Derived Hecke action on the trivial cohomology of division algebras

TL;DR

This paper extends Venkatesh's theorem on derived Hecke actions from imaginary quadratic fields to all number fields, showing the stable cohomology is a free module generated by the unit class, with a rational form preserving structures.

Contribution

It generalizes the structure theorem for derived Hecke actions to all number fields and introduces a rational form of the strict derived Hecke algebra.

Findings

Stable submodule is a free module generated by the unit class.

The strict derived Hecke algebra has a rational form preserving rational structures.

New results on the reduction map in K-theory of number fields.

Abstract

This article generalizes Venkatesh's structure theorem for the derived Hecke action on the Hecke trivial cohomology of a division algebra over an imaginary quadratic field to division algebras over all number fields. In particular, we show that the stable submodule of the Hecke trivial cohomology attached to a division algebra is a free module generated by the unit class for the action of the strict derived Hecke algebra. Moreover, the strict derived Hecke algebra possesses a rational form that preserves the canonical rational structure on the stable cohomology during the derived Hecke action. The main ingredients in our improvement are a careful study of the congruence classes in the torsion cohomology of the arithmetic manifold and the author's new result on the reduction map in the $K$-theory of the ring of integers in number fields.