Good Reductions of Shimura Varieties of Preabelian Type in Arbitrary Unramified Mixed Characteristic, I

Abstract

We prove the existence of weak integral canonical models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence of integral canonical models of Shimura varieties of preabelian (resp. of abelian) type in mixed characteristic $(0,p)$ with $p\Ge 3$ (resp. with $p=2$) and with respect to hyperspecial subgroups; if $p=3$ (resp. if $p=2$) we restrict in this part I either to the $A_n$, $C_n$, $D_n^{\dbH}$ (resp. $A_n$ and $C_n$) types or to the $B_n$ and $D_n^{\dbR}$ (resp. $B_n$, $D_n^{\dbH}$ and $D_n^{\dbR}$) types which have compact factors over $\dbR$ (resp. which have compact factors over $\dbR$ in some $p$-compact sense). Though the second application is new just for $p\Le 3$, a great part of its proof is new even for $p\Ge 5$ and corrects [Va1, 6.4.11] in most of the cases. The second application forms progress towards the proof of a conjecture of Milne. It also provides in arbitrary mixed characteristic the very first examples of general nature of projective varieties over number fields which are not embeddable into abelian varieties and which have N\'eron models over certain local rings of rings of integers of number fields.