The reciprocity conjecture of Khare and Wintenberger (Q2878712)

The goal of the present paper is to prove the reciprocity conjecture of \textit{C. Khare} and \textit{J.-P. Wintenberger} as found in [Int. Math. Res. Not. 2014, No. 1, 194--223 (2014; Zbl 1307.11117)]. Let \(p\) be an odd prime, \(F\) be a CM number field containing the \(p\)th roots of unity, \((\mathfrak q_1,\mathfrak q_2)\) a pair of primes of the maximal totally real subfield \(F^+\) of \(F\) that are inert in the cyclotomic \(\mathbb Z_p\) extension \(F_\infty^+/F^+\). The conjecture asserts the equality of two pro-cyclic subgroups of the Galois group of the maximal pro-\(p\) extension \(\mathcal M\) of \(F_\infty^+\) that is unramified outside \(p\) and abelian over \(F^+\). The first is the Frobenius line \(M_G\) defined as the intersection with \(\mathrm{Gal}(\mathcal M/F_\infty^+)\) of the closed subgroup of \(\mathrm{Gal}(\mathcal M/F^+)\) generated by the Frobenius elements of \(\mathfrak q_1\) and \(\mathfrak q_2\). The second is the ray class line \(N_G\) generated by the class of an exact sequence defining the minus part of the \(p\)-part of the ray class group of \(F_\infty\) of conductor \(\mathfrak q_1\mathfrak q_2\).