Finite non-metabelian Schur sigma-Galois groups of class field towers

TL;DR

This paper constructs specific finite p-groups with particular transfer kernel types and invariants, demonstrating their realization as Schur sigma-groups and linking these structures to class field towers of imaginary quadratic fields.

Contribution

It provides explicit examples of non-metabelian Schur sigma-groups with prescribed properties, advancing understanding of class field towers for primes 5 and 7.

Findings

Existence of 98 non-metabelian Schur sigma-groups for p=7

Infinite families of non-metabelian Schur sigma-groups for p=5

Identification of discriminants with class field tower length at least 3

Abstract

For each odd prime p>=5, there exist finite p-groups G with derived quotient G/D(G)=C(p)xC(p) and nearly constant transfer kernel type k(G)=(1,2,...,2) having two fixed points. It is proved that, for p=7, this type k(G) with the simplest possible case of logarithmic abelian quotient invariants t(G)=(11111,111,21,21,21,21,21,21) of the eight maximal subgroups is realized by exactly 98 non-metabelian Schur sigma-groups S of order 7^11 with fixed derived length dl(S)=3 and metabelianizations S/D(D(S)) of order 7^7. For p=5, the type k(G) with t(G)=(2111,111,21,21,21,21) leads to infinitely many non-metabelian Schur sigma-groups S of order at least 5^14 with unbounded derived length dl(S)>=3 and metabelianizations S/D(D(S)) of fixed order 5^7. These results admit the conclusion that d=-159592 is the first known discriminant of an imaginary quadratic field with 7-class field tower of precise length L=3, and d=-90868 is a discriminant of an imaginary quadratic field with 5-class field tower of length L>=3, whose exact length remains unknown.