Pure Maths Colloquium: Ranks of elliptic curves with prescribed torsion
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
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.
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