Iwasawa theory for elliptic curves over imaginary quadratic fields
Let \(E\) be an elliptic curve defined over \(\mathbb Q\), let \(K\) be a complex quadratic number field, and let \(K_\infty\) be a \(\mathbb Z_p\)-extension of \(K\). Let \(\Sigma\) be a set of primes of \(K\) containing the primes above \(p\) as well as the primes of bad reduction of \(E\), and let \(K_\Sigma\) denote the maximal algebraic extension of \(K\) that is unramified outside \(\Sigma\). For a subextension \(L/K\) of \(K_\Sigma/K\) we have the Selmer groups Sel\(_{p^n}(E/L)\); these form inverse and direct systems in a natural way, with limits \(S_p(E/L)\) and Sel\(_{p^\infty}(E/L)\), respectively. Now consider these Selmer groups over the \(n\)-th layer \(K_n\) of \(K_\infty\), and assume that \(E\) has complex multiplication by \(K\). Then it is conjectured that there is an integer \(r \in \{0, 1, 2\}\) depending on \(E\) and \(K_\infty/K\) such that the \(\mathbb Z_p\)-rank of \(S_p(E/K_n)\) (or, equivalently, the \(\mathbb Z_p\)-corank of Sel\(_{p^\infty}(E/L)\) is equal to \(r p^n + O(1)\)). The author provides evidence for this conjecture by proving it e.g. for the anticyclotomic \(\mathbb Z_p\)-extension of \(K\) under suitable assumptions guaranteeing the existence of Heegner points (in particular, the \(L\)-series \(L(E/K,s)\) should vanish to odd order at \(s = 1\)). In this case it turns out that we have \(r = 1\). The article also contains information on the structure of the cohomology groups\break \(H^i(K_\Sigma/K_\infty,E[p^\infty])\) for \(i = 1, 2\) viewed as discrete modules over the Iwasawa algebra of \(K_\infty/K\).
scientific article; zbMATH DE number 2081344
scientific article