Rank of elliptic curves with nontrivial torsion group (Q1424584)

In most cases the known examples of elliptic curves over \( \mathbb{Q}\) with large rank have the torsion subgroup of order 1 or 2. In the paper \textit{O. Lecacheux} [C. R. Acad. Sci., Paris, Sér. I, Math. 332, 1--6 (2001; Zbl 1004.11031)] the author constructed elliptic curves over \( \mathbb{Q}(T)\) with rank \(\geq 3\) and torsion subgroup isomorphic to \(\mathbb{Z}/5\mathbb{Z}\). The paper under review is devoted to the investigation of elliptic curves over \( \mathbb{Q}(T)\) with rank greater than 3 and non-zero torsion group. An elliptic curve over \( \mathbb{Q}(T)\) is constructed with non-constant modular invariant having three independent points over \( \mathbb{Q}(T)\) and with the torsion group isomorphic to one of following groups: \(\mathbb{Z}/6\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}, \mathbb{Z}/3\mathbb{Z}\times \mu_3\). The proofs are based on the investigation of elliptic fibration on fiber products of elliptic modular surfaces.