Axel Sarlin: The étale fundamental group, étale homotopy and anabelian geometry

In 1983 Grothendieck wrote a letter to Faltings, outlining what is today known as the anabelian conjectures. These conjectures concern the possibility to reconstruct curves and schemes from their étale fundamental group. Although Faltings never replied to the letter, his student Mochizuki began working on it. A major achievement by Mochizuki and Tamagawa was to prove several important versions of these conjectures.

In this thesis we will first introduce Grothendieck’s Galois theory with the aim to define the étale fundamental group and formulate Mochizuki’s result. After recalling some necessary homotopy theory, we will introduce the étale homotopy type, which is an extension of the étale fundamental group developed by Artin, Mazur and Friedlander. This is done in order to describe some recent work by Schmidt and Stix that improves on the results of Mochizuki and Tamagawa by extending them from étale fundamental groups to étale homotopy types of certain (possibly higher-dimensional) schemes.

Advisor: Fabien Morel (Munich)