Euler systems with local conditions

Euler systems are certain norm-compatible families of cohomology classes, which play a key role in studying the arithmetic Galois representations. We briefly survey known systems, and recall standard conjecture Perrin-Riou predicting what kind system one should expect for general representation. Surprisingly, several recent constructions do not seem to fit predictions this conjecture, we formulate more explains these extra objects. The novel aspect our is that it predicts there often be different ranks associated given representation, describe how objects related.