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


Authors: M. Bhargava, A. Shankar, T. Taniguchi, F. Thorne, J. Tsimerman and Y. Zhao
Journal: J. Amer. Math. Soc. 33 (2020), 1087-1099
MSC (2010): Primary 11G05, 11R29
DOI: https://doi.org/10.1090/jams/945
Published electronically: August 28, 2020
MathSciNet review: 4155220
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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.


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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.
Article copyright: © Copyright 2020 American Mathematical Society