Sporadic Cubic Torsion
Linfoot Number Theory Seminar
8th December 2020, 4:00 pm – 5:00 pm
Virtual Seminar, https://zoom.us/j/96415443765
In 1978, Mazur classified the torsion subgroups which can occur in the Mordell-Weil group of an elliptic curve over Q. His result was extended to elliptic curves over quadratic number fields by Kamienny, Kenku, and Momose, with the full classification being completed in 1992. What both of these cases have in common is that each subgroup in the classification occurs for infinitely many elliptic curves; however, this no longer holds for cubic number fields. In 2012, Najman showed that there exists a unique (up to Qbar-isomorphism) elliptic curve whose torsion subgroup over a particular cubic field is Z/21Z. This curve yielded the first sporadic example of a torsion subgroup.
In this talk, I will first recall previous literature concerning torsion subgroups of elliptic curves over number fields. Then, I will discuss recent joint work with A. Etropolski, M. Derickx, M. van Hoeij, and D. Zureick-Brown where we complete the classification of torsion subgroups of elliptic curves over cubic number fields by explicitly determining the cubic points on various modular curves.