The Carlitz Logarithm as a Period Morphism for Local $G$-Shtukas

TL;DR

This paper explores the analogy between local shtukas in function fields and p-divisible groups, demonstrating that the Carlitz logarithm functions as a period morphism in a specific moduli problem, with results on representability and explicit computation.

Contribution

It proves the ind-representability of the Rapoport-Zink functor for local shtukas in a particular case and explicitly computes the associated Rapoport-Zink space and period morphism.

Findings

Rapoport-Zink functor is ind-representable in a specific case

The Rapoport-Zink space is explicitly computed

The Carlitz logarithm serves as the period morphism

Abstract

Local shtukas are the function field analogs for $p$-divisible groups. Similar to the $p$-adic theory, one defines Rapoport-Zink functors and Rapoport-Zink spaces for these local shtukas. The associated Hodge-Pink structures are described uniquely by a morphism, called the period morphism of the moduli problem. We will prove the ind-representability of the Rapoport-Zink functor in a particular case and compute the corresponding Rapoport-Zink space as well as the corresponding period morphism. In this case, the period morphism is given by the Carlitz logarithm.