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Existence of non-elliptic mod l Galois representations for every l >5

Luis Dieulefait
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Abstract

For $\ell = 3$ and 5 it is known that every odd, irreducible, 2-dimensional representation of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ and determinant equal to the cyclotomic character must "come from" the $\ell$-torsion points of an elliptic curve defined over $\Q$. We prove, by giving concrete counter-examples, that this result is false for every prime $\ell >5$.

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Luis Dieulefait. 2004-04-02. Existence of non-elliptic mod l Galois representations for every l >5. https://arxiv.org/abs/math/0404025

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