Uniformly rigid spaces and N\'eron models of formally finite type

TL;DR

The paper introduces uniformly rigid spaces, a new category of non-archimedean analytic spaces with a uniform structure, and explores their properties, Néron models, and applications to abelian varieties.

Contribution

It defines uniformly rigid spaces, shows their relation to formal models, and constructs formal Néron models with applications to arithmetic geometry.

Findings

Uniformly rigid spaces coincide with their underlying rigid spaces at points.

The uniformly rigid generic fiber of a formal scheme is quasi-compact with bounded global functions.

Existence of formal Néron models for certain uniformly rigid groups is established.

Abstract

We introduce a new category of non-archimedean analytic spaces over a complete discretely valued field. These spaces, which we call uniformly rigid, may be viewed as classical rigid-analytic spaces together with an additional uniform structure: On the level of points, a uniformly rigid space coincides with its underlying rigid space, while its G-topology is coarser and its sheaf of holomorphic functions is smaller. A uniformly rigid structure is, locally, induced by an integral formal model of formally finite type. In fact, Berthelot's generic fiber functor factors naturally through the category of uniformly rigid spaces. The uniformly rigid generic fiber of a quasi-compact formal scheme of formally finite type is quasi-compact, and its global functions are bounded. We construct our new category by starting out from semi-affinoid algebras and nested rational coverings; we prove a uniformly rigid acyclicity theorem, and we study coherent modules. We then define formal N\'eron models for uniformly rigid spaces to be smooth formal schemes of formally finite type satisfying the universal N\'eron property. We establish the existence of formal N\'eron models for certain classes of uniformly rigid groups, and we discuss applications to Chai's program concerning the computation of the base change conductor for abelian varieties with potentially multiplicative reduction. This paper is a faithful copy of the author's doctoral thesis.