Infinitely many elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ and rank at least six
We show that there are infinitely many elliptic curves over $\mathbb{Q}$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ and rank at least $6$, answering the case $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ of Problem 2.6 in Dujella's list of open problems. The largest rank previously known to occur infinitely often with this torsion was $5$, due to Eroshkin. The curves are fibers of a one-parameter family of Dujella and Peral, which we show has rank exactly $3$ over the rational function field. The family is a base change of an elliptic K3 surface, and…
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