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  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, 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.

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Item Permalink: https://hdl.handle.net/21.11116/0000-0007-AFE8-F Version Permalink: https://hdl.handle.net/21.11116/0000-0007-AFE9-E
Genre: Journal Article

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 Creators:
Bhargava, M., Author
Shankar, A., Author
Taniguchi, T., Author
Thorne, F., Author
Tsimerman, J., Author
Zhao, Yongqiang1, Author           
Affiliations:
1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Number Theory
 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.

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Language(s): eng - English
 Dates: 2020
 Publication Status: Issued
 Pages: 13
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 1701.02458
DOI: 10.1090/jams/945
 Degree: -

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Title: Journal of the American Mathematical Society
  Abbreviation : J. Amer. Math. Soc.
Source Genre: Journal
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Publ. Info: American Mathematical Society
Pages: - Volume / Issue: 33 (4) Sequence Number: - Start / End Page: 1087 - 1099 Identifier: -