Let \(p\) be a prime number and denote by \(\mathbb{Q}(\mu_p)\) the cyclotomic field. The authors prove the following theorem: if there exists an elliptic curve \(E\) over \(\mathbb{Q}(\mu_p)\) such that the points of order \(p\) of \(E(\overline{\mathbb{Q}})\) are all \(\mathbb{Q}(\mu_p)\)-rational, then \(p=2,3,5,13\) or \(p> 1000\). The part of the result that concerns the case \(p\equiv 3\pmod 4\) has been obtained by L. Merel in [Duke Math. J. 110, 81-119 (2001; Zbl 1020.11041)]. In this paper the case \(p\equiv 1\pmod 4\) is considered. The authors first prove that under an additional hypothesis on \(p\), if the elliptic curve \(E\) over \(\mathbb{Q}(\mu_p)\) has the \(p\)-torsion \(\mathbb{Q}(\mu_p)\)-rational, each subgroup of \(E(\overline{\mathbb{Q}})\) of order \(p\) gives a \(\mathbb{Q}(\sqrt{p})\) defined point on \(X_0(p)\); on the other hand, they prove that this \(\mathbb{Q}(\sqrt{p})\)-rationality conclusion is contrary to fact.
The last part of the paper explains how the authors use the computer to verify that the additional hypothesis on \(p\), which involves the nonvanishing of a certain \(L\)-function, is satisfied for \(p=11\) and \(13< p< 1000\).