1 Erratum to: RACSAM DOI 10.1007/s13398-014-0199-x
The statements in the proofs of Proposition 7 and Proposition 8 that all elliptic curves with \(j\)-invariants \(0\) and \(1728\) belong to \(3\) and \(2\) families of quadratic twists is false. There are infinitely many families of quadratic twists with \(j\)-invariant \(0\) and infinitely many with \(j\)-invariant \(1728\).
Accordingly, Proposition 7 of the original paper should be as follows:
Proposition 1
Among all the elliptic curves with \(j\)-invariant \(0,\) there exists infinitely many with trivial torsion\(,\) infinitely many with torsion \(C_2,\) infinitely many with torsion \(C_3,\) and 1 with torsion \(C_6.\)
Proof
An elliptic curve with \(j\)-invariant \(0\) is of the form
It follows that \(E_D(\mathbb {Q})[2]\simeq \mathbb {Z}/2\mathbb {Z}\) if \(D\) is a cube and and \(E_D(\mathbb {Q})[2]\) is trivial otherwise. A computation using division polynomials proves that there is no \(4\)-torsion in \(E_D(\mathbb {Q})\).
By [1, Theorem 3] it follows that \(E_D(\mathbb {Q})[3]\simeq \mathbb {Z}/3\mathbb {Z}\) if \(D\) is a square and \(E_D(\mathbb {Q})[3]\) is trivial otherwise.
There can be no other torsion in \(E_D(\mathbb {Q})\) by [2, Proposition 1].
In Remark 2, it should say “infinitely many large torsion twists” instead of “4 large torsion twists” and the sentence “For example \(E_2\) has 4 large torsion twists” should be deleted.
Also, Proposition 8 of the original paper should be as follows:
Proposition 2
Among all the elliptic curves with \(j\)-invariant \(1728,\) there exists infinitely many with torsion \(C_2,\) infinitely many with torsion \(C_2\oplus C_2\) and \(1\) with torsion \(C_4.\)
Proof
An elliptic curve with \(j\)-invariant \(1728\) is of the form
It follows that \(E_D(\mathbb {Q})[2]\simeq \mathbb {Z}/ 2 \mathbb {Z}\) if \(D\) is not a square and \(E_D(\mathbb {Q})[2]\simeq \mathbb {Z}/ 2 \mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}\) if \(D\) is a square. A computation using division polynomials shows that the only elliptic curve with \(4\)-torsion in this family is \(E_{4}:y^2=x^3+4x\) with \(E_4(\mathbb {Q})_{tors}\simeq \mathbb {Z}/4\mathbb {Z}.\) By [1, Theorem 3], there is no \(p\)-torsion in \(E_D(\mathbb {Q})\) for any odd prime \(p\).
References
Dieulefait, L., González-Jiménez, E., Jiménez Urroz, J.: On fields of definition of torsion points of elliptic curves with complex multiplication. Proc. Am. Math. Soc. 139, 1961–1969 (2011)
Parish, J.L.: Rational torsion in complex-multiplication elliptic curves. J. Number Theory 33, 257–265 (1989)
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The online version of the original article can be found under doi:10.1007/s13398-014-0199-x.
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Najman, F. Erratum to: The number of twists with large torsion of an elliptic curve. RACSAM 109, 549–550 (2015). https://doi.org/10.1007/s13398-014-0208-0
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DOI: https://doi.org/10.1007/s13398-014-0208-0