Organizing the arithmetic of elliptic curves (Q2577006)

Let \(E\) be an elliptic curve over \({\mathbb Q}\), \(p\) a prime number and \(K\) a number field. Fixing this data \((p,K,E)\), let \(K_{\infty}/K\) denote the maximal \({\mathbb Z}_p\)- power extension of \(K\) and put \(G=\) Gal\((K_{\infty}/K)\). The arithmetic of \(E\) along intermediary layers of \(K_{\infty}\) has been studied in several recent works. In previous articles [Elliptic curves and class field theory. Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press, 185--195 (2002; Zbl 1036.11023); Pairings in the arithmetic of elliptic curves. Modular curves and Abelian varieties. Based on lectures of the conference, Bellaterra, Barcelona, July 15--18, 2002. Basel: Birkhäuser. Prog. Math. 224, 151--163 (2004; Zbl 1071.11028)], the authors showed that under certain assumptions, these various results could be encapsulated in a single skew-Hermitian matrix \(H\) with entries in the Iwasawa algebra \({\mathbb Z}_p[[G]]\). The authors use the terminology that `\(H\) organizes the arithmetic of \((p,K,E)\)'. Under the additional hypothesis that the \(p\)-torsion of the Tate-Shafarevich group of \(E/K\) is trivial, the matrix \(H\) is related to the matrix defining the \(p\)-adic height pairing on \(E(K)\). The main results of this paper provide a construction of such `organizing matrices' in a fairly general context. These results make crucial use of the work of Nekovař, which in turn uses work of Greenberg. The authors then give various numerical examples where the organizing matrix is described explicitly. They also provide a counter-example to a conjecture on regular primes in anti-cyclotomic \({\mathbb Z}_p\)-extensions, made in their earlier papers.