The universal Mumford curve, and its abelian differentials and periods in arithmetic formal geometry

TL;DR

This paper constructs the universal Mumford curve within arithmetic formal geometry, providing explicit formulas for abelian differentials and their periods, advancing understanding of degenerations in algebraic curves.

Contribution

It introduces a universal Mumford curve in arithmetic formal geometry and derives explicit formulas for abelian differentials and periods.

Findings

Construction of the universal Mumford curve as a family over deformation space

Explicit formulas for abelian differentials on the universal curve

Explicit formulas for periods of the universal Mumford curve

Abstract

We construct the universal Mumford curve of given genus as a family of Mumford curves over the deformation space of degenerate curves in the category of arithmetic formal geometry. Furthermore, we give explicit formulas of abelian differentials and their periods of the universal Mumford curve.