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キーワード:
Mathematics, Number Theory
要旨:
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.
