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

Bhargava, M.; Shankar, A.; Taniguchi, T.; Thorne, F.; Tsimerman, J.; Zhao, Yongqiang

doi:10.1090/jams/945

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学術論文

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;

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https://doi.org/10.1090/jams/945
(出版社版)

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arXiv:1701.02458.pdf
(プレプリント), 178KB

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引用

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.

引用: https://hdl.handle.net/21.11116/0000-0007-AFE8-F

要旨

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.