[Paper Review] The stabilization of the Frobenius--Hecke traces on the intersection cohomology of orthogonal Shimura varieties

Yihang Zhu|arXiv (Cornell University)|Jan 29, 2018

Advanced Algebra and Geometry21 references5 citations

TL;DR

This paper establishes the stabilization of Morel's formula for Frobenius–Hecke traces on the intersection cohomology of orthogonal Shimura varieties over Q, using the Langlands–Kottwitz method and endoscopic classification. The key result is a stabilized trace formula that enables explicit computation of the Hasse–Weil zeta function of the intersection cohomology in special cases via recent work of Arthur and Taïbi on automorphic representations.

ABSTRACT

We study Shimura varieties associated with special orthogonal groups over the field of rational numbers. We prove a version of Morel's formula for the Frobenius--Hecke traces on the intersection cohomology of the Baily--Borel compactification. Our main result is the stabilization of this formula. As an application, we compute the Hasse--Weil zeta function of the intersection cohomology in some special cases, using the recent work of Arthur and Taïbi on the endoscopic classification of automorphic representations of special orthogonal groups.

Motivation & Objective

• To extend Morel's formula for Frobenius–Hecke traces to the intersection cohomology of Baily–Borel compactifications of orthogonal Shimura varieties.
• To stabilize this trace formula using the Langlands–Kottwitz method and endoscopic techniques.
• To compute the Hasse–Weil zeta function of the intersection cohomology in special cases using the endoscopic classification of automorphic representations.
• To provide a cohomological framework for studying arithmetic invariants of orthogonal Shimura varieties through trace formula methods.

Proposed method

• Adapt the Langlands–Kottwitz method to the setting of intersection cohomology of orthogonal Shimura varieties.
• Use truncated Lie algebra cohomology and Kostant–Weyl terms to describe the geometric side of the trace formula.
• Apply Kottwitz's fixed point formula to relate traces on cohomology to stable orbital integrals.
• Construct integral models of Shimura varieties to ensure arithmetic compatibility in trace computations.
• Leverage the endoscopic classification of automorphic representations of special orthogonal groups (Arthur–Taïbi) to stabilize the spectral side.
• Establish a comparison between Kostant–Weyl terms and stable discrete series characters to match geometric and spectral terms.

Experimental results

Research questions

• RQ1 How can Morel’s formula for Frobenius–Hecke traces be extended to the intersection cohomology of Baily–Borel compactifications of orthogonal Shimura varieties?
• RQ2 What is the stabilized form of the trace formula for these cohomology groups, and how does it relate to endoscopic data?
• RQ3 How can the stabilized trace formula be used to compute the Hasse–Weil zeta function of the intersection cohomology in specific cases?
• RQ4 What is the precise relationship between Kostant–Weyl terms and stable discrete series characters in the context of orthogonal groups?
• RQ5 To what extent does the endoscopic classification of automorphic representations allow for explicit arithmetic computations in the cohomology of orthogonal Shimura varieties?

Key findings

• The paper proves a stabilized version of Morel’s formula for the Frobenius–Hecke traces on the intersection cohomology of orthogonal Shimura varieties over Q.
• The stabilization is achieved via the Langlands–Kottwitz method, with explicit matching of geometric and spectral terms using endoscopic data.
• The stabilized formula allows for the computation of the Hasse–Weil zeta function of the intersection cohomology in special cases, such as when the associated automorphic representations are discrete series.
• The Kostant–Weyl terms in the formula are shown to match stable discrete series characters, providing a cohomological realization of endoscopic transfer.
• The method establishes a precise link between the cohomology of orthogonal Shimura varieties and the endoscopic classification of automorphic representations, as developed by Arthur and Taïbi.
• The results provide a foundational framework for computing arithmetic zeta functions of Shimura varieties using trace formula techniques and automorphic L-functions.