On the strict endoscopic part of modular Siegel threefolds

TL;DR

This paper investigates the strict endoscopic components of the cohomology of modular Siegel threefolds, explicitly constructing parts via theta lifts and confirming a conjecture on their Hodge-theoretic description.

Contribution

It identifies a non-zero subspace of the strict endoscopic part constructed by theta lifts and proves a conjecture relating to their Hodge structures for level one threefolds.

Findings

Existence of a non-zero subspace constructed by theta lifts.

Explicit calculation of levels of lifted forms into GSp(4).

Proof of Faber and van der Geer's conjecture on Hodge structures.

Abstract

In this paper we study the non-holomorphic strict endoscopic parts of inner cohomology spaces of a modular Siegel threefold respect to local systems. First we show that there is a non-zero subspace of the strict endoscopic part such that it is constructed by global theta lift of automorphic froms of $(GL(2)\times GL(2))/G_{m}$. Secondly, we present an explicit analytic calculation of levels of lifted forms into GSp(4), based on the paramodular representations theory for $GSp(4; F)$. Finally, we prove the conjecture, by C. Faber and G. van der Geer, that gives a description of the strict endoscopic part for Betti cohomology and (real) Hodge structures in the category of mixed Hodge structures, in which the modular Siegel threefold has level structure one.