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

PDF

Related Concepts

Mathematics Algebraic number field Discriminant Elliptic curve Torsion (gastropod) Bounded function Supersingular elliptic curve Quartic function Pure mathematics Twists of curves Upper and lower bounds Schoof's algorithm Mathematical analysis Quarter period Medicine Computer science Surgery Artificial intelligence

Manjul Bhargava , Arul Shankar , Takashi Taniguchi , Frank Thorne , Jacob Tsimerman , Yongqiang Zhao ·

YOU? · · 2017 · Open Access · · DOI: https://doi.org/10.48550/arxiv.1701.02458 · OA: W2593483592

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 $O_ε(|{\rm Disc}(K)|^{1/2+ε})$ 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.