.

Findings

Drinfeld modular curves are organized in a tree structure.

Proved geometric irreducibility of (3,2)-type curves.

Characterized the function fields of these modular curves.

Abstract

We introduce normalized Drinfeld modular curves that parameterize rank $m$ Drinfeld modules compatible with a $T$-torsion structure arising from a given conjugacy class of nilpotent upper-triangular $n\times n$ matrices with rank $\geqslant n-m$ over a finite field $\mathbb{F}_q$. This creates a deep link connecting the classification of nilpotent upper-triangular matrices and the decomposition of Drinfeld modular curves. The conjugacy classes of nilpotent upper-triangular matrices one-to-one corresponds to certain $T$-torsion flags, and form a tree structure. As a result, the associated Drinfeld modular curves are organized in the same tree. This generalizes the tower structure introduced by Bassa, Beelen, Garcia, Stichtenoth, and others. Additionally,we prove the geometric irreducibility of $(3,2)$-type normalized Drinfeld modular curves, and characterize their associated function fields.