A collection of books, papers, preprints, surveys, and lectures notes of the AHGT members of the network.
``Panorama of recent progress in the arithmetic-geometry theory of Galois and homotopy groups and its ramifications. The program has now evolved beyond the classical group-theoretic legacy of Grothendieck to result in an autonomous project that exploits a new geometrization of the original insight and sketches new frontiers between homotopy geometry, homology geometry, and diophantine geometry. We "close the loop” by including the last twenty-year progress of the Japanese arithmetic-geometry school via Ihara’s program and Nakamura-Tamagawa-Mochizuki’s anabelian approach, which brings its expertise in terms of algorithmic, combinatoric, and absolute reconstructions. These methods supplement and interact with those from the classical arithmetic of covers and Hurwitz spaces and the motivic and geometric Galois representations [...].''
...See here for all the AHGT books and surveys
A snapshot of the most recent progress of the field and of the latest interest of the AHGT members.
...See here for some references prior to 2023
Reports of the ``Ateliers'', collaborative writings for a first contact with recent reserch topics.
The theory of perfectoid spaces, introduced by P. Scholze mainly in two articles has immediately shown its potency by proving new cases of the monodromy-weight conjecture (It is a conjecture by Deligne which relates the eigenspaces of the action of the Frobenius on the étale cohomology of a projective scheme over a complete discrete valuation ring with the unipotent action from the inertia group on this cohomology). It has since remained at the forefront of research in arithmetic geometry through its connections to central topics such as abelian varieties and their moduli spaces as well as p-divisible groups and Shimura varieties.
The Grothendieck-Teichmüller group GT, first introduced by Drinfel'd and Ihara in the 90s, follows Grothendieck's insight of ``Esquisse d'un programme'' to encapture the absolute Galois group of rational numbers in a combinatorial way, via topological group properties, within the outer automorphism group of a certain category of tower of étale fundamental groupoids. Recent breakthroughs were obtained, on the topological side (via operads, see Fresse-Horel ~2012) and on the arithmetic side (via combinatorial anabelian geometry, see Hoshi-Minamide-Mochizuki ~2017).
We study the arithmetical construction of the Grothendieck- Teichmüller group such as given by Ihara, then the anabelian and geometric methods used by Hoshi, Minamide and Mochizuki to provide recent results on GT.
By patching and gluing G-covers over a complete discretely valued field Harbater resolved the Regular Inverse Galois Problem for such fields and proved the Abhyankar’s conjecture. A framework for patching over fields is introduced by Harbater and Hartmann, which is further developed to produce local-global principles and various arithmetic results for function fields of curves over complete discretely valued fields. This method is then improved by Mehmeti using Berkovic analytification.