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SUMMARY:Ranks of elliptic curves with prescribed torsion over number field
 s - Johan Bosman (Warwick)
DTSTART:20120228T143000Z
DTEND:20120228T153000Z
UID:TALK35334@talks.cam.ac.uk
CONTACT:Tom Fisher
DESCRIPTION:Let _d_ be a positive integer\, and let _T<sub>d</sub>_ be the
  set of isomorphism\nclasses of groups that can occur as the torsion subgr
 oup of _E(K)_\, where _K_ is a number field of degree\n_d_ and _E_ is an e
 lliptic curve over _K_.  _T<sub>1</sub>_ is known by Mazur's theorem\,\n_T
 <sub>2</sub>_ is known as well\, and for _d_ equal to 3 or 4\, it is known
  which\ngroups occur infinitely often.\n\nWe shall study the following pro
 blem: given a _d_ <= 4 and a group _T_ in\n_T<sub>d</sub>_\, what are the 
 possibilities for the Mordell-Weil rank of _E_\, where\n_E_ is an elliptic
  curve over a number field _K_ of degree _d_ with the torsion subgroup of 
 _E(K)_\nisomorphic to _T_.  For _d_ = 2 and _T = *Z*/13*Z*_ or _T = *Z*/18
 *Z*_\, and also for _d_ = 4 and\n_T = *Z*/22*Z*_\, it turns out that the r
 ank is always even.  This will be\nexplained by a phenomenon we call "fals
 e complex multiplication".\n\nThis is joint work with Peter Bruin\, Andrej
  Dujella\, and Filip Najman.\n
LOCATION:MR13
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