Oddness of residually reducible Galois representations

We show that suitable congruences between polarized automorphic forms over a CM field always produce elements in the Selmer group for exactly the ±-Asai (aka tensor induction) representation that is critical in the sense of Deligne. For this we relate the oddness of the associated polarized Galois representations (in the sense of the Bella ̈ıche-Chenevier sign being +1) to the parity condition for criticality. Under an assumption similar to Vandiver’s conjecture this also provides evidence for the Fontaine-Mazur conjecture for polarized Galois representations of any even dimension.