Functiones et Approximatio Commentarii Mathematici

Arithmetic descent of specializations of Galois covers

Ryan Eberhart and Hilaf Hasson

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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.

Article information

Funct. Approx. Comment. Math., Volume 56, Number 2 (2017), 259-270.

First available in Project Euclid: 28 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11L10: Jacobsthal and Brewer sums; other complete character sums
Secondary: 14H30: Coverings, fundamental group [See also 14E20, 14F35] 11R32: Galois theory

Galois covers Galois groups curves arithmetic descent specializations Inverse Galois Problem


Eberhart, Ryan; Hasson, Hilaf. Arithmetic descent of specializations of Galois covers. Funct. Approx. Comment. Math. 56 (2017), no. 2, 259--270. doi:10.7169/facm/1613.

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