### 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 (q_{1}, 1_{2}) 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 q_{1} and q_{2}. 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1409-1424 |

Number of pages | 16 |

Journal | International Mathematics Research Notices |

Volume | 2014 |

Issue number | 5 |

DOIs | |

State | Published - Nov 1 2014 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)