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arXiv · 1906.04008

A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds

Pol van Hoften
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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 $\operatorname{GSp}_4$.

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Pol van Hoften. 2019-06-10. A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds. https://doi.org/10.1007/s00209-021-02756-0

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