Local Shtukas and Divisible Local Anderson Modules

Abstract

We develop the analog of crystalline Dieudonné 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 Dieudonné 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.

45 pages, v4: Final version. Appears in Canadian Journal of Mathematics. The present arXiv version contains a few more details and proofs; see page 1 bottom

References in corpus (2)

• Local Shtukas and Divisible Local Anderson Modules
• Local Shtukas, Hodge-Pink Structures and Galois Representations

Cited by in corpus (9)

• Local Shtukas and Divisible Local Anderson Modules
• Local Shtukas, Hodge-Pink Structures and Galois Representations
• Langlands-Rapoport Conjecture Over Function Fields
• Periods of Drinfeld modules and local shtukas with complex multiplication
• Compactification of Level Maps of Moduli Spaces of Drinfeld Shtukas
• Deformations and elements of deformation theory
• On Langlands program, related representation and $G$-shtukas
• The local Langlands correspondence for $\DeclareMathOperator{\GL}{GL}\GL_n$ over function fields
• On Langlands program, global fields and shtukas