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On the complexity of computing the 2-Selmer group of an elliptic curve. (English) Zbl 0915.11032
Let \(E\) 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.

MSC:
11G05 Elliptic curves over global fields
14Q05 Computational aspects of algebraic curves
14H52 Elliptic curves
Software:
ecdata
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References:
[1] Buchmann, Journal de Theorie des Nombres de Bordeaux 6 pp 221– (1994) · Zbl 0828.11075
[2] DOI: 10.1215/S0012-7094-77-04431-3 · Zbl 0376.14011
[3] Birch, J. Reine Angew. Math. 212 pp 7– (1963) · Zbl 0118.27601
[4] Thiel, Algorithmic Number Theory pp 234– (1994)
[5] Cassels, Lectures on elliptic curves 24 (1991)
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[7] DOI: 10.1006/jnth.1995.1044 · Zbl 0832.14016
[8] Cremona, Algorithms for modular elliptic curves (1992)
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[10] Silverman, The arithmetic of elliptic curves (1986) · Zbl 0585.14026
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