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Compactifications of Iwahori-level Hilbert modular varieties

Fred Diamond

TL;DR

The paper addresses the construction and analysis of compactifications for $p$-adic Hilbert modular varieties with level structures including Iwahori and refined $p$-level, unifying Deligne--Pappas, Pappas--Rapoport, and Lan frameworks. It extends toroidal and minimal compactifications to these levels, proves Kodaira--Spencer isomorphisms and cohomological vanishing in the toroidal setting, and develops the saving trace to connect compactifications with Hecke operators at primes over $p$. The work also develops a robust theory of $q$-expansions and Hecke actions across multiple level structures, including $U_0({ rak P})$ and $U_1({ rak P})$, over general base rings, and provides explicit local descriptions at cusps and clasps. The results enable precise control of modular forms in integral settings, yield a coherent framework for $p$-adic deformation and base-change, and have implications for Galois representations and arithmetic geometry of Hilbert modular varieties. Overall, the paper advances the arithmetic geometry of Hilbert modular varieties by integrating compactifications, level structures, and Hecke action in a flexible, base-ring compatible framework.”

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.

Compactifications of Iwahori-level Hilbert modular varieties

TL;DR

The paper addresses the construction and analysis of compactifications for -adic Hilbert modular varieties with level structures including Iwahori and refined -level, unifying Deligne--Pappas, Pappas--Rapoport, and Lan frameworks. It extends toroidal and minimal compactifications to these levels, proves Kodaira--Spencer isomorphisms and cohomological vanishing in the toroidal setting, and develops the saving trace to connect compactifications with Hecke operators at primes over . The work also develops a robust theory of -expansions and Hecke actions across multiple level structures, including and , over general base rings, and provides explicit local descriptions at cusps and clasps. The results enable precise control of modular forms in integral settings, yield a coherent framework for -adic deformation and base-change, and have implications for Galois representations and arithmetic geometry of Hilbert modular varieties. Overall, the paper advances the arithmetic geometry of Hilbert modular varieties by integrating compactifications, level structures, and Hecke action in a flexible, base-ring compatible framework.”

Abstract

We study minimal and toroidal compactifications of -integral models of Hilbert modular varieties. We review the theory in the setting of Iwahori level at primes over , 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 -expansions of Hilbert modular forms, especially the effect of Hecke operators at primes over over general base rings.
Paper Structure (41 sections, 16 theorems, 107 equations)

This paper contains 41 sections, 16 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Level prime to $p$
  3. The Deligne--Pappas model
  4. Cusps of level $U$
  5. Toroidal compactification of $\widetilde{Y}_-$
  6. Relation with Lan's definition
  7. Minimal compactification of $\widetilde{Y}_-$
  8. Minimal compactification of ${Y}_-$
  9. The Pappas--Rapoport model
  10. Toroidal compactification of $\widetilde{Y}$
  11. Minimal compactification of $\widetilde{Y}$
  12. Minimal compactification of $Y$
  13. Iwahori level at $p$
  14. The Iwahori-level model
  15. Cusps of level $U_0({\mathfrak{P}})$
  16. ...and 26 more sections

Key Result

Theorem 3.3.1

There is a normal scheme $\widetilde{Y}_0({\mathfrak{P}})^{{\operatorname{tor}}}$ over ${\mathcal{O}}$ and an open immersion $\widetilde{Y}_0({\mathfrak{P}}) \hookrightarrow\widetilde{Y}_0({\mathfrak{P}})^{{\operatorname{tor}}}$ with the following properties:

Theorems & Definitions (23)

  • Theorem 3.3.1
  • Theorem 3.5.1
  • Lemma 3.6.1
  • Proposition 3.7.1
  • Lemma 4.4.1
  • proof
  • Corollary 4.5.1
  • Proposition 4.5.2
  • Proposition 4.6.1
  • Theorem 5.1.1
  • ...and 13 more