Local Shtukas and Divisible Local Anderson Modules

TL;DR

This paper develops an analog of crystalline Dieudonne9 theory for function fields, replacing p-divisible groups with divisible local Anderson modules and Dieudonne9 modules with local shtukas, establishing an anti-equivalence of categories.

Contribution

It introduces the theory of local shtukas and divisible local Anderson modules, establishing their anti-equivalence and relation to other structures in function field arithmetic.

Findings

Categories of divisible local Anderson modules and local shtukas are anti-equivalent.

Clarifies relations with formal Lie groups, Drinfeld modules, and t-motives.

Discusses Verschiebung map and deformation theory for these objects.

Abstract

We develop the analog of crystalline Dieudonn\'e theory for p-divisible groups in the arithmetic of function fields. In our theory p-divisible groups are replaced by divisible local Anderson modules, and Dieudonn\'e modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson's abelian t-modules and t-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings's and Abrashkin's theory of strict modules, which we review to some extent.