A note on elliptic curves with a rational 3-torsion point
The paper presents a descent via isogenies on elliptic curves defined over a number field \(F\). The known descriptions for the images of the local Kummer maps and for the orders of the Selmer groups require many informations about the curve \(E\) like, for example, the reduction types, the kernel of the isogeny, the number of connected components and so on. The detailed computations of these data are carried out for a 3-isogeny \(\phi\) with \(F\)-rational kernel and curves of the type \(E_{a,b}\,:\;y^2 +axy+by=x^3\) to provide the orders of the Selmer groups \(\mathrm{Sel}^{(\phi)}(E/F)\). The informations on the Selmer groups are then used to find bounds for the rank of the elliptic curve \(E_{a,b}\,\).
scientific article; zbMATH DE number 5779826
scientific article