The reciprocity conjecture of Khare and Wintenberger

We prove a strengthening of the "reciprocity conjecture" of Khare and Wintenberger. The input to the original conjecture is an odd prime p, a CM number field F containing the pth roots of unity, and a pair (q₁, 1₂) of primes of the maximal totally real subfield F⁺ of F that are inert in the cyclotomic Z_p-extension F⁺ _∞/F⁺. In analogy to a statement about generalized Jacobians of curves, the conjecture asserts the equality of two procyclic subgroups of the Galois group of the maximal pro-p extension ℳ of F ⁺_∞ that is unramified outside p and abelian over F⁺. The first is the intersection with Gal(ℳ/F⁺ _∞) of the closed subgroup of Gal(ℳ/F⁺) generated by the Frobenius elements of q₁ and q₂. The second is generated by the class of an exact sequence defining the minus part of the p-part of the ray class group of F_∞ of conductor.