  As a natural generalization of the notion of `higher rank Euler system', we
develop a theory of `higher special elements' in the exterior power biduals of
the Galois cohomology of $p$-adic representations. We show, in particular, that
such elements encode detailed information about the structure of Galois
cohomology groups and are related by families of congruences involving natural
height pairings on cohomology. As a first concrete application of the approach,
we use it to refine, and extend, a variety of existing results and conjectures
concerning the values of derivatives of Dirichlet $L$-series.
