Towards an Eichler-Shimura decomposition for ordinary p-adic Siegel modular forms

There are two different ways to construct families of ordinary p-adic Siegel modular forms. One is by p-adically interpolating classes in Betti cohomology, first introduced by Hida and then given a more representation-theoretic interpretation by Emerton. The other is by p-adically interpolating classes in coherent cohomology, once again pioneered by Hida and generalised in recent years by Boxer and Pilloni. I will explain these two constructions and then discuss joint work in progress with James Newton and Juan Esteban Rodrıguez Camargo that aims to compare them.