Rice University Department of Mathematics Colloquium

Sporadic Torsion on Elliptic Curves

4:00 pm Thursday, August 25th, 2016
Anastassia Etropolski (Rice University)

Abstract: In Mazur's celebrated 1978 Inventiones paper, he classifies 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, but this no longer holds for cubic number fields. In 2012, Najman showed that there exists an elliptic curve whose torsion subgroup over a particular cubic field is Z/21. This was the first "sporadic" example, and there is a very precise way to understand and classify such examples using the arithmetic of modular curves. I will show that this completes the classification of cubic torsion, as well as discuss how one should think of these sporadic examples and how one might search for others. This project is joint work with David Zureick-Brown and Jackson Morrow.

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