Compactifications of Iwahori-level Hilbert modular varieties

TL;DR

This paper investigates the structure and properties of compactifications of Hilbert modular varieties with Iwahori level structure, extending existing theories and analyzing the impact on modular forms and Hecke operators.

Contribution

It extends the theory of compactifications to finer level structures and applies it to study $q$-expansions and Hecke operators at primes over $p$.

Findings

Extended compactification theory to finer level structures.

Proved new results on Kodaira--Spencer isomorphisms and cohomological vanishing.

Analyzed effects of Hecke operators on $q$-expansions over general base rings.

Abstract

We study minimal and toroidal compactifications of $p$-integral models of Hilbert modular varieties. We review the theory in the setting of Iwahori level at primes over $p$, and extend it to certain finer level structures. We also prove extensions to compactifications of recent results on Iwahori-level Kodaira--Spencer isomorphisms and cohomological vanishing for degeneracy maps. Finally we apply the theory to study $q$-expansions of Hilbert modular forms, especially the effect of Hecke operators at primes over $p$ over general base rings.