The low degree unramified coverings of affine line in positive characteristic

Pradipkumar Haribhau Keskar, Purdue University

Abstract

Let $L\sb k$ be the affine line over an algebraically closed field k of characteristic $p >$ 0, and let $\pi\sb A(L\sb k$) be the algebraic fundamental group of $L\sb k$, i.e, $\pi\sb A(L\sb k$) is the set of finite Galois groups of unramified coverings of $L\sb k$. It was conjectured by Abhyankar that $\pi\sb A(L\sb k$) = the set of all quasi p-groups, where by a quasi p-group, we mean a finite group which is generated by all of its Sylow p-subgroups. In this thesis, we reduce mod 5 an equation given by Matzat and modifying it slightly, we obtain an unramified covering of affine line in characteristic 5 whose Galois group is a transitive permutation group on 12 letters isomorphic to the alternating group $A\sb5$. Matzat's equation had Galois group $M\sb{12}$ over the field $\doubq(\sqrt{-5})(T$). In addition to this fact, we use the information about maximal subgroups of Mathieu groups in this process. Also we solve a problem asked by Abhyankar and Yie by showing that a simpler version of an equation given by Abhyankar and Yie for an unramified covering of affine line in characteristic 2 with Galois group $A\sb6$ also does that job. We also strengthen their result by giving a criterion for the Galois group of their equation over k(X) to be $A\sb6$ for an arbitrary field k of characteristic 2.

Degree

Ph.D.

Advisors

Abhyankar, Purdue University.

Subject Area

Mathematics

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