The order of the \(p\)-Selmer groups and the rank of elliptic curves (Q1365399)

Let \(K\) be an algebraic number field and let \(E\) be an elliptic curve defined over \(K\). The rank of the Mordell-Weil group \(E(K)\) of \(K\)-rational points is closely related to the order of the \(p\)-Selmer group \(S^{(p)}(E/K)\) for the multiplication-by-\(p\) map. In this paper, the author follows \textit{N. Aoki's} method [Comment. Math. Univ. St. Pauli 42, 209-229 (1993; Zbl 0834.11026)] in order to establish an upper bound for the order of \(S^{(p)}(E/K)\) (\(p\) prime) in terms of the ideal class group of a certain finite extension of \(K\). The resulting formulas are too technical to permit reproduction here.