Organizing the Arithmetic of Elliptic Curves

Fix the data (p,K,E) where p is a prime number, K a number field, and E an elliptic curve over Q. Let K∞/K denote the maximal Zp-power extension of K. Recent work provides, in some instances, detailed information about p-adic completions of Mordell-Weil groups and their associated p-adic height pairings, and the p-primary Shafarevich-Tate groups and their associated Cassels pairings, over intermediate fields in K∞/K. Added to this information we also have a constellation of conjectures telling us even more precisely how all this arithmetic should behave. In previous articles [MR1, MR2] we have considered the possibility that, under some not too stringent assumptions, much of this arithmetic data can be packaged efficiently in terms of a single skew-Hermitian matrix with entries drawn from the Iwasawa algebra of the Zp-power extension K∞/K. We say that such a matrix H organizes the arithmetic of (p,K,E) if it plays this role vis-à-vis the arithmetic of (p,K,E). For a detailed discussion of this, see §7 below. In the special case where there is no nontrivial p-torsion in the Shafarevich-Tate group of E overK, our skewHermitian matrix may be thought of as a (skew-Hermitian) lifting to the Iwasawa algebra of the matrix describing the p-adic height pairing on the Mordell-Weil group E(K).