Zeta functions, excision in cyclic cohomology and index problems

The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain an index formula for “abstract elliptic pseudodifferential operators” associated to spectral triples, in the spirit of the one of Connes and Moscovici. This formula is notably well adapted when the zeta function has multiple poles. The second part is devoted to give a concrete realization of this formula by deriving an index theorem on the simple, but significant example of Heisenberg elliptic operators on a trivial foliation, which are in general not elliptic but hypoelliptic. The formula obtained is an extension of an index formula due to Fedosov.