Organizing the arithmetic of elliptic curves

Abstract

Suppose that E is an elliptic curve defined over a number field K , p is a rational prime, and K ∞ is the maximal Z p -power extension of K . In previous work [B. Mazur, K. Rubin, Elliptic curves and class field theory, in: Ta Tsien Li (Ed.), Proceedings of the International Congress of Mathematicians, ICM 2002, vol. II, Higher Education Press, Beijing, 2002, pp. 185–195; B. Mazur, K. Rubin, Pairings in the arithmetic of elliptic curves, in: J. Cremona et al. (Eds.), Modular Curves and Abelian Varieties, Progress in Mathematics, vol. 224, 2004, pp. 151–163] we discussed the possibility that much of the arithmetic of E over K ∞ (i.e., the Mordell–Weil groups and their p -adic height pairings, the Shafarevich–Tate groups and their Cassels pairings, over all finite extensions of K in K ∞ ) can be described efficiently in terms of a single skew-Hermitian matrix with entries drawn from the Iwasawa algebra of K ∞ / K . 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.