Rigid geometry seeks to provide a framework for studying geometric-analytic objects in the non-archimedean (e.g., p-adic) setting. In essence it tries to create a theory similar to complex analytic geometry, but over a non-archimedean field. Rigid geometry is often a foundational tool used in many other areas of arithmetic geometry, such as p-adic Hodge theory and the Langlands program.
Adic spaces, Berkovich Spaces, Huber rings, formal schemes, and fundamental groups.
Specialization for the pro-etale fundamental group (joint with Piotr Achinger and Marcin Lara) || Compositio Mathematica / arXiv
In previous work of Bhatt and Scholze, there is defined the notion of a pro-étale fundamental group of a scheme to account for covers missed by the étale fundmanetal group of singular schemes. On the other hand, in previous work of de Jong, there is defined a fundamental group (that we call the de Jong fundamental group) of a rigid space, meant to capture richer analytic covering spaces missed by the étale fundmanetal group. In this article we show that these, ostensibly unrelated groups, are connected by construction a specialization morphism from the de Jong fundamental group to the pro-étale fundamental group.
Geometric arcs and fundamental groups of rigid spaces (joint with Piotr Achinger and Marcin Lara) || Journal für die reine und angewandte Mathematik (Crelle's journal) / arXiv
In this paper we develop a new notion of covering spaces, called geometric coverings, in rigid geometry. Our definition is modeled after the notion of geometric coverings developed by Bhatt and Scholze in their work on the pro-étale topology for schemes. This definition must be modified to account for the more subtle topology of rigid spaces. We show that geometric coverings are closed under composition, disjoint union, and are etale local on the target. We also show that the category of geometric coverings is a tame infinite Galois category, and so supports a notion of fundamental group.
Variants of the de Jong fundamental group (joint with Piotr Achinger and Marcin Lara) || (to appear in American Journal of Mathematics) arXiv
In this paper we give an example showing that de Jong covering spaces of a rigid space, developed in previous work of de Jong (where it is called an étale covering space), cannot be glued together in the analytic (i.e., normal topological open cover) topology. We then use our work in the article Geometric arcs and fundamental groups of rigid spaces to produce enlargements of de Jong's category fixing this issue, and show how they are related to the pro-étale topology of rigid spaces as developed in work of Scholze.
This is a survery article for the paper Specialization for the pro-etale fundamental group.