We determine the irreducible components of Igusa varieties for Shimura varieties of Hodge type and use that to determine the irreducible components of central leaves. In particular, we show that the discrete Hecke-orbit conjecture is false in general
. Our method combines recent work of DAddezio on monodromy of compatible local systems with a generalisation of a method of Hida, using the Honda-Tate theory for Shimura varieties of Hodge type developed by Kisin-Madapusi Pera-Shin. We also determine the irreducible components of Newton strata in Shimura varieties of Hodge type by combining our results with recent work of Zhou-Zhu.
We study the mod $p$-points of the Kisin-Pappas integral models of abelian type Shimura varieties with parahoric level structure. We show that if the group is quasi-split and unramified, then the mod $p$ isogeny classes are of the form predicted by t
he Langlands-Rapoport conjecture (c.f. Conjecture 9.2 of arXiv:math/0205022). We prove the same results for quasi-split and tamely ramified groups when their Shimura varieties are proper. The main innovation in this work is a global argument that allows us to reduce the conjecture to the case of a very special parahoric, which is handled in the appendix. This way we avoid the complicated local problem of understanding connected components of affine Deligne-Lusztig varieties for general parahoric subgroups. Along the way, we give a simple irreducibility criterion for Ekedahl-Oort and Kottwitz-Rapoport strata.
We study the Picard-Lefschetz formula for the Siegel modular threefold of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology with arbitrary automorphic coefficients. We give some applications to the Lan
glands programme: Using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet-Langlands correspondence between $operatorname{GSp}_4$ and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of $operatorname{GSp}_4$.
