On the cohomology of $GL_3$ of elliptic curves and Quillen's conjecture

TL;DR

This paper computes the cohomology of $GL_3$ over elliptic curve function rings, revealing complex torsion structures and providing insights into Quillen's conjecture in the context of arithmetic groups.

Contribution

It offers a detailed computation of cohomology and Chern-class modules for $GL_3$ over elliptic curves, advancing understanding of Quillen's conjecture in this setting.

Findings

Explicit torsion classes in cohomology

Failure of torsion-free quotient to be free

Insights into the structure of cohomology rings

Abstract

The paper provides a computation of the additive structure as well as a partial description of the Chern-class module structure of the cohomology of $GL_3$ over the function ring of an elliptic curve over a finite field. The computation is achieved by a detailed analysis of the isotropy spectral sequence for the action of $GL_3$ on the associated Bruhat-Tits building. This provides insights into the function field version of Quillen's conjecture on the structure of cohomology rings of arithmetic groups. The computations exhibit a lot of explicit classes which are torsion for the Chern-class ring. In some examples, even the torsion-free quotient of cohomology fails to be free. A possible variation of Quillen's conjecture is also discussed.