On the complexity of computing the 2-Selmer group of an elliptic curve.

*(English)*Zbl 0915.11032Let \(E\) be an elliptic curve over \(\mathbb{Q}\) given by
\[
E: Y^2= X^3+ AX+ B,
\]
and assume \(E\) has no points of order 2 defined over \(\mathbb{Q}\). The authors give an algorithm for computing the 2-Selmer group of \(E\) which in theory improves on the corresponding classical algorithm of B. J. Birch and H. P. F. Swinnerton-Dyer [J. Reine Angew. Math. 212, 7–25 (1963; Zbl 0118.27601)] in that its complexity of
\[
O(\exp[(\log D\cdot\log\log D)^{{1\over 2}}(c_1+ o(1))])
\]
is better than that of Birch and Swinnerton-Dyer (which is at least \(O(\sqrt D))\). Here, \(D\) denotes the absolute discriminant of the curve \(E\). However, as the authors admit, in practice the Birch and Swinnerton-Dyer algorithm will probably be much faster than their method, which is asymptotically faster as the former has seen many improvements over the years [e.g. see J. Cremona, J. Symb. Comput. 31, No. 1-2, 71–87 (2001; Zbl 0965.11025)].

The authors’ algorithm is unconditional, but is complexity estimate assumes the GRH and a standard conjecture on the distribution of smooth reduced ideals.

The authors’ algorithm is unconditional, but is complexity estimate assumes the GRH and a standard conjecture on the distribution of smooth reduced ideals.

Reviewer: R.J.Stroeker (Rotterdam)

##### MSC:

11G05 | Elliptic curves over global fields |

14Q05 | Computational aspects of algebraic curves |

14H52 | Elliptic curves |

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\textit{S. Siksek} and \textit{N. P. Smart}, Glasg. Math. J. 39, No. 3, 251--257 (1997; Zbl 0915.11032)

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##### References:

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[2] | DOI: 10.1215/S0012-7094-77-04431-3 · Zbl 0376.14011 |

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