Nonemptiness of Newton strata of Shimura varieties of Hodge type

Abstract

For a Shimura variety of Hodge type with hyperspecial level at a prime $p$, the Newton stratification on its special fiber at $p$ is a stratification defined in terms of the isomorphism class of the rational Dieudonné module of parameterized abelian varieties endowed with a certain fixed set of Frobenius-invariant crystalline tensors (“$G_{{\mathbb{Q}}_p}$-isocrystal”). There has been a conjectural group-theoretic description of the $F$-isocrystals that are expected to show up in the special fiber. We confirm this conjecture. More precisely, for any $G_{{\mathbb{Q}}_p}$-isocrystal that is expected to appear (in a precise sense), we construct a special point whose reduction has associated $F$-isocrystal equal to the given one.