Résume | . Howard, B. Mazur and K. Rubin proved that the existence of Kolyvaginsystems relies on a cohomological invariant, what they call the coreSelmer rank. When the core Selmer rank is one, they determine thestructure of the Selmer group completely in terms of a Kolyvaginsystem. However, when the Selmer core rank is greater than one such aprecision could not be achieved. In fact, one do not expect a similiarresult for the structure of the Selmer group in general, as areflection of the fact that Bloch-Kato conjectures do not in generalpredict the existence of special elements, but a regulator, to computethe relevant L-values.An example of a core rank greater than one situation arises if oneattempts to utilize the Euler system that would come from the Starkelements (whose existence were predicted by K. Rubin) over a totallyreal number field. This is what I will discuss in this talk. I willexplain how to construct, using Stark elements, Kolyvagin systems forcertain modified Selmer structures (that are adjusted to have corerank one) and relate them to appropriate ideal class groups, followingthe machinery of Kolyvagin systems and prove a Gras-type conjecture.Should time permit, I will also discuss how to extend our technique todeduce the main conjectures of Iwasawa theory over totally real numberfields. |