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-infinity 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-infinity (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 infinity) can be described efficiently in terms of a single skew-Hermitian matrix with entries drawn from the lwasawa algebra of K-infinity/K. In this paper, using work of Nekovar [J. Nekovar, 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.