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Visibility of ideal classes

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
130
Issue
12
Identifiers
DOI: 10.1016/j.jnt.2010.07.005
Keywords
  • Ideal Class Groups
  • Capitulation
  • Cyclotomic Fields
  • Shafarevich–Tate Group
  • Visibility

Abstract

Abstract Cremona, Mazur, and others have studied what they call visibility of elements of Shafarevich–Tate groups of elliptic curves. The analogue for an abelian number field K is capitulation of ideal classes of K in the minimal cyclotomic field containing K. We develop a new method to study capitulation and use it and classical methods to compute data with the hope of gaining insight into the elliptic curve case. For example, the numerical data for number fields suggests that visibility of non-trivial Shafarevich–Tate elements might be much more common for elliptic curves of positive rank than for curves of rank 0.

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