Recent progress in determining p-class field towers

Abstract

For a fixed prime p, the p-class tower F_p^∞K of a number field K is considered to be known if a pro-p presentation of the Galois group G = Gal (F_p^∞K ∕ K) is given. In the last few years, it turned out that the Artin pattern AP(K) = (τ(K), κ(K)) consisting of targets τ(K) = (Cl_pL) and kernels κ(K) = (ker J_L|K) of class extensions J_L|K: Cl_p K → Cl_pL to unramified abelian subfields L|K of the Hilbert p-class field F_p¹K only suffices for determining the two-stage approximation M = G ∕ G'' of G. Additional techniques had to be developed for identifying the group G itself: searching strategies in descendant trees of finite p-groups, iterated and multilayered IPADs of second order, and the cohomological concept of Shafarevich covers involving relation ranks. This enabled the discovery of three-stage towers of p-class fields over quadratic base fields K = ℚ(√ d) for p ∈ {2, 3, 5}. These non-metabelian towers reveal the new phenomenon of various tree topologies expressing the mutual location of the groups G and M.