The first edition of this book was the indispensable reference for researchers in the theory of pro-p groups. In this second edition the presentation has been improved and important new material has been added. The first part of the book is group-theoretic. It develops the theory of pro-p groups of finite rank, starting from first principles and using elementary methods. Part II introduces p-adic analytic groups: by taking advantage of the theory developed in Part I, it is possible to define these, and derive all the main results of p-adic Lie theory, without having to develop any sophisticated analytic machinery. Part III, consisting of new material, takes the theory further. Among those topics discussed are the theory of pro-p groups of finite coclass, the dimension subgroup series, and its associated graded Lie algebra. The final chapter sketches a theory of analytic groups over pro-p rings other than the p-adic integers.
We investigate in this paper the zeta function [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i"/] associated to a nilpotent group Γ introduced in [GSS]. This zeta function counts the subgroups H ≤ Γ whose profinite completion Ĥ is isomorphic to the profinite completion [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i"/]. By representing [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i"/] as an integral with respect to the Haar measure on the algebraic automorphism group G of the Lie algebra associated to Γ and by generalizing some recent work of Igusa [I], we give, under some assumptions on Γ, an explicit finite form for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i"/] in terms of the combinatorial data of the root system of G and information about the weights of various representations of G . As a corollary of this finite form we are able to prove (1) a certain uniformity in p confirming a question raised in [GSS]; and (2) a functional equation that the local factors satisfy [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i"/]. This functional equation is perhaps the most important result of the paper as it is a new feature of the theory of zeta functions of groups.
L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ However Higman predicted a more subtle behaviour of this function as you vary the prime p and fix n which starts to reveal itself in the above evidence for n = 5 and 6. This is encapsulated in what has become known as Higman's PORC conjecture: Conjecture 1.2. (PORC)-For fixed n there is an integer N and polynomials Pn,iQ^)for 0 ^ i ^ N-1 so that if p = i mod N then /(^)=P^). PORC stands for Polynomial On Residue Classes. Higman's PORC conjecture has withstood any attack since Higman's own contribution in [23] in which he proved that counting class 2 elementary abelian p by elementary abelian ^-groups was PORG.
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