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Abhyankar's conjectures in Galois theory: Current status and future directions


Authors: David Harbater, Andrew Obus, Rachel Pries and Katherine Stevenson
Journal: Bull. Amer. Math. Soc. 55 (2018), 239-287
MSC (2010): Primary 11-02, 14-02, 11G20, 12F12, 14G17, 14H30, 14H37, 14J50
DOI: https://doi.org/10.1090/bull/1594
Published electronically: October 23, 2017
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Abstract: In this paper we survey the major contributions of Abhyankar to the development of the theory of fundamental groups and Galois covers in positive characteristic. We first discuss the current status of four conjectures of Abhyankar about Galois covers in positive characteristic. Then we discuss research directions inspired by Abhyankar's work, including many open problems.


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Additional Information

David Harbater
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: harbater@math.upenn.edu

Andrew Obus
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: obus@virginia.edu

Rachel Pries
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
Email: pries@math.colostate.edu

Katherine Stevenson
Affiliation: Department of Mathematics, California State Northridge, Northridge, California 91330
Email: katherine.stevenson@csun.edu

DOI: https://doi.org/10.1090/bull/1594
Keywords: Abhyankar conjecture, Galois cover, fundamental group, ramification theory, lifting problem, finite characteristic
Received by editor(s): March 22, 2017
Published electronically: October 23, 2017
Additional Notes: The first author was partially supported by NSF FRG grants DMS-1265290 and DMS-1463733, and by NSA grant H98230-14-1-0145. The second author was partially supported by NSF FRG grant DMS-1265290 and NSF grant DMS-1602054. The third author was partially supported by NSA grant 131011 and NSF grant DMS-15-02227.
Dedicated: Dedicated to Yvonne Abhyankar
Article copyright: © Copyright 2017 American Mathematical Society

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