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Elementary 3-descent with a 3-isogeny. (English) Zbl 1253.11063

Let \(E/ \mathbb Q\) be an elliptic curve of the form \[ E:y^2=x^3+D(ax+b)^2, \] where \(D\) is a fundamental discriminant; then \(E\) has an isogeny of degree 3. The authors explain explicitly and in great detail how to preform a 3-isogeny descent on \(E\), which gives an upper bound on the rank of \(E(\mathbb Q)\). The procedure is in theory similar to the more studied 2-isogeny descent, but much harder in practice.
At the end of the paper, the authors perform a 3-isogeny descent on the family of curves \(y^2=x^3+kp\), where \(k=2^i\) for \(i=0,1,2\) and \(p\) is a prime and give explicit bounds on the rank of this elliptic curve, depending on \(k\) and congruence class of \(p\) modulo \(9\). Note that for some \(k\) and \(p\) this enables the authors to exactly determine the rank of elliptic curves for which 2-descent fails to give the exact rank (due to the non-triviality of \(\text{Ш}(E (\mathbb Q))[2])\) and to find the generators, which are in some cases huge, of the Mordell-Weil group of such elliptic curves.

MSC:

11G05 Elliptic curves over global fields
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