Congruences of Elliptic Curves Arising from Non-Surjective Galois Representations

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• Sam Frengley, Cambridge
• Tuesday 02 November 2021, 14:30-15:30
• MR13.

If you have a question about this talk, please contact Rong Zhou.

Elliptic curves E/K and E’/K are said to be N-congruent if their N-torsion subgroups are isomorphic as Galois modules. When N=p is an odd 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 the image of the corresponding mod p Galois representation is contained in the normaliser of a Cartan subgroup of GL_2(F_p), but not the Cartan subgroup itself. By considering the modular curves X_{ns}^+(p) Halberstadt gave examples of 2p-congruences over Q for p \in {5,7,11} .

We 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 example of a 48-congruence over Q. We also formulate a conjecture classifying N-congruences between quadratic twists of elliptic curves over Q.

This talk is part of the Number Theory Seminar series.

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