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Iwasawa theory of elliptic curves and Galois module structure. (English) Zbl 0802.11051
Let \(E\) be an elliptic curve with complex multiplication by the ring of integers of \(K\), an imaginary quadratic field. Assume the class number of \(K\) is 1, and \(E\) is defined over \(K\) and has everywhere good reduction over \(F\), an abelian extension of \(K\). Let \(p\) be an odd prime which splits as \(p= \pi\pi^*\) over \(K\). Let \({\mathcal B}_ i\) be the \({\mathcal O}_ F\)-Hopf algebra representing \(\ker [\pi^{*i}]: E(\mathbb{Q}^ c)\to E(\mathbb{Q}^ c)\), and \({\mathcal A}_ i\) the Cartier dual of \({\mathcal B}_ i\). For \(Q\) in \(E(F)\) let \(G_ Q(i)\) be the preimage of \(Q\) under \([\pi^{*i}]\), and let \(C_ Q(i)\) be the maximal \({\mathcal A}_ i\)-stable submodule of the integral closure of \(\mathbb{Z}\) in the Kummer algebra \(F_ Q(i)= \text{Map}_{\Omega_ F} (G_ Q(i), \mathbb{Q}^ c)\). Then, as M. Taylor [Ill. J. Math. 32, 428-452 (1988; Zbl 0648.14019)] shows, \(C_ Q(i)\) is a \({\mathcal B}_ i\)-principal homogeneous space, hence a locally free \({\mathcal A}_ i\)-module. Thus, for \({\mathcal M}_ i\) the maximal order of \({\mathcal A}_ i\), there is a map \(\psi_ i: E(F)/ [\pi^{*i} ]E(F)\to C\ell({\mathcal M}_ i)\) into the locally free class group of \({\mathcal M}_ i\), by \(\psi_ i(Q)= (C_ Q (i)\cdot {\mathcal M}_ i)\).
The author shows that if the algebraic \(p\)-adic height pairing for \(E/F\) of B. Perrin-Riou [Invent. Math. 70, 369-398 (1983; Zbl 0547.14025)] is non-degenerate modulo torsion, that the \(p\)-primary component of the Tate-Shafarevich group of \(E/F\) is finite and that the rank of a certain eigenspace obtained from \(E(F)\) is 1, then there is some point \(Q\) of infinite order with \(\psi_ i(Q)=0\) for all \(i\).

11R23 Iwasawa theory
11G05 Elliptic curves over global fields
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
Full Text: DOI
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