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If I am not mistaken, it proves that mu invariant of the Z_p times Z_p extension is 0 and this was Schneps's thesis. It is unfortunately not enough to show the conjecture of Iwasawa in this case even using the vanishing of anticyclotomic mu invariant proven by Hida.
Perhaps one should mention that Euler system of Stark units is conjectural. Also, that the Euler system of Heegner points is not directly related to p-adic L-functions.
No. At this moment it is fair to say that Coates raised them as questions based on some numerical evidence though he called them conjectures. There is a very weak numerical evidence for the second part. Some japanese mathematicians (Okazaki, Fukuda, Komatsu were the names mentioned in the talk of Coates) have shown that for p=2,3 the ideal class groups of fields in the $\mathbb{Z}_p$ extension of $\mathbb{Q}$ has no prime divisor less than a million (I guess).
Since you mentioned about the problem of infinitely many fields of class number one, I would like to mention the conjectures that Coates recently made in a talk in Kyoto. Take the extension of $\mathbb{Q}$ with Galois group $\widehat{\mathbb{Z}}$. Then he conjectures that the set class numbers of all fields in this $\widehat{\mathbb{Z}}$ extension is a bounded set. He also conjectures that for any prime $p$ the class number of all fields in the $\mathbb{Z}_p$ extension of $\mathbb{Q}$ is 1. This was conjectured by Weber for $p=2$.
Yes, the tensor is over $\mathbb{Z}_p$. I wanted to put it but somehow whenever I tried doing that I did not get the desired typesetting output. The fact that $V$ is finite dimensional does not use the theorem of Ferrero-Washington about vanishing of $\mu$ invariant (which is anyways available only over abelian extensions of $\mathbb{Q}$). It follows from the fact that $X$ is torsion $\Lambda(G)$-module which was proved by Iwasawa.
But I guess the main conjecture of Iwasawa theory does not say anything about this because we can ignore finite submodules. The advances in Iwasawa theory have only been on formulating main conjectures in more general situations and not very much on getting finer information about the Iwasawa modules (Kurihara's work on computing all Fitting ideals in some situations is an exception to this that I know but not enough for these kinds of question I think). Does anyone if ETNC says anything for such conjectures?
