A Shimura variety of Hodge type is a moduli space for abelian varieties equipped with a certain collection of Hodge cycles. We show that the Newton strata on such varieties are nonempty provided that the corresponding group G is quasisplit at p, confirming a conjecture of Fargues and Rapoport in this case. Under the same condition, we conjecture that every mod p isogeny class on such a variety contains the reduction of a special point. This is a refinement of Honda–Tate theory. We prove a large part of this conjecture for Shimura varieties of PEL type. Our results make no assumption on the availability of a good integral model for the Shimura variety. In particular, the group G may be ramified at p.
"Honda–Tate theory for Shimura varieties." Duke Math. J. 171 (7) 1559 - 1614, 15 May 2022. https://doi.org/10.1215/00127094-2021-0063