Weil–Châtelet divisible elements in Tate–Shafarevich groups I: The Bashmakov problem for elliptic curves over $ \mathbb{Q} $. (May 2013) - Çiperiani, Mirela; Stix, Jakob
Abstract For an abelian variety <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wm4d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$A$]]> </alternatives> over a number field <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wjqr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$k$]]> </alternatives> we discuss the maximal divisible subgroup of <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wm72" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\mathrm{H} }^{1} (k, A)$]]> </alternatives> and its intersection with the subgroup Ш<alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wjm3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$(A/ k)$]]> </alternatives>. The results are most complete for elliptic curves over <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wk3b" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$ \mathbb{Q} $]]> </alternatives>.