A geometric Jacquet–Langlands correspondence for paramodular Siegel threefolds

Abstract We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove weight-monodromy conjecture its middle degree inner cohomology. give some applications to Langlands programme: using Rapoport-Zink uniformisation supersingular locus special fiber, we construct a geometric Jacquet–Langlands correspondence between $${\text {GSp}}_4$$ GSp 4 definite form, proving Ibukiyama. also an integral version use it deduce lowering result cohomological cuspidal automorphic representations .