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Local Shtukas and Divisible Local Anderson Modules

  • Urs Hartl,
    Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
  • Rajneesh Kumar Singh,
    Ramakrishna Vivekananda University, PO Belur Math, Dist Howrah, West Bengal 711202, India
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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.
Keywords: local shtuka, formal Drinfeld module, formal t-module local shtuka, formal Drinfeld module, formal t-module
MSC Classifications: 11G09, 13A35, 14L05 show english descriptions Drinfel'd modules; higher-dimensional motives, etc. [See also 14L05]
Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure [See also 13B22]
Formal groups, $p$-divisible groups [See also 55N22]
11G09 - Drinfel'd modules; higher-dimensional motives, etc. [See also 14L05]
13A35 - Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure [See also 13B22]
14L05 - Formal groups, $p$-divisible groups [See also 55N22]
 

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