The reciprocity conjecture of Khare and Wintenberger

Abstract

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 of primes of the maximal totally real subfield E of F that are inert in the cyclotomic Z_p-extension of E. In analogy to a statement about generalized Jacobians of curves, the conjecture asserts the equality of two procyclic subgroups of the Galois group G of the maximal p-ramified abelian pro-p extension of E. The first is the intersection of the subgroup of G fixing the cyclotomic Z_p-extension with the closed subgroup of G generated by the two primes. 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 the cyclotomic Z_p-extension of F of conductor the product of the two primes of E.