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

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M. Bhargava, A. Shankar, T. Taniguchi, F. Thorne, J. Tsimerman and Y. Zhao
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## 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 $O_{\epsilon ,n}(|\mathrm {Disc}(K)|^{1/2+\epsilon })$ coming from the bound on the entire class group). 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.## References

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## Additional Information

**M. Bhargava**- Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey
- MR Author ID: 623882
- Email: bhargava@math.princeton.edu
**A. Shankar**- Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada
- Email: ashankar@math.toronto.edu
**T. Taniguchi**- Affiliation: Department of Mathematics, Graduate School of Science, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
- MR Author ID: 749172
- Email: tani@math.kobe-u.ac.jp
**F. Thorne**- Affiliation: Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, South Carolina 29208
- MR Author ID: 840724
- Email: thorne@math.sc.edu
**J. Tsimerman**- Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada
- MR Author ID: 896479
- Email: jacobt@math.toronto.edu
**Y. Zhao**- Affiliation: School of Science, Westlake University, Hangzhou, 310024, People’s Republic of China
- Email: zhaoyongqiang@westlake.edu.cn
- Received by editor(s): March 12, 2017
- Received by editor(s) in revised form: November 1, 2019, and November 20, 2019
- Published electronically: August 28, 2020
- Additional Notes: The first author was supported in part by a Simons Investigator Grant and NSF Grant DMS-1001828.

The third author was supported in part by the JSPS and KAKENHI Grants JP24654005 and JP25707002.

The fourth author was supported in part by NSF Grant DMS-1201330, by the National Security Agency under a Young Investigator Grant, and by the Simons Foundation under Grants No. 563234 and 586594. - © Copyright 2020 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**33**(2020), 1087-1099 - MSC (2010): Primary 11G05, 11R29
- DOI: https://doi.org/10.1090/jams/945
- MathSciNet review: 4155220