Hostname: page-component-5cf477f64f-fcbfl Total loading time: 0 Render date: 2025-04-02T20:47:38.942Z Has data issue: false hasContentIssue false
  • English
  • Français

The $\mathbb{Z}/p$ metabelian birational $p$-adic section conjecture for varieties

Part of: General field theory

Published online by Cambridge University Press:  04 May 2017

Florian Pop
Florian Pop*
Affiliation:
Department of Mathematics, University of Pennsylvania, DRL, 209 S 33rd Street, Philadelphia, PA 19104, USA email pop@math.upenn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalize the $\mathbb{Z}/p$metabelian birational $p$-adic section conjecture for curves, as introduced and proved in Pop [On the birational$p$-adic section conjecture, Compos. Math. 146 (2010), 621–637], to all complete smooth varieties, provided $p>2$. The condition $p>2$ seems to be of technical nature only, and might be removable.

Keywords

anabelian geometryabsolute Galois groupsmetabelian Galois groupsp-adically closed fieldsrational pointsvaluationsHilbert decomposition theory(birational) section conjecture

MSC classification

Secondary: 12E30: Field arithmetic
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 7 , July 2017 , pp. 1433 - 1445
Copyright
© The Author 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arason, J., Elman, R. and Jacob, B., Rigid elements, valuations, and realization of Witt rings , J. Algebra 110 (1987), 449–467.Google Scholar
Ax, J. and Kochen, S., Diophantine problems over local fields III. Decidable fields , Ann. of Math. (2) 83 (1966), 437–456.Google Scholar
Efrat, I., Construction of valuations from K-theory , Math. Res. Lett. 6 (1999), 335–343.CrossRefGoogle Scholar
Faltings, G., Curves and their fundamental groups (following Grothendieck, Tamagawa and Mochizuki) , Astérisque 252 (1998), , Séminaire Bourbaki, exposé 840.Google Scholar
Grothendieck, A., Letter to G. Faltings (1983); see [SL98].Google Scholar
Grothendieck, A., Esquisse d’un programme (1984); see [SL98].Google Scholar
Koenigsmann, J., From p-rigid elements to valuations (with a Galois-characterization of p-adic fields) , J. reine angew. Math. 465 (1995), 165–182; with an appendix by F. Pop.Google Scholar
Koenigsmann, J., On the ‘section conjecture’ in anabelian geometry , J. reine angew. Math. 588 (2005), 221–235.Google Scholar
Kuhlmann, F.-V., Pank, M. and Roquette, P., Immediate and purely wild extensions of valued fields , Manuscripta Math. 55 (1986), 39–67.Google Scholar
Pop, F., Galoissche Kennzeichnung p-adisch abgeschlossener Körper , J. reine angew. Math. 392 (1988), 145–175.Google Scholar
Pop, F., On the birational p-adic section conjecture , Compos. Math. 146 (2010), 621–637.CrossRefGoogle Scholar
Prestel, A. and Roquette, P., Formally p-adic fields, Lecture Notes in Mathematics, vol. 1050 (Springer, 1985).Google Scholar
Schneps, L. and Lochak, P. (eds), Geometric Galois actions I, London Mathematical Society Lecture Note Series, vol. 242 (Cambridge University Press, 1998).Google Scholar
Stix, J., Birational p-adic Galois sections in higher dimensions , Israel J. Math. 198 (2013), 49–61.Google Scholar
Szamuely, T., Groupes de Galois de corps de type fini (d’après Pop) , Astérisque 294 (2004), 403–431.Google Scholar
Topaz, A., Commuting-liftable subgroups of Galois groups II , J. reine angew. Math., doi:10.1515/crelle-2014-0115 (published online 2015-01-27).Google Scholar
Ware, R., Valuation rings and rigid elements in fields , Canad. J. Math. 33 (1981), 1338–1355.Google Scholar