Abstract
AbstractWe 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 theories of strict modules, which we review briefly.
Highlights
In the arithmetic of number elds, elliptic curves and abelian varieties are important objects. eir theory has been extensively developed in the last two centuries and their moduli spaces have played a major role in Faltings’s proof of the Mordell conjecture [Fal,CS ], the proof of Fermat’s Last eorem by Wiles and Taylor [Wil, TW, CSS ], and the proof of the Langlands correspondence for GLn over nonarchimedean local elds of characteristic zero by Harris and Taylor [HT ]
In the present article we call them z-divisible local Anderson modules as in the following de nition, and we develop this theory under the technical assumption that ζ ∈ R is nilpotent. is theory was already announced in [Har,Har,Har,HK ] and was used in [Har ]
In Section we present the above de nition of z-divisible local Anderson modules G and give equivalent de nitions
Summary
In the arithmetic of number elds, elliptic curves and abelian varieties are important objects. eir theory has been extensively developed in the last two centuries and their moduli spaces have played a major role in Faltings’s proof of the Mordell conjecture [Fal ,CS ], the proof of Fermat’s Last eorem by Wiles and Taylor [Wil , TW , CSS ], and the proof of the Langlands correspondence for GLn over nonarchimedean local elds of characteristic zero by Harris and Taylor [HT ]. The goal of crystalline Dieudonné theory in the arithmetic of function elds is to describe the analogs of p-divisible groups that correspond to e ective local shtukas. In the present article we call them z-divisible local Anderson modules as in the following de nition, and we develop this theory under the technical assumption that ζ ∈ R is nilpotent. E description of z-divisible local Anderson modules by e ective local shtukas is deduced from Abrashkin’s [Abr ] anti-equivalence between nite locally free strict. Anderson modules over R with G[z] radicial, and the category of z-divisible formal Fq[[z]]-modules G with G[z] representable by a nite locally free group scheme, such that locally on Spec R there is an integer d with (z − ζ)d = on ωG. We call such a sheaf a locally free sheaf of OS [[z]]-modules of rank r
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