p-adic families of modular forms for Hodge type Shimura varieties with non-empty ordinary locus

We generalize some of the results of Andreatta, Iovita, and Pilloni and the author to Hodge type Shimura varieties having non-empty ordinary locus. For any p-adic weight κ, we give a geometric definition of the space of overconvergent modular forms of weight κ in terms of sections of a sheaf. We show that our sheaves live in analytic families, interpolating the classical sheaves for integral weights. We define an action of the Hecke algebra, including a completely continuous operator at p. In some simple cases, we also build the eigenvariety.