Complex multiplication and parity in Iwasawa theory

Abstract

We give a new, somewhat elementary method for proving parity results about Iwasawa-theoretic Selmer groups and apply our method to certain Galois representations which are not self-dual. The main result is essentially that Iwasawa's λ-invariants for these representations over dihedral Z p d -extensions are even. Our approach is a specialization argument and does not make use of Nekovář's deformation-theoretic Cassels pairing, though Nekovář's theory implies our results. Examples of the representations we consider arise naturally in the study of CM abelian varieties defined over the totally real subfield of the reflex field of the CM type. We also discuss connections with “large Selmer rank” in the sense of Mazur–Rubin and give several examples in the context of abelian varieties and modular forms.