Let $E$ be an elliptic curve defined over $\Q$. A prime $p$ is called a **non-surjective** prime for $E$ if the mod-$p$ Galois representation
\[
\rho_{E,p}: \Gal(\overline{\Q}/\Q) \to \GL(2,\F_p)
\]
attached to $E$ is **not** surjective.

There are only finitely many non-surjective primes for any elliptic curve without CM by a result of Serre. Conjecturally, the largest possible non-surjective prime is $37$.

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- Last edited by John Jones on 2018-06-19 18:53:33

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- 2018-06-19 18:53:33 by John Jones (Reviewed)