Kodaira-Spencer isomorphisms and degeneracy maps on Iwahori-level Hilbert modular varieties: the saving trace

TL;DR

The work develops integral and mod $p$ theory for Hilbert modular varieties with Iwahori-level structure at primes over $p$ by establishing a Kodaira--Spencer isomorphism for the dualizing sheaf on level-$p$ models and proving a vanishing of higher direct images for degeneracy maps. This cohomological control yields flat Stein factorizations and Cohen–Macaulay models, enabling a direct construction of Hecke operators at $p$ via the saving trace, without derived-category arguments. The results generalize prior work to ramified $p$ in $F$, provide a robust framework for Galois representations attached to mod $p$ Hilbert modular eigenforms of arbitrary weights, and supply tools for integral and mod $p Hecke theory across primes over $p$. Overall, the paper delivers a coherent, self-contained approach to constructing Hecke operators and Galois representations in this arithmetic setting, with explicit local deformation analyses and a detailed understanding of degeneracy fibers. These advances contribute to the broader project of bridging automorphic forms, Galois representations, and arithmetic geometry in the $p$-adic and mod $p$ contexts.

Abstract

We consider integral models of Hilbert modular varieties with Iwahori level structure at primes over p, first proving a Kodaira-Spencer isomorphism that gives a concise description of their dualizing sheaves. We then analyze fibres of the degeneracy maps to Hilbert modular varieties of level prime to p and deduce the vanishing of higher direct images of structure and dualizing sheaves, generalizing prior work with Kassaei and Sasaki (for p unramified in the totally real field F). We apply the vanishing results to prove flatness of the finite morphisms in the resulting Stein factorizations, and combine them with the Kodaira-Spencer isomorphism to simplify and generalize the construction of Hecke operators at primes over p on Hilbert modular forms (integrally and mod p).

Theorems & Definitions (42)

• Theorem A
• Theorem B
• Theorem C
• Corollary D
• Proposition 2.4.1
• Theorem 3.1.1
• Theorem 3.1.2
• Corollary 3.1.3
• Remark 3.1.4
• Theorem 3.2.1