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

Abstract

Let $\varphi$ be a rank $r$ Drinfeld $\BF_q[T]$-module determined by ${\varphi }_T(X)=TX+g_1{X}^q+...+g_{r-1}{X}^{q^{r-1}}+X^{q^r}$, 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 ${\varphi }_N(X)$ 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.

arXiv commentNotice of replacement: The previous version contains a fatal flaw, briefly explained here. This article is now replaced by the longer paper arXiv:1503.06420 [math.NT], where the flaw is corrected

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