The Selmer groups of elliptic curves. (Q1566025)

Fix a prime \(p\) and let \(E\) be an elliptic curve with discriminant \(\Delta\), defined over some number field \(K\) containing a \(p\)-torsion point. Let \(S\) denote the set of all places of \(K\) dividing \(p\Delta\infty\), and let \(C_{K,S}\) denote the \(S\)-class group of \(K\). If \(E[p] \subset K\), then the \(p\)-Selmer group \(S^{(p)}(E/K)\) has order \(\# S^{(p)}(E/K) \leq p^{2 \# S}(\# C_{K,S}[p])^2\) (actually this bound is proved with \(S\) replaced by a slightly smaller set); if \(E(K)[p]\) has order \(p\), a similar bound is derived involving the \(S\)-class group of \(L = K(E[p])\) for a suitable set \(S\) of places in \(L\).