A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds

Abstract

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 ${\mathrm{GSp}}_4$ 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 ${\mathrm{GSp}}_4$.

Almost final version, to appear in Math. Z

References in corpus (2)

• Arthur's multiplicity formula for GSp${}_4$ and restriction to Sp${}_4$
• Serre weights for $GS{p}_4$ over totally real fields

Cited by in corpus (2)

• On the Bruhat-Tits stratification of a quaternionic unitary Shimura variety
• Serre weights for $GS{p}_4$ over totally real fields