Relation between the Selmer group and the ideal class group Let  $E/K$ be an elliptic curve defined over the number field $K$. Does exist any relation between the $p$-Selmer groups of $E/K$ and the ideal class group $Cl(K)$ of $K$?
 A: The standard reference is Schaefer. For more accessible examples see the articles by Eisenbeis, Frey and Ommerborn and by Washington; see also my answer to this question on MO.
A: Franz Lemmermeyer's answer points out to a connection. Let me point out in the converse direction that believable heuristics suggest that if $E(\mathbb Q)=0$, then $\operatorname{Sel}_p(E)$ should be random among commutative $p$-groups with a non-degenerate alternating bilinear pairing (to be a little more precise, the means that the weighted average of a reasonable function on all isomorphism classes of such groups should equal the same weighted average on $p$-Selmer groups of rank zero rational elliptic curves).
These heuristics are far from being established of course, but they are considered excellent guidelines and suggest that in some sense, everything can happen for $p$-Selmer groups when $E$ ranges over all elliptic curves over $\mathbb Q$, and anything does happen just as often as it should for random groups. This hints at a strong logical constraint on any potential connection between $\operatorname{Cl}(\mathcal O_K)[p^{\infty}]$ and $\operatorname{Sel}_p(E)$ for general $K$ and $E$.
Heuristics on Tate-Shafarevitch Groups of Elliptic Curves Defined over $\mathbb Q$ C.Delaunay, Experiment. Math. (2001).
