By Eyal Z. Goren
This publication is dedicated to yes facets of the speculation of $p$-adic Hilbert modular kinds and moduli areas of abelian kinds with genuine multiplication. the idea of $p$-adic modular types is gifted first within the elliptic case, introducing the reader to key rules of N. M. Katz and J.-P. Serre. it truly is re-interpreted from a geometrical perspective, that is built to offer the rudiments of an analogous thought for Hilbert modular varieties. the speculation of moduli areas of abelian types with actual multiplication is gifted first very explicitly over the advanced numbers. elements of the overall concept are then uncovered, specifically, neighborhood deformation thought of abelian kinds in optimistic attribute. The mathematics of $p$-adic Hilbert modular types and the geometry of moduli areas of abelian forms are comparable. This relation is used to review $q$-expansions of Hilbert modular varieties, at the one hand, and stratifications of moduli areas nonetheless. The publication is addressed to graduate scholars and non-experts. It makes an attempt to supply the mandatory historical past to all strategies uncovered in it. it could function a textbook for a sophisticated graduate direction.
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