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

TL;DR

This paper extends the theory of p-adic families of modular forms to Hodge type Shimura varieties with non-empty ordinary locus, providing geometric definitions, interpolation in families, and constructing eigenvarieties.

Contribution

It generalizes previous results to a broader class of Shimura varieties, defining overconvergent modular forms geometrically and constructing eigenvarieties in simple cases.

Findings

Sheaves of overconvergent modular forms interpolate classical sheaves in families.

An action of the Hecke algebra, including a p-adic operator, is established.

Eigenvarieties are constructed in certain cases.

Abstract

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 $\kappa$, we give a geometric definition of the space of overconvergent modular forms of weight $\kappa$ 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.