Section and towers

TL;DR

This paper explores the structure of towers of finite étale covers, discusses the correspondence between sections and towers, and examines the failure of Grothendieck's injectivity conjecture for abelian varieties.

Contribution

It clarifies the folklore correspondence between sections and towers, and analyzes the limitations of the Grothendieck section conjecture in specific cases.

Findings

The folklore correspondence between sections and towers is clarified.

The injectivity statement of Grothendieck's conjecture fails for abelian varieties.

The paper reinterprets anabelian concepts using model theory.

Abstract

We discuss the towers of finite \'etale covers which were essentially introduced by A.Tamagawa. The statement about correspondence between sections and cofinal towers is a folklore but perhaps not in a very explicit form. The last section explains how the "injectivity statement" of Grothendieck section conjecture fails for abelian varieties, which is also known in some form. The paper is based on an earlier article which was aimed to reinterpret anabelian setting in model theory terms.