Compact Support Cohomology of Picard Modular Surfaces

TL;DR

This paper computes the compact support cohomology of Picard modular surfaces using advanced trace formulas, linking Galois actions and Hecke algebra representations, building on foundational methods by Ihara, Langlands, and Kottwitz.

Contribution

It introduces a novel application of the Grothendieck-Lefschetz and Arthur-Selberg trace formulas to Picard modular surfaces, extending previous work by Laumon and Morel.

Findings

Explicit computation of cohomology as a virtual module

Integration of trace formulas with Galois and Hecke actions

Advancement in understanding the arithmetic of Picard modular surfaces

Abstract

We compute the cohomology with compact supports of a Picard modular surface as a virtual module over the product of the appropriate Galois group and the appropriate Hecke algebra. We use the method developed by Ihara, Langlands, and Kottwitz: comparison of the Grothendieck-Lefschetz formula and the Arthur-Selberg trace formula. Our implementation of this method takes as its starting point the works of Laumon and Morel.