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.

doi:10.1090/jams/945

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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