Eisenstein series — functions on locally symmetric spaces built by averaging over a parabolic subgroup — are fundamental objects in the theory of automorphic forms. At special values of their parameter, they have poles, and the residues of these poles contribute to the cohomology of arithmetic groups. These residual Eisenstein classes appear in specific cohomological degrees determined by the representation theory of the ambient group.
Mundy (arXiv:2603.12472) proves a vanishing theorem: for semisimple groups with maximal parabolic subgroups satisfying certain conditions, the residual Eisenstein cohomology vanishes in degrees outside a precise range. The proof constructs an explicit cochain — a primitive — for the cohomological classes in question, using regular (non-residual) Eisenstein series evaluated at nearby parameter values.
The construction works by taking a limit. Regular Eisenstein series at parameter s near the pole value sā have cohomological classes that are exact — they're boundaries of explicitly constructable cochains. As s approaches sā, the Eisenstein series develops a pole, but the cochain remains well-defined. The residual class, which is the coefficient of the pole, inherits the exactness in degrees where the cochain survives the limit. In degrees where it doesn't survive, the residual class can be nontrivial.
The precise determination of where the cochain survives requires understanding the intertwining operator — the linear map that relates Eisenstein series associated to different parabolic subgroups. Mundy shows this operator vanishes to order exactly 1 on a specific subrepresentation comprising two discrete series. This order-of-vanishing calculation is the technical heart: it determines the exact degree at which residual cohomology can exist, turning a general vanishing principle into a sharp result.