In this talk, we will explain (joint work with Antonio Cauchi and Francesco Lemma) how to construct classes in the middle-degree plus one cohomology of a Siegel Shimura variety of almost any dimension and how to compute their image under Beilinson-Deligne’s higher regulator in terms of Rankin-Selberg type automorphic integrals.

In the case of the Siegel sixfold, we relate the integral to a non-critical special value of a degree 8 spin L-function. If time permits, we will give an application to Iwasawa theory and the theory of Euler systems.