ON THE MODULARITY OF TORSION CLASSES IN THE COHOMOLOGY OF SHIMURA VARIETIES WITHOUT SEMISTABLE REDUCTION

Mathematical Applications of Science Fiction Abstract. We investigate the torsion cohomology of Shimura varieties of PEL type at primes of deepwild ramification where the integral model fails to exhibit semistable reduction. By constructing aderived infinite-level perfectoid cover of the local model and utilizing the spectral sequence of vanishingcycles for perverse sheaves on the special fiber, we prove that torsion classes in the cohomologyof the integral model lift to characteristic zero automorphic representations, provided the Heckealgebra acts via a pseudo-representation with sufficiently large image. We circumvent the lack ofsemistability by descending the test function from the infinite-level Igusa variety, establishing alocal-global compatibility result for the associated Galois representations at the bad prime p.