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The tamely ramified geometric quantitative minimal ramification problem

Part of: Field extensions Basic constructions Families, fibrations (Co)homology theory Algebraic number theory: global fields

Published online by Cambridge University Press:  09 November 2023

Mark Shusterman
Mark Shusterman*
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA mshusterman@math.harvard.edu
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Abstract

We prove a large finite field version of the Boston–Markin conjecture on counting Galois extensions of the rational function field with a given Galois group and the smallest possible number of ramified primes. Our proof involves a study of structure groups of (direct products of) racks.

Keywords

braided sethomogeneous subgroupslarge monodromyChebotarev's density theoremHurwitz spaces

MSC classification

Primary: 11R58: Arithmetic theory of algebraic function fields 12F12: Inverse Galois theory 14D22: Fine and coarse moduli spaces 14F20: Étale and other Grothendieck topologies and (co)homologies 54B40: Presheaves and sheaves
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 1 , January 2024 , pp. 21 - 51
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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