An example of a core rank greater than one situation arises if one
attempts to utilize the Euler system that would come from the Stark
elements (whose existence were predicted by K. Rubin) over a totally
real number field. This is what I will discuss in this talk. I will
explain how to construct, using Stark elements, many Kolyvagin systems for
certain modified Selmer structures (that are adjusted to have core
rank one) and relate them to appropriate ideal class groups, following
the machinery of Kolyvagin systems and prove a Gras-type conjecture.
At the end, I will discuss the Iwasawa theory of Stark units and
explain how to extend our technique to deduce the main conjectures of
Iwasawa theory over totally real number
fields.