Reconstructing function fields from rational quotients of mod-\(\ell \) Galois groups

Abstract

In this paper, we develop the main step in the global theory for the mod-\(\ell \) analogue of Bogomolov’s program in birational anabelian geometry for higher-dimensional function fields over algebraically closed fields. More precisely, we show how to reconstruct a function field K of transcendence degree \(\ge \)5 over an algebraically closed field, up-to inseparable extensions, from the mod-\(\ell \) abelian-by-central Galois group of K endowed with the collection of mod-\(\ell \) rational quotients.

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Acknowledgments

The author warmly thank Florian Pop and Thomas Scanlon for numerous technical discussions concerning the topics in this paper. The author also thanks Martin Hils and James Freitag for several helpful discussions. The manuscript was written during the MSRI semester on Model Theory, Arithmetic Geometry and Number theory in the spring of 2014. The author thanks MSRI and the organizers of this semester for their hospitality and for an excellent research environment. The author also thanks the referee for his/her extremely useful comments which helped improve the paper in many ways.

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Correspondence to Adam Topaz.

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This research was supported by NSF postdoctoral fellowship DMS-1304114.

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Topaz, A. Reconstructing function fields from rational quotients of mod-\(\ell \) Galois groups. Math. Ann. 366, 337–385 (2016). https://doi.org/10.1007/s00208-015-1327-4

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Mathematics Subject Classification

  • 12F10
  • 12G05
  • 12J10
  • 19D45