Level lowering on Siegel modular threefold of paramodular level

TL;DR

This paper proves level lowering results for cuspidal automorphic representations on Siegel modular threefolds with paramodular level, using geometric methods and a comparison of vanishing cycles.

Contribution

It introduces a geometric approach to level lowering for Siegel modular threefolds, adapting Ribet's method to a new setting involving quaternionic Shimura varieties.

Findings

Established level lowering results for specific automorphic representations.

Connected the geometry of Siegel threefolds with quaternionic Shimura varieties.

Provided a new proof technique based on vanishing cycles comparison.

Abstract

In this article we prove several level lowering results for cuspidal automorphic representations occurring in the cohomology of the Siegel modular threefold with paramodular level structure by adapting a method of Ribet in his proof of the Serre's epsilon conjecture. The proof is purely geometric and relies on the description of the supersingular locus of certain quaternionic unitary Shimura variety and an arithmetic level raising result on this Shimura variety. The heart of the proof is a comparison of the dimension of the space of vanishing cycles on the paramodular Siegel modular threefold with that on the quaternionic unitary Shimura variety.