Elliptic mod \ell Galois representations which are not minimally elliptic

Details Elliptic mod \ell Galois representations which are not minimally elliptic Elliptic mod \ell Galois representations which are not minimally elliptic

2004-09-07

arxiv.org/abs/math/0409115v2

Mathematics

Number Theory

Scientific paper

In a recent preprint, F. Calegari has shown that for $\ell = 2, 3, 5$ and 7 there exist 2-dimensional surjective representations $\rho$ of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ coming from the $\ell$-torsion points of an elliptic curve defined over $\Q$, but not minimally, i.e., so that any elliptic curve giving rise to $\rho$ has prime-to-$\ell$ conductor greater than the (prime-to-$\ell$) conductor of $\rho$. In this brief note, we will show that the same is true for any prime $\ell >7$, concretely, we will show that for any such $\ell$ the elliptic curve $$E^\ell: \qquad Y^2 = X (X- 3^\ell ) (X - 3^\ell - 1) $$ is semistable, has bad reduction at 3, the associated $\mod \ell$ Galois representation $\rho$ is surjective, unramified at 3, and there is no elliptic curve with good reduction at 3 whose associated $\mod \ell$ representation is isomorphic to $\rho$.

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