Galois groups associated to generic Drinfeld modules and a conjecture of Abhyankar

TL;DR

This paper proves that the Galois group of a polynomial associated with a generic Drinfeld module is isomorphic to a general linear group, confirming a conjecture by Abhyankar in the context of function field arithmetic.

Contribution

It establishes the Galois group structure for polynomials from generic Drinfeld modules, confirming Abhyankar's conjecture for these modules.

Findings

Galois group of $\phi_N(X)$ is isomorphic to $\GL_r(fq[T]/Nfq[T])$

Settles Abhyankar's conjecture in the context of Drinfeld modules

Demonstrates the algebraic independence of parameters over $fq(T)$

Abstract

Let $\phi$ be a rank $r$ Drinfeld $\BF_q[T]$-module determined by $\phi_T(X) = TX+g_1X^q+...+g_{r-1}X^{q^{r-1}}+X^{q^r}$, where $g_1,...,g_{r-1}$ are algebraically independent over $\BF_q(T)$. Let $N\in\BF_q[T]$ be a polynomial, and $k/\BF_q$ an algebraic extension. We show that the Galois group of $\phi_N(X)$ over $k(T,g_1,...,g_{r-1})$ is isomorphic to $\GL_r(\BF_q[T]/N\BF_q[T])$, settling a conjecture of Abhyankar.