Non-emptiness of Newton strata of Shimura varieties of Hodge type

TL;DR

This paper proves the non-emptiness of Newton strata in Shimura varieties of Hodge type by confirming a conjecture through two methods, constructing points with prescribed F-isocrystals and producing Kottwitz triples, thus advancing understanding of their structure.

Contribution

It confirms a conjecture on the group-theoretic description of F-isocrystals in Shimura varieties of Hodge type using two distinct approaches.

Findings

Confirmed the conjectural description of F-isocrystals in the special fiber.

Constructed points with prescribed F-isocrystals in the reduction.

Produced Kottwitz triples corresponding to expected F-isocrystals.

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 Dieudonne module of parameterized abelian varieties endowed with a certain fixed set of Frobenius-invariant crystalline cycles ("$F$-isocrystal with $G_{\mathbb{Q}_p}$-structure"). 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 by two different methods. More precisely, for any $F$-isocrystal with $G_{\mathbb{Q}_p}$-structure that is expected to appear (in a precise sense), first we construct a special point which has good reduction and whose reduction has associated $F$-isocrystal equal to given one. Secondly, we produce a Kottwtiz triple (with trivial Kottwitz invariant) with the $F$-isocrystal component being the given one. According to a recent result of Kisin which establishes the Langlands-Rapoport conjecture, such Kottwitz triple arises from a point in the reduction.