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Register a new userHida theory for Shimura varieties of Hodge type
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Xiaoyu Zhang
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In this article, we generalize the work of H.Hida and V.Pilloni to construct $p$-adic families of $mu$-ordinary modular forms on Shimura varieties of Hodge type $Sh(G,X)$ associated to a Shimura datum $(G,X)$ where $G$ is a connected reductive group over $mathbb{Q}$ and is unramified at $p$, such that the adjoint quotient $G^mathrm{ad}$ has no simple factors isomorphic to $mathrm{PGL}_2$.
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Let k be a perfect field of characteristic p>0. We prove the existence of ascending and descending slope filtrations for Shimura p-divisible objects over k. We use them to classify rationally these objects over bar k. Among geometric applications, we mention two. First we formulate Manin problems for Shimura varieties of Hodge type. We solve them if either pGe 3 or p=2 and two mild conditions hold. Second we formulate integral Manin problems. We solve them for certain Shimura varieties of PEL type.
Let $(G,X)$ be a Shimura datum of Hodge type, and $mathscr{S}_K(G,X)$ its integral model with hyperspecial level structure. We prove that $mathscr{S}_K(G,X)$ admits a closed embedding, which is compatible with moduli interpretations, into the integral model $mathscr{S}_{K}(mathrm{GSp},S^{pm})$ for a Siegel modular variety. In particular, the normalization step in the construction of $mathscr{S}_K(G,X)$ is redundant. In particular, our results apply to the earlier integral models constructed by Rapoport and Kottwitz, as those models agree with the Hodge type integral models for appropriately chosen Shimura data.
We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic $(0,p)$ of integral canonical models of projective Shimura varieties of Hodge type with respect to h--hyperspecial subgroups as pro-etale covers of Neron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.
We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type.
This is a survey of the three main methods developed in the last 15 years to prove the existence of integral canonical models of Shimura varieties of Hodge type. The only new part is formed by corrections to results of Kisin.
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