Cohomological Vanishing of Torsion Classes in Non-Compact Shimura Varieties via Perfectoid Methods

We establish a vanishing theorem for torsion classes in the cohomology of non-compact Shimura varieties of PEL type, generalizing the results of Caraiani and Scholze to the non-compact setting. By passing to the infinite level at p, we utilize the perfectoid structure of the associated Shimura variety and the geometry of the Hodge-Tate period map HT. We analyze the boundary contributions via the toroidal compactification and the associated stratification of the perfectoid space, invoking the Calegari-Geraghty method to control the completed cohomology. Specifically, we prove that under a suitable genericity assumption on the Hecke eigenvalues, the torsion cohomology vanishes outside the middle degree range. This result has significant implications for the modularity of torsion Galois representations associated with locally symmetric spaces that do not admit a Shimura variety structure but appear in the boundary of higher-rank Shimura varieties.