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

Abstract

We prove the existence of a new structure on the first Galois cohomology of generic families of symplectic self-dual $p$-adic representations of $G_{\mathbb{Q}_p}$ of rank two (a local sign decomposition): a functorial decomposition into free rank one Lagrangian submodules which encodes the $p$-adic variation of Bloch--Kato subgroups via completed epsilon constants, mirroring a symplectic structure. The local sign decomposition has diverse local as well as global arithmetic consequences. This includes compatibility of the Mazur--Rubin arithmetic local constant and completed epsilon constants, answering a question of Mazur and Rubin. The compatibility leads to new cases of the $p$-parity conjecture for Hilbert modular forms at supercuspidal primes $p$. We also formulate and prove an analogue of Rubin's conjecture over ramified quadratic extensions of $\mathbb{Q}_p$. Using it, we construct an integral $p$-adic $L$-function for anticyclotomic deformation of a CM elliptic curve at primes $p$ ramified in the CM field.