Oddness of residually reducible Galois representations

TL;DR

This paper establishes a link between congruences of automorphic forms over CM fields, the oddness of associated Galois representations, and the non-triviality of certain Selmer groups, providing evidence for deep conjectures in number theory.

Contribution

It relates the oddness of polarized Galois representations to the criticality of Asai representations and connects this to the Fontaine-Mazur conjecture under certain assumptions.

Findings

Congruences produce elements in Selmer groups for critical Asai representations.

Oddness of Galois representations correlates with parity conditions for criticality.

Provides evidence for Fontaine-Mazur conjecture assuming a Vandiver-like conjecture.

Abstract

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\"iche-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.