A Note on Systems of Realizations on Shimura Varieties

Abstract

Let $(G, \Omega)$ be a Shimura datum of abelian type. It is well known that the corresponding Shimura variety $\mathrm{Sh}(G, \Omega)$ should be a moduli space of abelian motives equipped with some additional structures. In this half-expository note, we give under some simplifying assumptions a moduli interpretation of $\mathrm{Sh}(G, \Omega)$ over the reflex field purely in terms of systems of realizations. The main purpose is to introduce some convenient formalism that can be used to avoid the technicalities about dealing with various notions of families of motives.