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CATEGORIES:Number Theory Seminar
SUMMARY:Congruences of Elliptic Curves Arising from Non-Su
rjective Galois Representations - Sam Frengley\, C
ambridge
DTSTART;TZID=Europe/London:20211102T143000
DTEND;TZID=Europe/London:20211102T153000
UID:TALK162211AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/162211
DESCRIPTION:Elliptic curves E/K and E'/K are said to be N-cong
ruent if their N-torsion subgroups are isomorphic
as Galois modules. When N=p is an odd prime Halber
stadt and Cremona--Frietas showed that an elliptic
curve E/K admits a p-congruence with a nontrivial
quadratic twist if and only if the image of the c
orresponding mod p Galois representation is contai
ned in the normaliser of a Cartan subgroup of GL_2
(F_p)\, but not the Cartan subgroup itself. By con
sidering the modular curves X_{ns}^+(p) Halberstad
t gave examples of 2p-congruences over Q for p \\i
n {5\,7\,11} .\n\n\nWe discuss how these results m
ay be extended to composite N. By constructing cer
tain modular curves we find an infinite family of
36-congruences and an example of a 48-congruence o
ver Q. We also formulate a conjecture classifying
N-congruences between quadratic twists of elliptic
curves over Q.\n\n
LOCATION:MR13
CONTACT:Rong Zhou
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