On Higher regulators of Siegel varieties

TL;DR

This paper constructs motivic cohomology classes for Siegel varieties, computes their regulators via automorphic integrals, and generalizes previous constructions for specific symplectic groups, advancing understanding of higher regulators.

Contribution

It introduces a new method to construct higher motivic cohomology classes for Siegel varieties of arbitrary dimension, extending previous work on specific cases.

Findings

Constructed motivic cohomology classes in middle degree plus one.

Computed Beilinson's regulator images using Rankin-Selberg integrals.

Generalized constructions from GSp(4) and GSp(6) to higher dimensions.

Abstract

We construct classes in the middle degree plus one motivic cohomology of the Siegel Shimura variety of almost any dimension. We compute their image by Beilinson's higher regulator in terms of Rankin-Selberg type automorphic integrals. Our construction generalises the one for $\mathrm{GSp}(4)$ and for $\mathrm{GSp}(6)$. For Siegel varieties associated to small genus symplectic groups, we also show how these integrals unfold.