We prove that the generic part of the $\mathrm{mod}\, \ell$ cohomology of Shimura varieties associated to quasi-split unitary groups of even dimension is concentrated above the middle degree, extending our previous work to a non-compact case. The result applies even to Eisenstein cohomology classes coming from the locally symmetric space of the general linear group, and has been used in joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Taylor and Thorne to get good control on these classes and deduce potential automorphy theorems without any self-duality hypothesis. Our main geometric result is a computation of the fibers of the Hodge–Tate period map on compactified Shimura varieties, in terms of similarly compactified Igusa varieties.

我们证明与偶维准分裂酉群相关的 Shimura 簇的 $\mathrm{mod}\, \ell$ 上同调的泛型部分集中在中度以上，将我们之前的工作扩展到非紧情况。该结果甚至适用于来自一般线性群的局部对称空间的爱森斯坦上同调类，并且已与 Allen、Calegari、Gee、Helm、Le Hung、Newton、Taylor 和 Thorne 联合使用，以更好地控制这些类并在没有任何自对偶假设的情况下推导出潜在的自同构定理。我们的主要几何结果是根据类似的致密 Igusa 品种计算致密 Shimura 品种的 Hodge-Tate 周期图的纤维。