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The aim of this paper is to show how zeta functions and excision in cycliccohomology may be combined to obtain index theorems. In the first part, weobtain a local index formula for "abstract elliptic pseudodifferentialoperators" associated to spectral triples. This formula is notably well adaptedwhen the zeta function has multiple poles. The second part is devoted to give aconcrete realization of this formula by deriving an index theorem on thesimple, but significant example of Heisenberg elliptic operators on a trivialfoliation, which are in general non-elliptic but hypoelliptic. The last partcontains a discussion on manifolds with conic singularity, more precisely aboutthe regularity of spectral triples in this context.

Zeta Functions, Excision in Cyclic Cohomology and Index Problems