Norm-Compatible Systems of Galois Cohomology Classes for ${\mathbf{GSp}}_6$

We construct global cohomology classes in the middle degree plus one cohomology group of the Shimura variety of the symplectic group ${\mathbf{GSp}}_6$ compatible when one varies the level at $p$. These classes are expected constituents of an Euler system for the Galois representations appearing in the middle degree étale cohomology groups of the aforementioned variety. 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.