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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
##### Keywords:
elliptic curve; algorithm; 2-Selmer group; complexity estimate
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##### References:
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