On Higher regulators of Siegel varieties

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.

30 pages. A mistake in the first version was corrected and a hypothesis on a second version was suppressed. Results on the particular case of $\mathrm{GSp}(6)$ have been moved elsewhere. Added unfolding calculations of the Rankin-Selberg integrals for small genus symplectic groups