A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds

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 Langlands 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 $\operatorname{GSp}_4$.Comment: Almost final version, to appear in Math.

THE LANGLANDS-RAPOPORT CONJECTURE (Automorphic forms, Automorphic representations, Galois representations, and its related topics)

We give a brief introduction to the Langlands-Rapoport conjecture, which describes the mod p points of Shimura varieties. We overview known results and explain what is missing to deal with the general case

On the ordinary Hecke orbit conjecture

We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge type at primes of good reduction. We make use of the global Serre-Tate coordinates of Chai as well as recent results of D'Addezio about the $p$-adic monodromy of isocrystals. The new ingredients in this paper are a general monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge type, and a rigidity result for the formal completions of ordinary Hecke orbits. Along the way we show that classical Serre--Tate coordinates can be described using unipotent formal groups, generalising results of Howe.Comment: 38 pages; v2 is a significantly revised version of v1; main results unchange