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Department of Mathematical Sciences

Seminar Archives

On this page you can find information about seminars in this and previous academic years, where available on the database.

Pure Maths Colloquium: Ranks of elliptic curves with prescribed torsion

Presented by Peter Bruin, University of Warwick

11 November 2013 16:00 in CM221

This talk is about a remarkable connection between ranks and
torsion subgroups of elliptic curves over number fields. No advanced
knowledge of elliptic curves will be assumed.

Let $E$ be an elliptic curve over a number field $K$. The Mordell-Weil
theorem states that the group of $K$-rational points of $E$ is the
direct sum of a finite group (the torsion group of $E$) and a free
Abelian group of finite rank. It turns out that over number fields $K$
of small degree, the presence of certain torsion groups forces this rank
to be even. This occurs for 13-torsion over quadratic fields, for
example.

The reason behind this is a phenomenon that we call "false complex
multiplication", which is closely related to the arithmetic of modular
curves. I will explain this phenomenon and try to indicate the wider
context in which it can be situated.

This is joint work with Johan Bosman, Andrej Dujella and Filip Najman.

Contact anna.felikson@durham.ac.uk for more information