\(2^n\)-descent on elliptic curves for all \(n\) (Q2840162)

Let \(k\) be an algebraic number field and NEWLINE\[NEWLINE E:\; Y^2=X^3+a_2X^2+a_4X+a_6=: f(X) NEWLINE\]NEWLINE be an elliptic curve defined over \(k\). Usually, to perform 2-descent on \(E\) we need to work with a splitting field \(K:=k(\alpha_{1},\alpha_{2},\alpha_{3})\) of the polynomial \(f\). In a classical paper \textit{J. W. S. Cassels} [Proc. Lond. Math. Soc. (3) 12, 259--296 (1962; Zbl 0106.03705)], has shown how to perform 4-descent working only with \(K\). In the paper under review the author is interested in finding all \(2^n\)-coverings of the curve \(E\) or equivalently (according to the terminology in the paper) performing \(2^n\)-descent on \(E\). More precisely, the author shows how to perform \(2^n\)-descents for all \(n\) without a further field extension. The results presented in the paper generalize the mentioned work of Cassels and also the later results of the same author in [J. Reine Angew. Math. 494, 101--127 (1998; Zbl 0883.11028)].