Ghost classes in $\mathbb{Q}$-rank two orthogonal Shimura varieties

TL;DR

This paper investigates the existence of ghost classes in certain orthogonal Shimura varieties, using cohomological and Eisenstein cohomology techniques, and establishes non-existence results for specific cases.

Contribution

It provides new non-existence results for ghost classes in orthogonal Shimura varieties of signature (2, n) for n=4,5, and derives restrictions on ghost class weights in other cases.

Findings

Non-existence of ghost classes for n=4,5 in most irreducible representations.

Strong restrictions on weights of ghost classes in remaining cases.

Ghost classes satisfy the weak middle weight property.

Abstract

In this article, the existence of ghost classes for the Shimura varieties associated to algebraic groups of orthogonal similitudes of signature (2, n) is investigated. We make use of the study of the weights in the mixed Hodge structures associated to the corresponding cohomology spaces and results on the Eisenstein cohomology of the algebraic group of orthogonal similitudes of signature (1, n-1). For the values of n = 4, 5 we prove the non-existence of ghost classes for most of the irreducible representations (including most of those with an irregular highest weight). For the rest of the cases, we prove strong restrictions on the possible weights in the space of ghost classes and, in particular, we show that they satisfy the weak middle weight property.