Specialization results in Galois theory
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- by Pierre Dèbes and François Legrand PDF
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Abstract:
The central topic of this paper is this question: is a given $k$-étale algebra $\prod _lE_l/k$ the specialization of a given $k$-cover $f:X\rightarrow B$ of the same degree at some unramified point $t_0\in B(k)$? We reduce it to finding $k$-rational points on a certain $k$-variety, which we then study over various fields $k$ of diophantine interest: finite fields, local fields, number fields, etc. We have three main applications. The first one is the following Hilbert-Grunwald statement. If $f:X\rightarrow \mathbb {P}^1$ is a degree $n$ $\mathbb {Q}$-cover with monodromy group $S_n$ over $\overline {\mathbb {Q}}$, and finitely many suitably large primes $p$ are given with partitions $\{d_{p,1}, \ldots , d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points $t_0\in \mathbb {Q}$ that are degree $n$ field extensions with residue degrees $d_{p,1}, \ldots , d_{p,s_p}$ at each prescribed prime $p$. The second one provides a description of the separable closure of a PAC field $k$ of characteristic $p\not =2$: it is generated by all elements $y$ such that $y^m-y\in k$ for some $m\geq 2$. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers.References
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Additional Information
- Pierre Dèbes
- Affiliation: Laboratoire Paul Painlevé, Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France
- ORCID: 0000-0001-9506-1380
- Email: Pierre.Debes@math.univ-lille1.fr
- François Legrand
- Affiliation: Laboratoire Paul Painlevé, Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France
- Email: Francois.Legrand@math.univ-lille1.fr
- Received by editor(s): September 4, 2011
- Received by editor(s) in revised form: January 13, 2012, and January 26, 2012
- Published electronically: March 25, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5259-5275
- MSC (2010): Primary 11R58, 12E30, 12E25, 14G05, 14H30; Secondary 12Fxx, 14Gxx, 14H10
- DOI: https://doi.org/10.1090/S0002-9947-2013-05800-X
- MathSciNet review: 3074373