Non-vanishing mod $p$ of derived Hecke algebra of the multiplicative group over number field

TL;DR

This paper proves that the degree one part of the derived Hecke action on the cohomology of an arithmetic manifold related to the multiplicative group over a number field does not vanish modulo p, using the Grunwald--Wang theorem.

Contribution

It establishes the non-vanishing modulo p of the degree one derived Hecke action for the multiplicative group over a number field, under mild assumptions.

Findings

Degree one derived Hecke action is non-vanishing mod p.

Uses Grunwald--Wang theorem as a key ingredient.

Provides new insights into the structure of cohomology in arithmetic manifolds.

Abstract

We investigate the derived Hecke action on the cohomology of an arithmetic manifold associated to the multiplicative group over a number field. The degree one part of the action is proved to be non-vanishing modulo $p$ under mild assumptions. The main ingredient is the Grunwald--Wang theorem.