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

DOI10.1090/JAMS/945zbMATH Open1456.11101arXiv1701.02458OpenAlexW3081065211MaRDI QIDQ5131071

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Publication date: 2 November 2020

Published in: (Search for Journal in Brave)

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 $O_{epsilon}(|mDisc(K){|}^{1/2+epsilon})$ 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.

Full work available at URL: https://arxiv.org/abs/1701.02458

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