Moduli spaces of Mumford curves over Z

Abstract: Schottky uniformization is the description of an analytic curve as the quotient of an open dense subset of the projective line by the action of a Schottky group. All complex curves admit this uniformization, as well as some $p$-adic curves, called Mumford curves. In this talk, I present a construction of universal Mumford curves, analytic spaces that parametrize both archimedean and non-archimedean uniformizable curves of a fixed genus. This result relies on the existence of suitable moduli spaces for marked Schottky groups, that can be built using the theory of Berkovich spaces over rings of integers of number fields due to Poineau.

After introducing Poineau's theory from scratch, I will describe universal Mumford curves and explain how these can be used as a framework to study the Tate curve and to give higher genus generalizations of it. This is based on joint work with Jérôme Poineau.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca