Geometric coverings of rigid spaces

Abstract: From Tate's uniformization of elliptic curves onwards, the notion of 'covering space', and consequently the notion of fundamental groups, has played a guiding role in the development of rigid geometry. A huge leap forward in our understanding of what exactly covering space/fundamental group might mean in this context was carried out by de Jong in the mid 90s where he was able to form a fundamental group that encompassed both the topological coverings (e.g. those appear in Tate's uniformization) and finite etale coverings. In our current work we propose an extension of those covering spaces considered by de Jong, which not only provides a more conceptual framework for talking about covering spaces as a whole, but also is closed under many of the natural geometric operations that de Jong's covering spaces are not (e.g. disjoint unions and etale localization). Along the way we address some questions posed in de Jong's article, as well as giving a concrete description of the locally constant sheaves in the pro-etale topology which appears in Scholze's work on p-adic Hodge theory.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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