Local Shtukas and Divisible Local Anderson Modules
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.
Cite
@article{arxiv.1511.03697,
title = {Local Shtukas and Divisible Local Anderson Modules},
author = {Urs Hartl and Rajneesh Kumar Singh},
journal= {arXiv preprint arXiv:1511.03697},
year = {2019}
}
Comments
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