The 2-class groups of cubic fields and 2-descents on elliptic curves
Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) with no rational point of order two and \(K\) the cubic field generated by a point of order two. Under some local conditions, there is an exact sequence which relates the 2-class group \(C(K)\) of \(K\) to the 2-Selmer group of \(E/\mathbb{Q}\). Combining the 2-descent exact sequence of \(E/\mathbb{Q}\), one can connect \(C(K)\) with the Mordell-Weil group \(E(\mathbb{Q})\) of \(E/\mathbb{Q}\), and give an inequality on their ranks. The topic of the simplest cubic fields is emphasized in this paper. For such fields, it is possible to describe more precisely the relation of two groups \(C(K)\) and \(E(\mathbb{Q})\) by means of the twists of elliptic curves, and a better estimation of the 2-rank of \(C(K)\) is obtained.
scientific article; zbMATH DE number 203562
scientific article