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SUMMARY:Unfaking the fake Selmer group - Ronald van Luijk (Leiden)
DTSTART:20101123T143000Z
DTEND:20101123T153000Z
UID:TALK26450@talks.cam.ac.uk
CONTACT:Tom Fisher
DESCRIPTION:Let _C_ be a smooth projective curve over a global field _k_\n
 with Jacobian _J_. Then the Mordell-Weil group _J(k)_ of _k_-rational\npoi
 nts on _J_ is finitely generated. Knowing the torsion subgroup\,\nwhich is
  usually relatively easy to find\, the rank of _J(k)_ can be\nread off fro
 m the size of the finite group _J(k)/2J(k)_. This quotient\ninjects into t
 he so called Selmer group\, which is abstractly defined\nas a certain subg
 roup of the cohomology group _H[1](k\,J [ 2 ])_. The\nSelmer group is fini
 te\, so the image of _J(k)/2J(k)_ in it can be\ndetermined by deciding for
  each element of the Selmer group separately\nwhether or not it is in the 
 image of _J(k)/2J(k)_. Unfortunately\, the\nabstract definition of the Sel
 mer group is not very amenable to\nexplicit computations\, which are in pr
 actice done with the fake Selmer\ngroup instead. In general the fake Selme
 r group is isomorphic to a\nquotient of the Selmer group by a subgroup of 
 order 1 or 2. In\nthis talk we will define all the groups just mentioned a
 nd we will\nintroduce a new group\, equally amenable to explicit computati
 ons as\nthe fake Selmer group\, that is always isomorphic to the Selmer gr
 oup.\nThis is joint work with Michael Stoll.
LOCATION:MR13
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