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

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 K (the trivial bound being O ϵ , n ( | D i s c ( K ) | 1 / 2 + ϵ ) 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 A_4 -quartic fields of bounded discriminant.