On the existence of families of elliptic curves of rank four with all two-torsion points (Q2765946)

In this impressive paper, the author constructs families of elliptic curves of Mordell-Weil rank four with torsion structure \(\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2 \mathbb{Z}\). The main ingredient she employs here is the universal variety \(V_n\) of dimension three, constructed in [\textit{H. Yamagishi}, Pac. J. Math. 191, 189-200 (1999; Zbl 1041.11042)], which parametrizes all elliptic curves of rank \(n\). By the very universality of \(V_n\), the condition that the Mordell-Weil group of an elliptic curve contains all two-torsion points gives rise to a surface \(W_n\subset V_n\) which parametrizes elliptic curves of rank \(n\) with the torsion structure \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2 \mathbb{Z}\). Thereafter she focuses on the case \(n=4\) and shows that \(W_4\) is birational to the Kummer surface associated to the product of two elliptic curves. This enables her to obtain the above-mentioned result.NEWLINENEWLINENEWLINEThere are several other articles which concern the construction of elliptic curves of high rank with a given torsion structure; the characteristics of her method, however, is that everything is quite explicit and natural.