Organizing the arithmetic of elliptic curves

In this paper, using work of Nekovár˘ [J. Nekovár˘, Selmer complexes. Preprint available at 〈http://www.math.jussieu.fr/∼nekovar/pu/〉], we show that under not-too-stringent conditions such an “organizing” matrix does in fact exist. We also work out an assortment of numerical instances in which we can describe the organizing matrix explicitly.