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 $K$?
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4$\begingroup$ There is a pairing on a certain subgroup of the $p$-Selmer group with values in the $p$-torsion subgroup of the class group. See for instance Mazur-Tate "Canonical Height Pairings via biextensions". For actual points it is essentially the square root of the denominator ideal of the $x$-ccordinate, but the point needs to have good reduction at all places. $\endgroup$– Chris WuthrichCommented Dec 18, 2020 at 22:51
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2$\begingroup$ Another connection arises, when $E[p]\subset E(K)$. Or more generally if there is a $p$-isogeny whose kernel $E[\phi]\cong \mu[p]$. Then the $\phi$-Selmer group is in the same cohomology group as the class group. It may happen that the local conditions are the same (or closely related). In a joint paper with Jean Gillibert, we looked at both of these connections. $\endgroup$– Chris WuthrichCommented Dec 18, 2020 at 22:57
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$\begingroup$ @ChrisWuthrich Many thanks for your helpful comments. I'll read carefully these your interesting results. $\endgroup$– A. MaarefparvarCommented Dec 19, 2020 at 6:32
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2 Answers
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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.
answered Dec 29, 2021 at 10:26
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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).
