Is there a difference between Euler systems and Kolyvagin systems - or do they refer to the same thing? For example there is the Heegner point Euler system, but you don't really see a Heegner point Kolyvagin system. 


Also, Rubin-Mazur prove that the space of Kolyvagin system is free of rank one (under certain conditions) - is something similar true for Euler systems? 


In particular I would like some clarification on Mazur-Rubin "Kolyvagin Systems" Section 3.2 (e.g. in Theorem 3.2.4 what do these technical assumptions really mean)



What's the difference between Euler systems and Kolyvagin systems?
Euler systems and Kolyvagin systems are closely related but quite different beasts nevertheless.


According to their respective definitions, Euler systems are systems of classes $\{c(n)\in H^1(G\_{\mathbb Q(\zeta\_n)},V)\}\_n$ for $V$ a $p$-adic $G\_{\mathbb Q}$-representation verifying certain properties, especially with respect to corestriction from $\mathbb Q(\zeta\_n\ell)$ to $\mathbb Q(\zeta\_n)$, to whereas Kolyvagin systems are systems of classes $\{\kappa(n)\in H^1(G\_{\mathbb Q},T/p^N T)\}$ for some power $N$ and $T\subset V$ is a $G\_{\mathbb Q}$-stable $\mathbb Z\_p$-lattice inside $V$ verifying certain properties, especially with respect to localizations at $\ell$ of $\kappa(n\ell)$ and $\kappa(n)$. In the above, $G\_K$ denote the absolute Galois group $\operatorname{Gal}(\bar{K}/K)$ of $K$ and the definitions can of course be generalized to general base number fields. Hence, purely formally, Euler systems "live" in (the Galois cohomology of) many different extensions and are $\mathbb Z\_p$-linear or $\mathbb Q\_p$-linear objects whereas Kolyvagin systems "live" in (the Galois cohomology of) the base extension only but are torsion objects.


That said, and putting axiomatics aside for a moment, every time we can get hold of an Euler system in some sense, it is possible to manufacture a Kolyvagin system in some sense by the Kolyvagin's derivative operation or one of its variants, and in particular there certainly is a Kolyvagin system of Heegner points as is very apparent already in *Euler systems* (V.Kolyvagin, 1990) but as was axiomatized in the language of Mazur-Rubin in *The Heegner point Kolyvagin system* (B.Howard,2004).



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> Also, Rubin-Mazur prove that the space of Kolyvagin system is free of rank one (under certain conditions) - is something similar true for Euler systems?
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First all, please note that this theorem of Mazur-Rubin is actually due to B.Howard, as *Kolyvagin systems* makes clear. To answer your question, the answer is yes but in quite deep sense: it was understood in *Iwasawa theory and p-adic Hodge theory* (K.Kato,1993) that the existence of Euler systems in some sense is a formal consequence of the compatibility with proper base change of conjectures on special values of partial $L$-functions of motives and more precisely that Euler systems are actually bases of some spaces of motivic cohomology $\Delta$ expressing these special values. These spaces are y construction of rank 1 so that the "space" of Euler systems is of rank 1 in some sense. However, it also clear in Mazur-Rubin that Kolyvagin systems in their axiomatics are of rank 1 only under rather stringent hypotheses on the fixed part under complex conjugation, and it was understood at least since *Euler systems, Iwasawa theory and Selmer groups* (K.Kato,99) and axiomatized in recent years by K.Rubin and B.Mazur that the correct version of the space of Kolyvagin systems (the notion making it always of rank 1) has to reflect the aforementioned properties of $\Delta$.


Regarding your last question, I thus think that you would be better off reading the most recent literature of Mazur-Rubin, which makes an explicit link between the fact that Euler systems are always of rank 1 in some sense with the comparable property for the space of Kolyvagin systems, rather than delving into the specific axiomatic choices of Theorem 3.2.4 (whose meaning is simply that Kolyvagin's derivative applied to an Euler system yields a Kolyvagin system). 


