The Taylor-Wiles method for coherent cohomology, II

TL;DR

This work extends the Taylor–Wiles method to coherent cohomology of non-compact Shimura varieties $S_K(G,X)$ attached to unitary groups with smooth integral models, leveraging toroidal compactifications and Lan–Suh vanishing to control interior cohomology. By integrating Nakajima’s freeness results for group actions with a derived-category, diamond-operator framework, the authors prove that, after localizing at a non-Eisenstein maximal ideal, coherent and de Rham cohomology are free over the localized Hecke algebra and that the boundary contributions are Eisenstein. They establish that congruence ideals attached to coherent cohomological realizations are independent of signatures, and they verify a general Gorenstein property for ordinary $p$-adic families, enabling $p$-adic $L$-function constructions within EHLS. Consequently, these results generalize previous compact-case work to non-compact unitary Shimura varieties and provide structural insights into automorphic Galois representations and their $p$-adic interpolations. The paper thus advances automorphy-lifting-type conclusions for coherent cohomology and strengthens the bridge between geometric boundary analysis and arithmetic applications.

Abstract

We show that the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\CF$ , when $S$ has a smooth model over a $p$-adic integer ring. This generalizes the main results of the article \cite{H13}, which treated the case when $S$ is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle $\CF$ and the prime $p$. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to étale coverings of appropriate toroidal compactifications. The result is applied to show that, when the Taylor-Wiles method applies, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which $G$ is attached. We also show that the Gorenstein hypothesis used to construct $p$-adic $L$-functions in \cite{EHLS} is valid under rather general hypotheses.

Theorems & Definitions (63)

• Definition 9
• Proposition 10
• Remark 11
• Theorem 12
• Theorem 15
• Proposition 17
• proof
• Corollary 19
• Proposition 22
• Theorem 24