English

$2^\infty$-Selmer groups, $2^\infty$-class groups, and Goldfeld's conjecture

Number Theory 2017-06-08 v2

Abstract

We prove that the 22^\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 22^\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)o(N).

Keywords

Cite

@article{arxiv.1702.02325,
  title  = {$2^\infty$-Selmer groups, $2^\infty$-class groups, and Goldfeld's conjecture},
  author = {Alexander Smith},
  journal= {arXiv preprint arXiv:1702.02325},
  year   = {2017}
}

Comments

84 pages, comments welcome

R2 v1 2026-06-22T18:12:27.859Z