Normalization in the integral models of Shimura varieties of abelian type

Abstract

Let $(G,X)$ be a Shimura datum of Hodge type, and $\mathscr{S}_K(G,X)$ its integral model with hyperspecial (resp.~parahoric, assuming the group is unramified) 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. More generally, for integral models of abelian type Shimura varieties, since the normalization step occurs only in the associated Hodge type constructions, our result also removes normalization from the construction of general abelian type integral models. In particular, our results apply to the earlier integral models constructed by Rapoport, Kottwitz etc.~(resp.~Rapoport-Zink etc.) and various other constructions in existing literature, as those models agree with the Hodge (or abelian) type integral models for appropriately chosen Shimura data.

Moreover, combined with a result of Lan's on the boundary components of toroidal compactifications of integral models of Shimura varieties, our result also implies that there exist closed embeddings of toroidal compactifications of integral models of Hodge type into toroidal compactifications of Siegel integral models, for suitable choices of cone decompositions.

In the case of PEL type integral models of Shimura varieties, we show that finiteness of the Hodge morphism boils down to finiteness of certain $H^1_{\mathrm{fppf}}$. Moreover, we give a simple down-to-earth proof for the Hodge embedding for (potentially exotic) PEL type integral models. Such results apply to certain exotic models not covered by Kisin (resp.~Kisin-Pappas) models.