Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

@article{Bhargava2017BoundsO2,
  title={Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves},
  author={Manjul Bhargava and Arul Shankar and Takashi Taniguchi and Frank Thorne and Jacob Tsimerman and Yongqiang Zhao},
  journal={arXiv: Number Theory},
  year={2017}
}
We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_{\epsilon}(|{\rm Disc}(K)|^{1/2+\epsilon})$ by Brauer--Siegel). This yields corresponding improvements to: 1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves; 2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves; 3) bounds on the sizes of 2-Selmer… 
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