$2^\infty$-Selmer groups, $2^\infty$-class groups, and Goldfeld's conjecture
$2^\infty$-Selmer groups, $2^\infty$-class groups, and Goldfeld's conjecture
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发表时间：
2017-02
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通讯作者：
Alexander D. Smith
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作者：
Alexander D. Smith
我们证明了虚二次域的2^\infty $-类群具有Cohen-Lenstra启发式预测的分布。给定一条具有满有理2-挠率且无四阶有理循环子群的椭圆曲线E/Q，我们类似地证明了E的二次挠率的2^\infty $-塞尔默群具有Delaunay启发式所预言的分布.特别是，在具有|D| < N，排名至少为二的曲线的数量为$o（N）$。
We prove that the $2^\infty$-class groups of the imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra heuristic. Given an elliptic curve E/Q with full rational 2-torsion and no rational cyclic subgroup of order four, we analogously prove that the $2^\infty$-Selmer groups of the quadratic twists of E have distribution as predicted by Delaunay's heuristic. In particular, among the twists E^d with |d| < N, the number of curves with rank at least two is $o(N)$.