They were defined by the speaker of this talk in his recent work. So I geuss the best place to start is math.uni-bonn.de/people/scholze .

You will have to tell us more about your $E$ if you hope that we can help. Stated this way, your question is too vague.

Update : I asked Karl Rubin and John Coates and both seem to think that the problem is open. Maybe it is within reach, I don't know.

So you wish to understand the field over which cm-curves achieve good reduction everywhere. Maybe Serre-Tate's "Good reduction of abelian varieties" helps.

it looks to me as if he thought that there were functions that vanish identically but have one non-zero value.

Interesting. Unfortunately, - as far as I see- he considers non-cyclotomic extensions over the base-field $k$. I am interested in cyclotomic extensions of a $K$, which is itself an abelian extension of $k$.

If you know how to compute the $p$-adic logarithm on the formal group for your $J$, then you could just check by computing with sufficient precision if the image of the generators of $J(\mathbb{Q})$ are independent in the Lie algebra.

In matwbn.icm.edu.pl/ksiazki/aa/aa76/aa7626.pdf, the authors Coray, Manoil, call the surjectivity of $\pi^{*}$ the "BigPic" property and discuss it in more details.

I meant Shinishi Kobayashi math.nagoya-u.ac.jp/~shinichi, look at his Inventiones 03 paper. ... and so I say "hello, j!".

Then you should look at papers of Kabayashi and Iovita-Pollack. Also Perrin-Riou has computed the Euler characteristic of the dual of Selmer group over a cyclotomic $\mathbb{Z}_p$-extension in her Asterisque book. But you will find "Arithmétiques des courbes elliptiques à réduction supersingulière en p" a better place to start. ps: Are you jvo ?

Did you really mean to define $\mathfrac{S}(A/K)$ the way you did ? The compact Selmer group, usually denoted with $\mathfrak{S}(A/K)$, is in the Galois cohomology of $T_p A$, not $A_{p^{\infty}}$. If you change this in the definition then your remarks afterwards are correct. If you want to keep this definition, then the answer of Remke Kloosterman tells you what should be true.

An idea that does not help : It is equivalent to ask to check if the conic $x^2+y^2 = n$ has a rational solution. But unless I factor $n$, I would not know at which finitely many primes $p$ I have to check locally solubility.

Is the last $\cong$ in your question, just any isomorphism of groups or do you want to take the one induced by an isogeny from $A$ to its dual ? If $p$ divides the degree of this isogeny then it might be that only in one direction the isogeny gives an isomorphism (and maybe none, I don't know), but I have no examples at hand.

No, the point is that in your EDIT 2, you have $A = B$ as you start with a subset of endomorphisms.

They have different $j$-invariant. Yes, as right modules. I don't see why the fact that this ideal is principal will imply that they should be isomorphic.