The growth of Selmer ranks of an abelian variety with complex multiplication
Let \(A\) be an abelian variety over a field \(F\) having complex multiplication by integers of \(K\), where \(F/K\) is a Galois extension and \(K\) is a CM field. The authors study the variation of the \({\mathcal P}\)-ranks of the Selmer groups in pro-\(p\) algebraic extensions \(\{F_n/F\} \) where \({\mathcal P}\) is a prime in \(K\) over odd \(p\). After studying the \(\mathbb{Z} _p\) extension case obtaining an analogue of Mazur's control theorem, they consider the case where \(A=E\) is an elliptic curve, \(K\) is an imaginary quadratic field, and \(F_n \) is the \(n\)th layer of the infinite \(p\)-Hilbert class field tower \(F_\infty /F\), and prove under certain hypotheses that the \({\mathcal P}\)-rank of the Selmer group of \(E\) over \(F_n\) is unbounded as \(n\rightarrow \infty \), and that if furthermore the \(p\)-primary part \(E(F)_p=0\) then the \({\mathcal P}\)-rank of the Selmer group of \(E\) over \(F_\infty\) is infinite, so is the Mordell-Weil rank or the \({\mathcal P}\)-rank of the Shafarevich-Tate group. Similar problems were studied by Mazur, Coates, and especially \textit{R. Greenberg} [Introduction to Iwasawa theory for elliptic curves, IAS/Park City Math. Ser. 9, 407--464 (2001; Zbl 1002.11048)]. Note that the second case here is not covered by Greenberg's theory since the \(p\)-class field tower is not \(p\)-adic analytic.
scientific article; zbMATH DE number 5074493