Explicit Drinfeld moduli schemes and Abhyankar's generalized iteration conjecture

TL;DR

This paper proves Abhyankar's conjecture by explicitly constructing Drinfeld moduli schemes and analyzing the Galois groups of Drinfeld modules, establishing a clear link between algebraic structures and Galois representations.

Contribution

It provides the first explicit construction of Drinfeld moduli schemes of level $tn$ and confirms Abhyankar's generalized iteration conjecture regarding Galois groups.

Findings

Galois group of $\psi_n(X)$ is isomorphic to $ ext{GL}_r(\mathbb{F}_q[t]/n ext{)}$

Explicit construction of Drinfeld moduli schemes of level $tn$

Resolution of Abhyankar's conjecture in this context

Abstract

Let $k$ be a field containing $\mathbb{F}_q$. Let $\psi$ be a rank $r$ Drinfeld $\mathbb{F}_q[t]$-module determined by $\psi_t(X) = tX+a_1X^q+\cdots+a_{r-1}X^{q^{r-1}}+X^{q^r}$, where $t,a_1,\ldots,a_{r-1}$ are algebraically independent over $k$. Let $n\in\mathbb{F}_q[T]$ be a monic polynomial. We show that the Galois group of $\psi_n(X)$ over $k(t,a_1,\ldots,a_{r-1})$ is isomorphic to $\mathrm{GL}_r(\mathbb{F}_q[t]/n\mathbb{F}_q[t])$, settling a conjecture of Abhyankar. Along the way we obtain an explicit construction of Drinfeld moduli schemes of level $tn$.