The generic monodromy of Drinfeld modular varieties in special characteristic

TL;DR

This paper proves the surjectivity of the monodromy representation on the special fibre of Drinfeld modular varieties with certain level conditions, and demonstrates irreducibility of Kronecker factors in specific cases.

Contribution

It establishes a general surjectivity result for monodromy representations in Drinfeld modular varieties and applies it to irreducibility of Kronecker factors in rank $r$.

Findings

Monodromy representation is surjective on the special fibre.

Kronecker factors of Drinfeld modular polynomials are irreducible.

Results apply to Drinfeld $ extbf{F}_q[t]$-modules in level $t$.

Abstract

By combining theorems of Drinfeld and Strauch, we show that the monodromy representation on the special fibre of a Drinfeld modular variety, with level not divisible by the characteristic, is surjective. We illustrate this result in the special case of Drinfeld $\mathbb{F}_q[t]$-modules in level $t$, and apply this to show that the Kronecker factors of a Drinfeld modular polynomial in rank $r$ are irreducible.