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CATEGORIES:Number Theory Seminar
SUMMARY:Completely faithful Selmer groups over GL(2)-exten
sions - Gergely Zábrádi (Eötvös Loránd University)
DTSTART;TZID=Europe/London:20140218T161500
DTEND;TZID=Europe/London:20140218T171500
UID:TALK49523AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/49523
DESCRIPTION:Let E and A be two elliptic curves\, both defined
over Q\, and p>3 be a good ordinary prime for E. A
ssume that A has no complex multiplication so that
the Galois group G of the extension F_\\infty/Q=Q
(A[p^{\\infty}])/Q is an open subgroup of GL_2(Z_p
). The aim of the talk is to investigate the dual
Selmer group of E over F_\\infty. Under certain te
chnical hypotheses we prove that its characteristi
c element satisfies a functional equation. Assume
further that there exists a prime q (different fro
m p) such that (i) A has potentially multiplicativ
e reduction at q and (ii) all the p-power division
points of E are defined over the completion of F_
\\infty at a prime above q. As a consequence we sh
ow that the dual Selmer X(E/F_\\infty) cannot be a
nnihilated by any element in the centre of \\Lambd
a(G). In particular\, if in addition the \\Lambda(
H)-rank of X(E/F_\\infty) equals 1 then X(E/F_\\in
fty) is completely faithful. Unfortunately\, this
latter condition is never satisfied if E=A\, but w
e do have examples of completely faithful Selmer g
roups when E is different from A. This is joint wo
rk in progress with T. Backhausz.
LOCATION:MR13
CONTACT:James Newton
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