Tempered currents and Deligne cohomology of Shimura varieties, with an application to $\mathrm{GSp}_6$

TL;DR

This paper introduces a new way to describe Deligne-Beilinson cohomology for Shimura varieties using tempered currents, facilitating regulator computations and advancing the understanding of Beilinson conjectures.

Contribution

It provides a novel description of Deligne-Beilinson cohomology for Shimura varieties via tempered currents, enabling explicit regulator calculations and applications to special values of L-functions.

Findings

Constructed classes in motivic cohomology of Siegel sixfolds.

Computed their images under Beilinson regulators using automorphic integrals.

Linked integrals to special values of Spin L-functions as predicted by Beilinson.

Abstract

We provide a new description of Deligne-Beilinson cohomology for any Shimura variety in terms of tempered currents. This is particularly useful for computations of regulators of motivic classes and hence to the study of Beilinson conjectures. As an application, we construct classes in the middle degree plus one motivic cohomology of Siegel sixfolds and we compute their image by Beilinson higher regulator in terms of Rankin-Selberg type automorphic integrals. Using results of Pollack and Shah, we relate the integrals to noncritical special values of the degree $8$ Spin $L$-functions, as predicted by Beilinson conjectures.