An algorithm for determining torsion growth of elliptic curves
Computational Results pdf
Torsion growth over degree d
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\(\Phi_{\mathbb Q}(d)\) is the set of groups, up to isomorphism, that occur as torsion groups \(E(K)_{tors}\) of an elliptic curve \(E/\mathbb Q\) base change to a number field \(K\) of degree d.
- Given \(G\in\Phi(1)\), let \(\Phi_{\mathbb Q}(d,G)\) the subset of \(\Phi_{\mathbb Q}(d)\) consisting of all torsion groups \(E(K)_{tors}\) of an elliptic curve \(E/\mathbb Q\) such that \(E(\mathbb Q)_{tors}=G\) base change to a number field \(K\) of degree d.
- \(\mathcal{H}_{\mathbb Q}(d,E)\) the set of torsion configurations of degree \(d\) of the elliptic curve \“E/\mathbb Q\). That is, the multiset formed by \(E(\mathbb Q)_{tors}\) together with the groups \(H\) such that there exists a number field \(K\) of degree dividing \(d\) such that \(E\) has primitive torsion growth over \(K\) and
\(E(K)_{tors}=H\).
- \(\mathcal{H}_{\mathbb Q}(d)\) the set of \(\mathcal{H}_{\mathbb Q}(d,E)\) as \(E\) runs over all elliptic curves defined over \(\mathbb Q\) such that \(\mathcal{H}_{\mathbb Q}(d,E)\ne \{E(\mathbb Q)_{tors}\}\).
- \(N_\mathbb Q(d)\) the maximum of \(N_\mathbb Q(S)\) for all \(S\in\mathcal{H}_\mathbb Q(d)\), where \(N_\mathbb Q(S)\) denotes the minimum conductor \(N_\mathbb Q(E)\) such that \(\mathcal{H}_{\mathbb Q}(d,E)=S\).
- \(h_{\mathbb Q}(d)\) he maximum number of field extensions of degrees dividing \(d\) where there is primitive torsion growth. That is, the maximum of the cardinality of the sets \(S\in \mathcal{H}_{\mathbb Q}(d)\).
Only Primitive torsion growth over degree d
- \(\Psi_{\mathbb Q}(d)\subseteq \Phi_{\mathbb Q}(d)\) the set of groups, up to isomorphism, that appear as primitive torsion growth of an elliptic curve defined over \(\mathbb Q\) over a number field of degree \(d\).
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\(\Psi_{\mathbb Q}(d,G)\), \(\mathcal{G}_{\mathbb Q}(d,E)\), \(\mathcal{G}_{\mathbb Q}(d)\), \(g_{\mathbb Q}(d)\), \(M_{\mathbb Q}(d)\) analogously to \(\Phi_{\mathbb Q}(d,G)\), \(\mathcal{H}_{\mathbb Q}(d,E)\), \(\mathcal{H}_{\mathbb Q}(d)\), \(h_{\mathbb Q}(d)\), \(N_{\mathbb Q}(d)\), respectively.
Last modified: 14/4/2019