The distribution of second p-class groups on coclass graphs

Abstract

For a given prime p, Leedham-Green, Newman, and Eick have defined the structure of a directed graph G(p) on the set of all isomorphism classes of finite p-groups. Two vertices are connected by an edge G → H if G is isomorphic to the last lower central quotient H/γc(H) where c = cl(H) denotes the nilpotency class of H. If the condition |H| = p|G| is imposed on the edges, G(p) is partitioned into countably many disjoint subgraphs G(p, r), r ≥ 0, called coclass graphs of p-groups G of coclass r = cc(G) = n− cl(G) where |G| = p. A coclass graph G(p, r) is a forest of finitely many coclass trees T (Gi) with roots Gi, each with a single infinite main line having a pro-p-group of coclass r as its inverse limit, and additionally contains finitely many sporadic groups outside of coclass trees: G(p, r) = (∪i T (Gi)) ∪ G0(p, r). By Artin’s reciprocity law, the second p-class groups Gp(K) = Gal(F 2 p(K)|K) of algebraic number fields K, where Fp(K) denotes the second Hilbert p-class field of K, are vertices of the metabelian skeleton of G(p). Our aim is firstly to provide a general algorithm for determining the structure of Gp(K) for a given number field K by means of number theoretical invariants of the intermediate fields K ≤ N ≤ Fp(K) between K and its first Hilbert p-class field Fp(K) and secondly to show that the arithmetic of special types of base fields K gives rise to selection rules for Gp(K), e.g. • If p = 2 and K is complex quadratic of type (2, 2), there are no selection rules and G(2, 1) is entirely populated by the G2(K), apart from the isolated group C4. • If p = 3 and K is complex quadratic of type (3, 3) or real quadratic of type (3, 3) without total principalization, then either G3(K) is sporadic or lies on an even branch B2k of a coclass tree of an even coclass graph G(3, 2j). • If p ≥ 3, K is quadratic of type (p, p), and Gp(K) is of coclass 1, then K must be real quadratic and Gp(K) lies on an odd branch B2k+1 of the unique coclass tree T (Cp × Cp) of G(p, 1). Our aforementioned new algorithm is based on the family of transfers Vi : G/G ′ → Ui/U ′ i from a metabelian p-group G to all intermediate groups G′ ≤ Ui ≤ G. We prove that the main lines of coclass trees, and all other coclass families arising from the periodicity of branches, share a common transfer kernel type κ(G) = (ker(Vi)) and that κ(G) is determined by the transfer target type τ(G) = (str(Ui/U ′ i)) where str(A) denotes the multiplet of type invariants of an abelian p-group A. Consequently, the structure of Gp(K) and the principalization type of a number field K is determined by the structures of the p-class groups Clp(Ni) of all intermediate fields K ≤ Ni ≤ Fp(K), according to the Artin reciprocity law. We have implemented this algorithm in PARI/GP to determine the structure of the second 3-class groups G3(K) = Gal(F 2 3(K)|K) of the 4 596 quadratic number fields K = Q( √ D) with discriminant −10 < D < 10 and 3-class group Cl3(K) of type (3, 3) and to analyze their distribution on the coclass graphs G(3, r), 1 ≤ r ≤ 6. References. [1] D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. (2011), DOI 10.1007/s00605-010-0277-x. [2] D. C. Mayer, The second p-class group of a number field (preprint 2010). [3] D. C. Mayer, Principalization algorithm via class group structure (preprint 2011).