Filip Najman

math.NT — Number Theory

Torsion of elliptic curves over number fields of degree $6$, $7$, $8$ and $9$

Contributed by Filip Najman

We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree $d$ for $d=6$, $7$, $8$ and $9$. Apart from the groups that occur infinitely often, exactly $4$ groups occur in degree $6$, $3$ in degree $7$, $8$ in degree $8$ and $9$ in degree $9$, and among them the groups $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}$ over sextic fields, $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/15\mathbb{Z}$ over octic fields and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/26\mathbb{Z}$ over nonic fields were not known to occur. For $13$ of these $24$ pairs of a degree and a group we also determine all the elliptic curves that realize them. The proofs combine the global method of Derickx, Hecke sieves, formal immersion criteria at the prime $2$ with Hecke operators, and Riemann-Roch computations on modular curves and their quotients over finite fields.