Ranks of elliptic curves in cyclic sextic extensions of \(\mathbb{Q}\) (Q6585096)

In this paper, the authors study the rank of the Mordell-Weil groups \(E(K)\) as K range over cyclic sextic extensions of \(\mathbb{Q}\). They show that for any elliptic curve \(E/\mathbb{Q}\), there are infinitely many primitive sextic characters \(\chi\), and hence infinitely many cyclic sextic extensions \(K = K_\chi\) of \(\mathbb{Q}\), for which \(\chi\)-component of \(E(K_{\chi})\) has positive rank and so \(L(E/\mathbb{Q}, 1, \chi) = 0\). For certain curves \(E/\mathbb{Q}\), they show that the number of such fields \(K\) of conductor less than \(X\) is \(\gg \sqrt{X}\), which up to logarithmic factors is the conjectured frequency.