Iwasawa \(\mu\)-invariants of elliptic curves and their symmetric powers.
This paper is a continuation of the author's earlier paper [J. Number Theory 93, No. 1, 1--22 (2002; Zbl 1014.11065)]. The main theme of these papers is to understand when and how the \(\mu\)-invariant of the Selmer group of an ordinary \(p\)-adic Galois representation over the cyclotomic \({\mathbb Z}_p\)-extension (\(p\) is always odd) is positive. The two main representations considered are those associated to an elliptic curve and the symmetric powers of this representation. In his earlier paper, the author introduced invariants \(m_k(A)\) and \(\delta_k(A)\); here \(A\) is the associated discrete module for the Galois representation and \(k\) is any subfield of the cyclotomic extension of the base field. For simplicity, let us assume that the representation is one associated to an elliptic curve defined over \({\mathbb Q}\) so that \(A\) is the Galois module usually denoted by \(E_{p^{\infty}}\). Let \(E\) have good, ordinary or split multiplicative reduction at \(p\) and suppose that \(E[p]\) is irreducible as a \(G_{\mathbb Q}\)-module. The curve \(E\) satisfies ``Hypothesis \textbf{N}'' if the image of the representation is the normalizer of a split Cartan subgroup and the subfield of \({\mathbb Q}(E[p])\) fixed by the ramified Cartan subgroup is real. The main theorem gives a series of equivalent conditions which relates the condition that \(E\) satisfies Hypothesis \textbf{N} to statements on existence of fields \(K\) containing the base field such that the `\(m\)-invariant' \(m_{\mathbb Q}(E_{\mid_{K}})\) is positive. A conjecture relating the \(m\)-invariant to the classical \(\mu\)-invariant is formulated for representations for which the Selmer group is definable and has the property that it is co-torsion as a module over the Iwasawa algebra.
scientific article; zbMATH DE number 2025725
scientific article