A Golod-Shafarevich Equality and p-Tower Groups

TL;DR

This paper establishes a Golod-Shafarevich equality for analytic pro-p-groups and applies it to show that certain p-groups associated with quadratic imaginary fields are necessarily very large, refining understanding of p-class field towers.

Contribution

It introduces a Golod-Shafarevich equality for analytic pro-p-groups and applies it to classify and constrain the size of p-groups in quadratic imaginary fields.

Findings

Finite p-groups of certain types are necessarily very large.

The Golod-Shafarevich equality refines previous inequalities.

Applications to p-class field towers of quadratic imaginary fields.

Abstract

All current techniques for showing that a number field has an infinite p-class field tower depend on one of various forms of the Golod-Shafarevich inequality. Such techniques can also be used to restrict the types of p-groups which can occur as Galois groups of finite p-class field towers. In the case that the base field is a quadratic imaginary number field, the theory culminates in showing that a finite such group must be of one of three possible presentation types. By keeping track of the error terms arising in standard proofs of Golod-Shafarevich type inequalities, we prove a Golod-Shafarevich equality for analytic pro-p-groups. As an application, we further work of Skopin, showing that groups of the third of the three types mentioned above are necessarily tremendously large.