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June 2017 Arithmetic descent of specializations of Galois covers
Ryan Eberhart, Hilaf Hasson
Funct. Approx. Comment. Math. 56(2): 259-270 (June 2017). DOI: 10.7169/facm/1613

Abstract

Given a $G$-Galois branched cover of the projective line over a number field $K$, we study whether there exists a closed point of $\mathbb{P}^1_K$ with a connected fiber such that the\break $G$-Galois field extension induced by specialization ``arithmetically descends'' to $\mathbb{Q}$ (i.e., there exists\break a $G$-Galois field extension of $\mathbb{Q}$ whose compositum with the residue field of the point is equal to the specialization). We prove that the answer is frequently positive (whenever $G$ is regularly realizable over $\mathbb{Q}$) if one first allows a base change to a finite extension of $K$. If one does not allow base change, we prove that the answer is positive when $G$ is cyclic. Furthermore, we provide an explicit example of a Galois branched cover of $\mathbb{P}^1_K$ with no $K$-rational points of arithmetic descent.

Citation

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Ryan Eberhart. Hilaf Hasson. "Arithmetic descent of specializations of Galois covers." Funct. Approx. Comment. Math. 56 (2) 259 - 270, June 2017. https://doi.org/10.7169/facm/1613

Information

Published: June 2017
First available in Project Euclid: 28 March 2017

zbMATH: 06864158
MathSciNet: MR3660963
Digital Object Identifier: 10.7169/facm/1613

Subjects:
Primary: 11L10
Secondary: 11R32 , 14H30

Keywords: arithmetic descent , curves , Galois covers , Galois groups , inverse Galois problem , specializations

Rights: Copyright © 2017 Adam Mickiewicz University

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Vol.56 • No. 2 • June 2017
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