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SUMMARY:Congruences of Elliptic Curves Arising from Non-Surjective Galois 
 Representations - Sam Frengley\, Cambridge
DTSTART:20211102T143000Z
DTEND:20211102T153000Z
UID:TALK162211@talks.cam.ac.uk
CONTACT:Rong Zhou
DESCRIPTION:Elliptic curves E/K and E'/K are said to be N-congruent if the
 ir N-torsion subgroups are isomorphic as Galois modules. When N=p is an od
 d prime Halberstadt and Cremona--Frietas showed that an elliptic curve E/K
  admits a p-congruence with a nontrivial quadratic twist if and only if th
 e image of the corresponding mod p Galois representation is contained in t
 he normaliser of a Cartan subgroup of GL_2(F_p)\, but not the Cartan subgr
 oup itself. By considering the modular curves X_{ns}^+(p) Halberstadt gave
  examples of 2p-congruences over Q for p \\in {5\,7\,11} .\n\n\nWe discuss
  how these results may be extended to composite N. By constructing certain
  modular curves we find an infinite family of 36-congruences and an exampl
 e of a 48-congruence over Q. We also formulate a conjecture classifying N-
 congruences between quadratic twists of elliptic curves over Q.\n\n
LOCATION:MR13
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