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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Prime-to-$ p$ étale fundamental groups of punctured projective lines over strictly Henselian fields


Authors: Hilaf Hasson and Jeffrey Yelton
Journal: Trans. Amer. Math. Soc. 373 (2020), 3009-3030
MSC (2010): Primary 11S20, 14G20, 14G32, 14H30
DOI: https://doi.org/10.1090/tran/7865
Published electronically: February 21, 2020
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Abstract: Let $ K$ be the fraction field of a strictly Henselian DVR of residue characteristic $ p \geq 0$ with algebraic closure $ \bar {K}$, and let $ \alpha _{1}, \ldots , \alpha _{d} \in \mathbb{P}_{K}^{1}(K)$. In this paper, we give explicit generators and relations for the prime-to-$ p$ étale fundamental group of $ \mathbb{P}_K^1\smallsetminus \{\alpha _1,\ldots ,\alpha _d\}$ that depend (solely) on their intersection behavior. This is done by a comparison theorem that relates this situation to a topological one. Namely, let $ a_{1}, \ldots , a_{d}$ be a distinct power series in $ \mathbb{C}[[x]]$ with the same intersection behavior as the $ \alpha _i$'s, converging on an open disk centered at 0, and choose a point $ z_{0} \neq 0$ lying in this open disk. We compare the natural action of $ \operatorname {Gal}(\bar {K} / K)$ on the prime-to-$ p$ étale fundamental group of $ \mathbb{P}_{\bar {K}} \smallsetminus \{\alpha _{1}, \ldots , \alpha _{d}\}$ to the topological action of looping $ z_0$ around the origin on the fundamental group of $ \mathbb{P}_{\mathbb{C}}^1\smallsetminus \{a_1(z_0),\ldots ,a_d(z_0)\}$. This latter action is, in turn, interpreted in terms of Dehn twists. A corollary of this result is that every prime-to-$ p$ $ G$-Galois cover of $ \mathbb{P}_{\bar K}^1\smallsetminus \{\alpha _1,\ldots ,\alpha _d\}$ satisfies that its field of moduli (as a $ G$-Galois cover) has degree over $ K$ dividing the exponent of $ G / Z(G)$.


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Additional Information

Hilaf Hasson
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: hilaf@math.umd.edu

Jeffrey Yelton
Affiliation: Department of Mathematics “Federigo Enriques”, University of Milan, Milan, MI 20133, Italy
Email: jeffery.yelton@unimi.it

DOI: https://doi.org/10.1090/tran/7865
Received by editor(s): July 12, 2017
Received by editor(s) in revised form: July 20, 2017, August 9, 2018, and December 30, 2018
Published electronically: February 21, 2020
Article copyright: © Copyright 2020 American Mathematical Society