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We prove a strengthening of the "reciprocity conjecture" of Khare andWintenberger. The input to the original conjecture is an odd prime p, a CMnumber field F containing the pth roots of unity, and a pair of primes of themaximal totally real subfield E of F that are inert in the cyclotomicZ_p-extension of E. In analogy to a statement about generalized Jacobians ofcurves, the conjecture asserts the equality of two procyclic subgroups of theGalois group G of the maximal p-ramified abelian pro-p extension of E. Thefirst is the intersection of the subgroup of G fixing the cyclotomicZ_p-extension with the closed subgroup of G generated by the two primes. Thesecond is generated by the class of an exact sequence defining the minus partof the p-part of the ray class group of the cyclotomic Z_p-extension of F ofconductor the product of the two primes of E.

The reciprocity conjecture of Khare and Wintenberger