A Golod-Shafarevich Equality and p-Tower Groups

Details A Golod-Shafarevich Equality and p-Tower Groups A Golod-Shafarevich Equality and p-Tower Groups

2010-08-18

arxiv.org/abs/1008.3002v1

A Golod-Shafarevich Equality and p-Tower Groups. J. Number Theory 129 (2009), no. 11, 2808--2819

Mathematics

Number Theory

12 pages, pre-reviewer version

Scientific paper

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.

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