We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology. 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 \({\text {GSp}}_4\) 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 \({\text {GSp}}_4\).

我们研究了Siegel模数级三重模的Picard–Lefschetz公式，并证明了其重度单调猜想的中度内部同调性。我们为Langlands程序提供了一些应用程序：使用特殊纤维超奇异轨迹的Rapoport-Zink均匀化，我们构造\（{\ text {GSp}} _ 4 \）与确定的内部形式之间的几何Jacquet-Langlands对应关系，证明了Ibukiyama的猜想。我们还证明了权重单论猜想的积分形式，并用它来推导\（{\ text {GSp}} _ 4 \）的同位尖峰自同构表示的水平降低结果。