김완수 교수

We use Lau’s classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial.

Let O K be a 2-adic discrete valuation ring with perfect residue field k. We classify p-divisible groups and p-power order finite flat group schemes over O K in terms of certain Frobenius modules over S := W (k) [[u]]. We also show the compatibility with crystalline Dieudonn theory and associated Galois representations. Our approach differs from Lau's generalization of display theory, who independently obtained our result using display theory.

When $p>2$ , we construct a Hodge-type analogue of Rapoport–Zink spaces under the unramifiedness assumption, as formal schemes parametrizing ‘deformations’ (up to quasi-isogeny) of $p$ -divisible groups with certain crystalline Tate tensors. We also define natural rigid analytic towers with expected extra structure, providing more examples of ‘local Shimura varieties’ conjectured by Rapoport and Viehmann.

Assume that <f>$p > 2$</f>, and let <f>$\\mathscr {O} _K$</f> be a <f>$p$</f>-adic discrete valuation ring with residue field admitting a finite <f>$p$</f>-basis, and let <f>$R$</f> be a formally smooth formally finite-type <f>$\\mathscr {O} _K$</f>-algebra. (Indeed, we allow slightly more general rings <f>$R$</f>.) We construct an anti-equivalence of categories between the categories of <f>$p$</f>

We review the analog of Fontaine's theory of crystalline $p$-adic Galois representations and their classification by weakly admissible filtered isocrystals in the arithmetic of function fields over a finite field. There crystalline Galois representations are replaced by the Tate modules of so-called local shtukas. We prove that the Tate module functor is fully faithful. In addition to this étale realization of a local shtuka we discuss also the de Rham and the crystalline cohomology realizations

When $p>2$, we construct a Hodge-type analogue of Rapoport-Zink spaces under the unramifiedness assumption, as formal schemes parametrising "deformations" (up to quasi-isogeny) of $p$-divisible groups with certain crystalline Tate tensors. We also define natural rigid analytic towers with expected extra structure, providing more examples of "local Shimura varieties" conjectured by Rapoport and Viehmann.

Let K be a finite extension of Q_p, and choose a uniformizer pi in K. Choose pi_{n+1} such that pi_1:=pi and pi_{n+1}^p=pi_n, and let K_infty denote the field extension of K obtained by adjoining pi_{n+1} for all n. We introduce a new technique using restriction to Gal(Kbar/K_infty) to study deformations and mod p reductions in p-adic Hodge theory. One of our main results in deformation theory is the existence of deformation rings for Gal(Kbar/K_infty)-representations "of height <= h" for any po

We show that the integral canonical models of Hodge-type Shimura varieties at odd good reduction primes admits ‘ $p$ -adic uniformization’ by Rapoport–Zink spaces of Hodge type constructed in Kim [ Forum Math. Sigma 6 (2018) e8, 110 MR 3812116].

We use E. Lau's classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial. We apply this to prove the Tate conjecture for K3 surfaces in characteristic 2.

We review the analog of Fontaine's theory of crystalline $p$-adic Galois representations and their classification by weakly admissible filtered isocrystals in the arithmetic of function fields over a finite field. There crystalline Galois representations are replaced by the Tate modules of so-called local shtukas. We prove that the Tate module functor is fully faithful. In addition to this étale realization of a local shtuka we discuss also the de Rham and the crystalline cohomology realizations

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Kisin and Pappas constructed integral models of Hodge-type Shimura varieties with parahoric level structure at $p&gt;2$, such that the formal neighbourhood of a mod~$p$ point can be interpreted as a deformation space of $p$-divisible group with some Tate cycles (generalising Faltings' construction). In this paper, we study the central leaf and the closed Newton stratum in the formal neighbourhoods of mod~$p$ points of Kisin-Pappas integral models with parahoric level structure; namely, we obtain