Torsion growth of rational elliptic curves in sextic number fields
For some years now, an interest has arisen in the problem of studying the following problem: given an elliptic curve \(E\), defined over \({\mathbb Q}\), study how its torsion subgroup \(E_{tors} ({\mathbb Q})\) changes when the base field is expanded to a number field \(K\). This line of research has been pioneered by E. González-Jiménez and F. Najman, who have tackled themselves (also with a number of co-authors) many of the challenging problems arising in this context. The paper under review follows the footsteps of these previous works by giving a(n almost) complete result concerning the case where \(K\) is a sextic degree field. In particular, it narrows down the list of possible torsion groups appearing under these conditions to exactly \(34\) of them (\(18\) of them cyclic, the rest of them the direct sum of the form \(C_n \oplus C_m\)). All but one of the groups are known to occur, the sole exception being \(C_3 \oplus C_{18}\), which we still do not know if it appears. The paper is well-written, and the account on the previous landmarks is a nice introduction to the subject (even for non-specialists). The techniques are pretty much standard in this line of work, but it requires a high level of mastery to apply them in the way they are done here.
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