Galois descent for affine Bost-Connes boundaries: capitulation, the obstruction cocycle, and the emergence of the class group

Abstract

Let p be a non-principal prime of a number field K, h the order of its class, H the Hilbert class field, G = Gal(H/K) = Cl_K. By the principal ideal theorem p capitulates, pO_H = (Pi), and p splits into |Cl_K|/h primes of residue degree h, the Frobenius being the class itself. The capitulated affine system over H is constructed on the product of the local completions with unit-extended scaling monoid - the extension is forced, since sigma(Pi) = u_sigma Pi - and three descent theorems are proved. The obstruction cocycle sigma -> sigma(Pi)/Pi represents the image of [p] under a canonical injection of Cl_K into H^1(G, O_H^*) built from Hilbert 90 and the principal ideal theorem: the full class group is exactly the group of affine descent obstructions, and an invariant generator exists iff p is principal. The bottom scaling is the norm of the top scaling up to units, and Green imprimitivity plus Julg reduces the equivariant K-theory to the inert local model crossed by Z/h. The ladder of unit classes is explicit: ord[1] = q^h - 1 at the bottom and q^{|Cl_K|} - 1 at the top - the order of the class at the bottom, the class number at the top, the group structure in the cocycles. The integral equivariant computation of the reduction, the fixed-point versus crossed-product Morita question, and the separation of the unit-torus footprint are recorded as the remaining problems.