# A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds

## TL;DR

This paper proves the weight-monodromy conjecture for a specific Siegel threefold, constructs a geometric Jacquet-Langlands correspondence, and applies these results to automorphic forms and the Langlands program.

## Contribution

It establishes the weight-monodromy conjecture for paramodular Siegel threefolds and constructs a geometric Jacquet-Langlands correspondence using Rapoport-Zink uniformisation.

## Key findings

- Proved the weight-monodromy conjecture for the middle degree cohomology.
- Constructed a geometric Jacquet-Langlands correspondence for GSp(4).
- Derived a level lowering result for automorphic representations.

## 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 $\operatorname{GSp}_4$ 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 $\operatorname{GSp}_4$.