Bounds on $2$-torsion in class groups of number fields and integral points on elliptic curves
Bounds on $2$-torsion in class groups of number fields and integral points on elliptic curves
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椭圆曲线上数域和积分点的类群中 $2$ 扭转的界限
DOI：
10.1090/jams/945
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发表时间：
2020
影响因子：
3.9
通讯作者：
Zhao Y.
中科院分区：
文献类型：
--
作者：
Bhargava M.;Shankar A.;Taniguchi T.;Thorne F.;Tsimerman J.;Zhao Y.
本文证明了三次及更高次数域类群的2-扭子群的大小的第一个已知的非平凡界（平凡界来自于整个类群上的界）。这产生了相应的改进：（1）界的Brumer和克雷默的大小和秩的2-塞尔默群的椭圆曲线，（2）界的Helfgott和Venkatesh的整数点的数量的椭圆曲线，（3）界的大小的2-塞尔默群和秩的Jacobian的超椭圆曲线，和（4）界的Bethnic和Wong的数量的四次域的有界判别式。引用
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(the trivial bound beingcoming 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-quartic fields of bounded discriminant. References