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SUMMARY:Completely faithful Selmer groups over GL(2)-extensions - Gergely 
 Zábrádi (Eötvös Loránd University)
DTSTART:20140218T161500Z
DTEND:20140218T171500Z
UID:TALK49523@talks.cam.ac.uk
CONTACT:James Newton
DESCRIPTION:Let E and A be two elliptic curves\, both defined over Q\, and
  p>3 be a good ordinary prime for E. Assume that A has no complex multipli
 cation so that the Galois group G of the extension F_\\infty/Q=Q(A[p^{\\in
 fty}])/Q is an open subgroup of GL_2(Z_p). The aim of the talk is to inves
 tigate the dual Selmer group of E over F_\\infty. Under certain technical 
 hypotheses we prove that its characteristic element satisfies a functional
  equation. Assume further that there exists a prime q (different from p) s
 uch that (i) A has potentially multiplicative reduction at q and (ii) all 
 the p-power division points of E are defined over the completion of F_\\in
 fty at a prime above q. As a consequence we show that the dual Selmer X(E/
 F_\\infty) cannot be annihilated by any element in the centre of \\Lambda(
 G). In particular\, if in addition the \\Lambda(H)-rank of X(E/F_\\infty) 
 equals 1 then X(E/F_\\infty) is completely faithful. Unfortunately\, this 
 latter condition is never satisfied if E=A\, but we do have examples of co
 mpletely faithful Selmer groups when E is different from A. This is joint 
 work in progress with T. Backhausz.
LOCATION:MR13
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