Counting Drinfeld modules with prescribed local conditions

TL;DR

This paper provides asymptotic counts for Drinfeld modules over function fields with specific local conditions, linking the problem to counting points on weighted projective spaces and offering new insights into their distribution.

Contribution

It introduces a novel approach to counting Drinfeld modules with prescribed local conditions by relating it to point counting on weighted projective spaces over function fields.

Findings

Asymptotic formulas for Drinfeld modules with local conditions

Connections established between Drinfeld modules and weighted projective spaces

Results may inform future research on point counting in algebraic geometry

Abstract

We give asymptotics for the number of Drinfeld $\mathbb{F}_q[T]$-modules over $\mathbb{F}_q(T)$ of a given height, which satisfy prescribed sets of local conditions. This is done by relating our problem to a problem about counting points on weighted projective spaces. Our results for counting points of bounded height on weighted projective spaces over global function fields may be of independent interest.