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

Florian Breuer

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

Number Theory 2015-08-20 v3

Authors: Florian Breuer

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.

Keywords

Drinfeld module Galois group Abhyankar conjecture algebraic independence polynomial algebraic extension general linear group

Cite

@article{arxiv.1303.2334,
  title  = {Galois groups associated to generic Drinfeld modules and a conjecture of Abhyankar},
  author = {Florian Breuer},
  journal= {arXiv preprint arXiv:1303.2334},
  year   = {2015}
}

Comments

Notice 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

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

Abstract

Keywords

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