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.

V3: 42 pages; Final version; purely expository changes; numbering changed in section 8. Will appear in Math. Annalen

References in corpus (4)

• Commuting elements in Galois groups of function fields
• Commuting-Liftable Subgroups of Galois Groups II
• Introduction to birational anabelian geometry
• Combinatorial geometries of field extensions