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 Zp-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.
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