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Journal Article

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

MPS-Authors
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Zhao,  Yongqiang
Max Planck Institute for Mathematics, Max Planck Society;

External Ressource

https://doi.org/10.1090/jams/945
(Publisher version)

Fulltext (public)

arXiv:1701.02458.pdf
(Preprint), 178KB

Supplementary Material (public)
There is no public supplementary material available
Citation

Bhargava, M., Shankar, A., Taniguchi, T., Thorne, F., Tsimerman, J., & Zhao, Y. (2020). Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves. Journal of the American Mathematical Society, 33(4), 1087-1099. doi:10.1090/jams/945.


Cite as: http://hdl.handle.net/21.11116/0000-0007-AFE8-F
Abstract
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 groups and ranks of Jacobians of hyperelliptic curves; and 4) bounds of Baily and Wong on the number of $A_4$-quartic fields of bounded discriminant.