On torsion subgroups of elliptic curves with integral \(j\)-invariant over imaginary cyclic quartic fields
The author considers the problem of determining the torsion group of an elliptic curve \(E\) with integral modular invariant \(j\) over a number field \(K\). He solves the problem for imaginary cyclic quartic fields \(K\). However, he has to impose the additional condition that the residue degrees \(f_2\) and \(f_3\) of a prime of \(K\) lying over 2 resp. 3 both are less than 4. Under this condition, he obtains ten groups as possible torsion groups of \(E\) over \(K\), and all these groups do really occur. The problem was solved earlier by the reviewer and his coauthors if \(K\) is a quadratic or cubic or real biquadratic or totally imaginary biquadratic field. The real biquadratic case is treated in the later paper with the title ``Computing the torsion group of elliptic curves by the method of Gröbner bases'' [\textit{A. Pethő}, \textit{J. Stein}, \textit{Th. Weis}, and \textit{H. G. Zimmer}, Prog. Comput. Sci. Appl. 15, 245-265 (1998)]. On the other hand, the cubic case is divided into the subcases, where \(K\) is pure cubic or cyclic cubic [see the reviewer and \textit{A. Pethő}, On a system of norm-equations over cyclic cubic number fields, Publ. Math. 53, 317-332 (1998; Zbl 0911.11020)] or general cubic [see the reviewer, \textit{A. Pethő}, \textit{J. Stein} and \textit{Th. Weis}, Int. J. Algebra Comput. 7, 353-413 (1997; Zbl 0868.11030)]. The integrality condition on \(j\) is stipulated here in order to exclude multiplicative reduction of \(E\) over \(K\) when the order of the torsion group depends on \(E\) over \(K\). Of course, the integrality condition on \(j\) is removed by Mazur for \(K=\mathbb{Q}\) and by Kamienny, Kenku and Momose if \(K\) is quadratic over \(\mathbb{Q}\). We mention in this connection also a paper of \textit{L. Merel} [Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124, 437-449 (1996)] in which the famous boundedness conjecture for the torsion group of \(E\) over an arbitrary number field \(K\) is proved. The author of the present paper follows closely the methods of reduction theory, parametrizations and norm equations used by the reviewer and his coauthors for elliptic curves \(E\) over quadratic number fields. Moreover, he has to employ modifications of the equations found by Reichert for the modular curves \(X_1(11)\), \(X_1(13)\), \(X_1(15)\) and \(X_1(16)\). Even though the present paper is more complicated than its predecessors, its outcome is almost the same as for elliptic curves \(E\) with integral modular invariant \(j\) in a quadratic number field \(K\), the exceptions being the groups isomorphic to \(\mathbb{Z}/7\mathbb{Z}\), \(\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\) and \(\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}\). Indeed these curves do occur only over imaginary quadratic fields which are known to be not contained in imaginary cyclic quartic fields \(K\). Of course, one would like to get rid of the extra condition on \(K\) that \(f_2<4\) and \(f_3<4\). The author devotes the concluding section 5 to this condition: In order to delete this condition one would have to look at torsion groups of \(E\) over \(K\) containing (up to isomorphism) one of the groups \(\mathbb{Z}/17\mathbb{Z}\), \(\mathbb{Z}/19\mathbb{Z}\) or \(\mathbb{Z}/23\mathbb{Z}\). These latter groups of high order are not accessible by Merel's theorem. Their order is too high for the above direct calculations. Perhaps the method of Gröbner bases helps here. In conclusion of this report, we quote two publications not mentioned in the reference section of the paper under review. One is by \textit{Soonhak Kwon} who was not aware of the contributions of the reviewer and his coauthors [J. Number Theory 62, 144-162 (1997; Zbl 0876.11027)]. Here the behaviour of torsion groups of elliptic curves over \(\mathbb{Q}\) is studied under transition from \(\mathbb{Q}\) to quadratic extensions of \(\mathbb{Q}\). (The corresponding (much more difficult) problem concerning the rank of \(E\) over \(\mathbb{Q}\) is being dealt with, e.g., by U. Schneiders and the reviewer.) The second is a dissertation with the title ``Torsion groups of elliptic curves with integral \(j\)-invariant over the normal closure of pure cubic fields'', written at the University of Crete by Thanasis Vessis under the guidance of Jannis Antoniadis. The method used here follows the one employed by the reviewer and his coauthors for elliptic curves over pure cubic fields. In fact, in the finiteness cases, which are the majority of all cases, the curves \(E\) and the fields \(K\) so that \(E\) over \(K\) has one of the (finitely many) given torsion groups can all be listed. In the dissertation of Thanasis Vessis, however, this is not true of the torsion group isomorphic to \(\mathbb{Z}/4\mathbb{Z}\). It seems that, due to the natural limitations with respect to the amount of computations, the topic of the paper under review is rather exhausted. One should now perhaps look at the problem without imposing the condition of integrality of \(j\).
scientific article; zbMATH DE number 1130798