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.

中文翻译：

---

椭圆曲线上数域和积分点的类群中2-扭转的上界

我们证明了第一个已知的关于三次和更高阶数域 $K$ 的类群的 2-扭转子群的大小的非平凡边界（平凡边界是 $O_{\epsilon}(|{\rm Disc}(K )|^{1/2+\epsilon})$ 来自 Brauer--Siegel）。这对以下方面产生了相应的改进：1) Brumer 和 Kramer 在 2-Selmer 群的大小和椭圆曲线的等级上的界限；2) Helfgott 和 Venkatesh 关于椭圆曲线上积分点数的界限；3) 2-Selmer 群的大小和超椭圆曲线的雅可比行列的界限；和 4) Baily 和 Wong 在有界判别式的 $A_4$-四次域的数量上的边界。