Neumann problems for p -harmonic functions, and induced nonlocal operators in metric measure spaces

Abstract: We show that, under certain specific hypotheses, 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 $\mathcal{F}$, when $S$ has a smooth model over a $p$-adic integer ring. As an application, we show that, when the hypotheses are satisfied, 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 the second author's article with Eischen, Li, and Skinner, as elements of Hida's ordinary Hecke algebra, is valid rather generally. The present paper generalizes the main results of an earlier paper by the second author, 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 $\mathcal{F}$ 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 \'etale coverings of appropriate toroidal compactifications.