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Weil–Châtelet divisible elements in Tate–Shafarevich groups I: The Bashmakov problem for elliptic curves over $ \mathbb{Q} $

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 February 2013

Mirela Çiperiani  and
Jakob Stix
Mirela Çiperiani
Affiliation:
Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712, USA email mirela@math.utexas.edu
Jakob Stix
Affiliation:
Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany email stix@mathi.uni-heidelberg.de
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Abstract

For an abelian variety $A$ over a number field $k$ we discuss the maximal divisible subgroup of ${\mathrm{H} }^{1} (k, A)$ and its intersection with the subgroup Ш$(A/ k)$. The results are most complete for elliptic curves over $ \mathbb{Q} $.

Keywords

elliptic curveabelian varietySelmer groupWeil–Châtelet groupTate–Shafarevich group

MSC classification

Secondary: 11G05: Elliptic curves over global fields 11G10: Abelian varieties of dimension $> 1$
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 5 , May 2013 , pp. 729 - 753
Copyright
© The Author(s) 2013 

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