Reconstructing function fields from rational quotients of mod- $$\ell $$ ℓ Galois groups

Abstract

In this paper, we develop the main step in the global theory for the mod-ℓ 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 ≥ 5 over an algebraically closed field, up-to inseparable extensions, from the mod-ℓ abelian-by-central Galois group of K endowed with the collection of mod-ℓ rational quotients.

The Galois action on geometric lattices and the mod- $$\ell $$ ℓ I/OM

“…2 for this terminology). The majority of the work is then devoted to showing that σ is compatible with all such onedimensional geometric subgroups, and these include the rational subgroups considered in [34]. One then concludes, along similar lines to the mod-global theory from loc.…”

“…We will then use techniques from the mod-abelian-by-central variant of Bogomolov's Program, including both the mod-local theory [26,33,36] and the mod-global theory [34].…”

“…These fundamental differences between the pro-and mod-context were described in detail in the introduction of [34], and we refer the reader there for these details. Nevertheless, we mention here that, in the pro-context, one eventually uses the Fundamental Theorem of Projective Geometry applied to an infinite-dimensional k-projective space embedded in H 1 (K , Z (1)), where K is a function field over k. The main difficulty in the mod-context is that H 1 (K , Z/ (1)) contains no such kprojective space.…”

“…See Sect. 2 for a detailed summary of the proof of the mod-I/OM, and see the introduction of [34] for more on the comparison between the pro-and mod-contexts. We now introduce the mod-abelian-by-central context in detail.…”

“…Assertion (2) follows from the Künneth formula along with the fact that G a is isomorphic to a direct power of Z/ ; see [34,Fact 8.1] and the surrounding discussion for more details. Assertion (3) is the standard "duality" between the commutator and the cup-product.…”