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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Bounds on $2$-torsion in class groups of number fields and integral points on elliptic curves
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by M. Bhargava, A. Shankar, T. Taniguchi, F. Thorne, J. Tsimerman and Y. Zhao HTML | PDF
J. Amer. Math. Soc. 33 (2020), 1087-1099 Request permission

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.
  • © 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