math.NT — Number Theory

Infinitely many elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ and rank at least six

Contributed by Manvir Jaswal

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 quadratic twists of that surface give it three further independent points over the function field of an elliptic curve of rank one. Congruences then carry the independence of the six points, and the torsion $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$, from one rational point of that curve to infinitely many. The result is unconditional. It contradicts the prediction of Park, Poonen, Voight and Wood that only finitely many curves with this torsion have rank greater than $5$, although the curves form a sparse family. A second family, over an elliptic curve of rank two, gives another proof.