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Let $E$ be an elliptic curve defined over $\Q$ and let $K$ be a number field. We say that there is torsion growth from $\Q$ to $K$ if the torsion subgroup $E(K)_{\rm tor}$ of $E(K)$ is strictly larger than $E(\Q)_{\rm tor}$.

If there is torsion growth in a field $K$ then obviously the torsion also grows in every extension of $K$. We say that the torsion growth in $K$ is primitive if $E(K)_{\rm tor}$ is strictly larger than $E(K')_{\rm tor}$ for all proper subfields $K' \subsetneq K$.

For every elliptic curve $E$ there is torsion growth in at least one field of degree $2$, $3$, or $4$, and torsion can only grow in fields whose degree is divisible by $2$, $3$, $5$ or $7$: see Theorem 7.2 of [10.1090/mcom/3478 ]. Additionally, there is no primitive torsion growth in fields of degrees $22$ or $26$: see Lemma 2.11 of [10.1080/10586458.2020.1771638 ]. Hence the only degrees less than $24$ in which primitive torsion growth occurs are $2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21$.

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• Last edited by John Cremona on 2020-10-12 05:52:59
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