Weil–Châtelet divisible elements in Tate–Shafarevich groups I: The Bashmakov problem for elliptic curves over $ \mathbb{Q} $. (May 2013)

• Record Type: Journal Article
• Authors: Çiperiani, Mirela; Stix, Jakob
• Is Part Of: Compositio mathematica. Volume 149:Number 5(2013)
• Page Start: 729
• Page End: 753
• Publication Date: 2013-05
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X12000747
• Languages: English
• ISSNs: 0010-437X

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>For an abelian variety <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wm4d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$A$]]></tex-math></alternatives></inline-formula> over a number field <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wjqr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$k$]]></tex-math></alternatives></inline-formula> we discuss the maximal divisible subgroup of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wm72" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathrm{H} }^{1} (k, A)$]]></tex-math></alternatives></inline-formula> and its intersection with the subgroup Ш<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wjm3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$(A/ k)$]]></tex-math></alternatives></inline-formula>. The results are most complete for elliptic curves over <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wk3b" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$ \mathbb{Q} $]]></tex-math></alternatives></inline-formula>.</p> </abstract>