University of Cambridge > Talks.cam > Number Theory Seminar > Ranks of elliptic curves with prescribed torsion over number fields

Ranks of elliptic curves with prescribed torsion over number fields

Add to your list(s) Download to your calendar using vCal

If you have a question about this talk, please contact Tom Fisher.

Let d be a positive integer, and let Td be the set of isomorphism classes of groups that can occur as the torsion subgroup of E(K), where K is a number field of degree d and E is an elliptic curve over K. T1 is known by Mazur’s theorem, T2 is known as well, and for d equal to 3 or 4, it is known which groups occur infinitely often.

We shall study the following problem: given a d <= 4 and a group T in Td, what are the possibilities for the Mordell-Weil rank of E, where E is an elliptic curve over a number field K of degree d with the torsion subgroup of E(K) isomorphic to T. For d = 2 and T = Z/13Z or T = Z/18Z, and also for d = 4 and T = Z/22Z, it turns out that the rank is always even. This will be explained by a phenomenon we call “false complex multiplication”.

This is joint work with Peter Bruin, Andrej Dujella, and Filip Najman.

This talk is part of the Number Theory Seminar series.

Tell a friend about this talk:

This talk is included in these lists:

Note that ex-directory lists are not shown.

 

© 2006-2019 Talks.cam, University of Cambridge. Contact Us | Help and Documentation | Privacy and Publicity