K-theory of the affine boundary quotient: the return of the norm and the trace

Abstract

For the affine control system attached to a principal prime p=(pi) of a number field K of degree n, with q = N(p), the boundary quotient is stably isomorphic to C_0(K_p) crossed by (O_K[1/pi] semidirect Z), and an iterated Pimsner-Voiculescu computation over the exterior-power tower with integral adjugate transfers yields both K-groups. The norm survives: the class of the unit generates a torsion summand Z/(q-1) of exact order q-1, attaining the known divisibility bound. The splitting data survives: the intermediate exterior layers contribute torsion of order governed by det(q - Lambda^r pi); for real quadratic fields this is Z/|q+-1-Tr(pi)|, so the trace of the uniformizer enters the K-theory. Unexpectedly, the invariant hears the unit orbit of the uniformizer (computed explicitly in Q(sqrt 2)), hence is not canonical in the prime; the class group is located at the entrance of the theory, obstructing global uniformizers for non-principal primes. Three problems are recorded honestly: the ray-class repair, the extension data for higher degree and several primes, and the q-primary bookkeeping.