Asymptotic distribution of CM points on the reduction of the Drinfeld modular curve

TL;DR

This paper investigates how rank 2 CM Drinfeld modules are distributed among the components of the reduced Drinfeld modular curve, using spectral and geometric methods.

Contribution

It provides the first detailed asymptotic distribution analysis of CM points on Drinfeld modular curves over function fields.

Findings

Describes the asymptotic distribution of CM Drinfeld modules.

Utilizes properties of the building map and spectral decomposition.

Analyzes the reduction of Drinfeld modular curves via Bruhat-Tits trees.

Abstract

We study a distribution problem over global function fields. More precisely, we describe the asymptotic distribution of rank $2$ CM Drinfeld modules among the irreducible components of the analytic reduction of the Drinfeld modular curve. Our approach relies on the properties of the building map and the spectral decomposition of the adjacency operator on a quotient of the Bruhat-Tits tree.