We construct classes in the middle degree motivic cohomology of the Siegel variety of almost any dimension. We compute their image by Beilinson's higher regulator in terms of Rankin-Selberg type automorphic integrals. In the case of GSp( 6), using results of Pollack and Shah, we relate the integral to a non-critical special value of a degree 8 spin L-function.

We construct global cohomology classes in the middle degree cohomology of the Shimura variety of the symplectic group GSp 6 compatible when one varies the level at p. These classes are expected constituents of an Euler system for the Galois representations appearing in these cohomology groups. As an application, we show how these classes provide elements in the Iwasawa cohomology of these representations and, by applying Perrin-Riou's machinery, p-adic L-functions associated to them.

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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 non-critical special values of the degree 8 Spin L-functions. Along the way, by defining and studying complexes of tempered currents on smooth projective complex varieties endowed with a normal crossings divisor, we provide a new description of Deligne-Beilinson cohomology for any Shimura variety. This is particularly useful for computations of higher regulators and fills a gap in the literature on the subject.

We describe work of Faltings on the construction of étale cohomology classes associated to symplectic Shimura varieties and show that they satisfy certain trace compatibilities similar to the ones of Siegel units in the modular curve case. Starting from those, we construct a two variable family of trace compatible classes in the cohomology of a unitary Shimura variety.