Motivic action on coherent cohomology of Hilbert modular varieties

TL;DR

This paper introduces a motivic cohomology action on the coherent cohomology of Hilbert modular varieties, extending existing conjectures and supported by both theoretical and numerical evidence.

Contribution

It proposes a new motivic cohomology action on Hilbert modular varieties' cohomology, extending conjectures and providing evidence for integral lifts.

Findings

The action is described on cohomology modulo p and over complex numbers.

Conjecture supported by Stark's conjecture and numerical evidence.

Provides a framework for understanding cohomology actions in arithmetic geometry.

Abstract

We propose an action of a certain motivic cohomology group on the coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo $p$ and over $\mathbb C$, and we conjecture that they both lift to an action on cohomology with integral coefficients. The conjecture is supported by theoretical evidence based on Stark's conjecture on special values of Artin $L$-functions, and by numerical evidence in base change cases.