In recent years there has been increasing interest in the study of elements that lie in the higher exterior powers of the cohomology of p-adic Galois representations that are defined over a number field.

 

In the best case, one hopes that a collection of such elements that are defined over a family of abelian extensions constitutes a ‘higher rank Euler system’ is explicitly related to special values of an L-series associated to the representation and leads to Galois structural information about the Selmer modules of the representation. 

 

In this talk, I will survey the recent development of the theory initiated by several projects of D.Burns and his collaborators. I will also describe how this theory gives new resolution and refinement even in a classical setting.