Iwasawa theory for the anticyclotomic extension
Let \(E\) be an elliptic curve over \(\mathbb Q\) with complex multiplication by the ring of integers of a quadratic imaginary field \(K\), and let \(p\) be a prime, different from 2 and 3, which splits in \(K\). The author determines the \(\Lambda\)-module structure of the local units modulo elliptic units for the anticyclotomic \(\mathbb Z_p\)-extension \(K^{-}_{\infty}\) of \(K\). To be more precise, choose a prime \(\mathfrak p\) of \(K\) dividing \(p\), let \(K_n\) denote the \(n\)th layer of \(K^-_{\infty}\) and let \(U_ n\) denote the group of principal units of the completion of \(K_ n\) at the prime above \(\mathfrak p\). Define the elliptic units \(C_ n\) of \(K_ n\) to be the norms to \(K_ n\) of the elliptic units of the field \(K(E_{p^{n+1}})\) generated over \(K\) by the coordinates of the \(p^{n+1}\)-division points of \(E\). Let \(\overline{C_n}\) denote the module generated over \(\mathbb Z_p\) by \(C_n\), and set \(\overline{C_{\infty}}=\lim_{\leftarrow}\overline{C_ n}\) and \(U_{\infty}=\lim_{\leftarrow}U_ n\). The author shows that the module \(Y_{\infty}=U_{\infty}/\overline{C_{\infty}}\) is \(\Lambda\)-isomorphic to the quotient of \(\Lambda\) by an explicitly described ideal. The author also points out that the same techniques will yield a similar result for any \(\mathbb Z_p\)-extension contained in \(K(E_{p^{\infty}})\) in which \(\mathfrak p\) is infinitely ramified.
anticyclotomic \(\mathbb Z_p\)-extension
scientific article; zbMATH DE number 3972990