The reciprocity conjecture of Khare and Wintenberger

TL;DR

This paper proves a strengthened version of the reciprocity conjecture by Khare and Wintenberger, relating Galois groups and class groups in cyclotomic extensions of CM fields, confirming a deep number-theoretic symmetry.

Contribution

It provides a proof of a strengthened form of the reciprocity conjecture, clarifying the relationship between Galois groups and class groups in specific cyclotomic extensions.

Findings

Confirmed the equality of two procyclic subgroups of the Galois group.

Established the connection between Galois group intersections and class group sequences.

Extended the conjecture to a stronger, more general setting.

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.