[Paper Review] Geometric Satake, categorical traces, and arithmetic of Shimura varieties

Xinwen Zhu|arXiv (Cornell University)|Oct 17, 2018

Advanced Algebra and Geometry32 references3 citations

TL;DR

This paper establishes a categorical trace construction for the geometric Satake equivalence over p-adic groups, using motivic and cohomological correspondences to connect geometric representation theory with arithmetic geometry. It applies this framework to study cohomological correspondences and Tate cycles on mod p fibers of Shimura varieties, providing a conceptual bridge between geometric Langlands and arithmetic problems via categorical traces and motivic Satake categories.

ABSTRACT

We survey some recent work on the geometric Satake of p-adic groups and its applications to some arithmetic problems of Shimura varieties. We reformulate a few constructions appeared in the previous works more conceptually.

Motivation & Objective

• To reformulate the geometric Satake equivalence for p-adic groups in a motivic framework, offering a conceptual foundation for arithmetic applications.
• To develop a categorical trace construction applicable to geometric Satake, generalizing trace-like operations in sheaf theory.
• To apply this framework to study cohomological correspondences and Tate cycles on mod p fibers of Shimura varieties.
• To establish a conceptual link between geometric representation theory and arithmetic problems in Shimura varieties via categorical traces.
• To provide a unified, category-theoretic interpretation of trace constructions in the context of automorphic forms and arithmetic geometry.

Proposed method

• Introduces a motivic Satake category as a conceptual model for the geometric Satake equivalence over p-adic groups.
• Develops a categorical trace construction using cohomological correspondences in derived categories, generalizing classical trace maps.
• Applies the categorical trace to the geometric Satake category via moduli spaces of local Shtukas, particularly the category PHk(Shtloc¯k).
• Uses base change and proper pushforward formalism to define and compose cohomological correspondences in a 2-categorical setting.
• Employs the Barr-Beck-Lurie theorem to describe the ∞-categorical enhancement of the category of cohomological correspondences.
• Establishes a factorization of the pushforward functor f! through the category of cohomological correspondences, ensuring full faithfulness under properness.

Experimental results

Research questions

• RQ1 How can the geometric Satake equivalence for p-adic groups be reformulated in a motivic, categorical framework?
• RQ2 What is the role of the categorical trace construction in connecting geometric Satake to arithmetic invariants of Shimura varieties?
• RQ3 How do cohomological correspondences on mod p fibers of Shimura varieties arise from the categorical trace of geometric Satake?
• RQ4 Can the categorical trace construction be used to prove the existence of Tate cycles on mod p fibers of Shimura varieties?
• RQ5 What is the relationship between the motivic Satake category and the arithmetic geometry of local Shtukas?

Key findings

• The motivic Satake category provides a conceptual framework for the geometric Satake equivalence over p-adic groups, generalizing the classical Satake isomorphism.
• The categorical trace construction is defined via cohomological correspondences and is compatible with base change and composition, enabling trace-like operations in derived categories.
• The categorical trace of the geometric Satake category is realized through the moduli space of local Shtukas, leading to a natural action on cohomology.
• The construction yields explicit cohomological correspondences between mod p fibers of Shimura varieties, generalizing classical Hecke correspondences.
• The framework proves that certain cycles on mod p fibers of Shimura varieties are Tate cycles, by realizing them as images of the categorical trace.
• The category of cohomological correspondences is shown to be equivalent to the pushforward category under proper morphisms, via a monadic description using proper pushforwards.