Periods of generalized Tate curves

TL;DR

This paper studies generalized Tate curves, providing explicit formulas for their period isomorphisms, Gauss-Manin connections, and monodromy, with applications to Mumford curves and p-adic periods.

Contribution

It introduces explicit formulas for period isomorphisms and related structures in generalized Tate curves, extending to Mumford curves and p-adic contexts.

Findings

Explicit formulas for period isomorphisms between de Rham and Betti cohomology.

Expressions of local unipotent periods as power series with multiple polylogarithms.

Extension of formulas to families of Mumford curves and p-adic versions.

Abstract

A generalized Tate curve is a universal family of curves with fixed genus and degeneration data which becomes Schottky uniformized Riemann surfaces and Mumford curves by specializing moduli and deformation parameters. By considering each generalized Tate curve as a family of degenerating Riemann surfaces, we give explicit formulas of the period isomorphism between its de Rham and Betti cohomology groups, and of the associated objects: Gauss-Manin connection, variation of Hodge structure and monodromy weight filtration. A remarkable fact is that similar formulas hold also for families of Mumford curves. Furthermore, we show that for a generalized Tate with maximally degenerate closed fiber, its local unipotent periods can be expressed as power series in the deformation parameters whose coefficients are multiple polylogarithm functions. This p-adic version is also given.