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Let $E$ be an elliptic curve defined over a number field $F$ and suppose $E$ has CM by $\mathcal{O}_K$ where $K$ is an imaginary quadratic field. What can we say about the non-anomalous primes of such an elliptic curve $E$? Do we know if for infinitely many primes, $E$ has non-anomalous reduction at $p$?

By definition, we say $v$ is anomalous for $E/F$ if $|\tilde{E}_v(k_w)|$ is divisible by $p$; otherwise, we say that $v$ is non-anomalous for $E/F$. If all $v\mid p$ are non-anomalous for $E/F$, then we say that $p$ is non-anomalous for $E/F$.

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  • $\begingroup$ For rank 0 elliptic curves over $\mathbb{Q}$ with CM, it is mentioned in a paper of Mazur (1972, Inventiones) that density 1/2 primes have non-anomalous reduction. $\endgroup$
    – debanjana
    Feb 13, 2019 at 5:28

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