A local sign decomposition for symplectic self-dual Galois representations of rank two

We present a new structure on the first Galois cohomology of families of symplectic self-dual p-adic representations of $G_Qp$ of rank two. This is a functorial decomposition into free rank one Lagrangian submodules encoding Bloch-Kato subgroups and epsilon factors, mirroring an underlying symplectic structure.   This local sign decomposition has local as well as global applications, including compatibility of the Mazur-Rubin arithmetic local constants and epsilon factors, and new cases of the parity conjecture. It also leads to a formulation and proof of an analogue of Rubin's conjecture over ramified quadratic extensions of Qp, which initiates an integral Iwasawa theory for CM elliptic curves at primes of additive reduction. (Joint with S. Kobayashi, K. Nakamura, and K. Ota.)