Bounds on 2torsion in class groups of number fields and integral points on elliptic curves
Abstract
We prove the first known nontrivial bounds on the sizes of the 2torsion 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 BrauerSiegel). This yields corresponding improvements to: 1) bounds of Brumer and Kramer on the sizes of 2Selmer 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 2Selmer 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.02458
 Bibcode:
 2017arXiv170102458B
 Keywords:

 Mathematics  Number Theory;
 11R29;
 11G05
 EPrint:
 12 pages