Lifting non-ordinary cohomology classes for SL(3)

TL;DR

This paper extends the theory of lifting eigenclasses in the cohomology of SL(3) from ordinary to non-ordinary and non-critical slope classes, broadening the scope of rigid analytic cohomology constructions.

Contribution

It generalizes Pollack and Pollack's theorem to include non-ordinary and non-critical slope classes in the cohomology of SL(3), providing new lifting results.

Findings

Proves lifting of non-ordinary eigenclasses under tighter conditions.

Extends the construction of overconvergent classes to non-critical slopes.

Broadens the applicability of cohomology class lifting in arithmetic groups.

Abstract

In this paper, we present a generalisation of a theorem of David and Rob Pollack. In 'A construction of rigid analytic cohomology classes for congruence subgroups of SL(3,Z)', they give a very general argument for lifting ordinary eigenclasses (with respect to a suitable operator) in the group cohomology of certain arithmetic groups. With slightly tighter conditions, we prove the same result for non-ordinary classes. Pollack and Pollack apply their results to the case of p-ordinary classes in the group cohomology of congruence subgroups for SL3, constructing explicit overconvergent classes in this setting. As an application of our results, we give an extension of their results to the case of non-critical slope classes in the same setting.