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 $E(K)_{\rm tors}$ of $E(K)$ is strictly larger than $E(\Q)_{\rm tors}$.

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 arXiv:1609.02515).

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 $E(K)_{\rm tors}$ is strictly larger than $E(K')_{\rm tors}$ for all proper subfields $K' \subset K$.

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- Last edited by John Jones on 2018-06-19 15:25:49

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- 2018-06-19 15:25:49 by John Jones (Reviewed)