Calegari–Geraghty freeness for quasi-split unitary groups

Establish sufficient hypotheses under which the localized top-degree cuspidal cohomology of the quasi-split unitary group U_E is free of rank one over its localized Hecke algebra, thereby proving the assumption $(H = _U)$ used in the period-divisibility theorem.

Background

The principal period-divisibility theorem is conditional on the freeness assumption $(H = _U)$ for localized top-degree cuspidal cohomology of U_E. Such freeness is known or expected to follow from Calegari–Geraghty patching methods under hypotheses such as minimality, Fontaine–Laffaille conditions at p, and sufficiently large residual image.

The paper suggests that adapting Calegari–Geraghty theory to quasi-split unitary groups could provide sufficient conditions, but does not establish those conditions or prove the required freeness.