Marcus du Sautoy

Analytic properties of zeta functions and subgroup growth

It has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group G. Although the story began much before, the recent activity grew out of a paper by Dan Segal in [36]. The story so far has been well-documented in Lubotzky’s subsequent survey paper in [30]. In [24] the second author of this article, Segal and Smith introduced the zeta function of a group as a tool for understanding this growth of subgroups. Let an(G) be the number of subgroups of index n in the finitely generated group G and sN (G) = a1(G) + · · ·+ aN (G) be the number of subgroups of index N or less. The zeta function is defined as the Dirichlet series with coefficients an(G) and has a natural interpretation as a noncommutative generalization of the Dedekind zeta function of a number field:

New horizons in pro-p groups

Lie methods in the theory of pro-p groups, A. Shalev on the classification of prop-p groups and finite p-groups, C.R. Leedham-Green, S. McKay prop-p trees and applications, L. Ribes, P. Zalesskii just infinite branch groups, R. I. Grigorchuk on just infinite abstract and profinite groups, J.S. Wilson the Nottingham group, R. Camina on groups satisfying the Golod-Shafarevich condition, E. Zelmanov sub-group growth in prop-p groups, A. Mann zeta functions of groups, M. du Sautoy, D. Segal where the wild things are - ramification groups and the Nottingham group, M. du Sautoy, I. Fesenko p-adic Galois representations and prop-p Galois groups, N. Boston the cohomology of p-adic analytic groups, P. Symonds, T. Weigel. Appendix: further problems.

A nilpotent group and its elliptic curve: Non-uniformity of local zeta functions of groups

A nilpotent group is defined whose local zeta functions counting subgroups and normal subgroups depend on counting points modp on the elliptic curvey2=x3−x. This example answers negatively a question raised in the paper of F. J. Grunewald, D. Segal and G. C. Smith where these local zeta functions were first defined. They speculated that local zeta functions of nilpotent groups might be finitely uniform asp varies. A proof is given that counting points on the elliptic curvey2=x3−x are not finitely uniform, and hence the same is true for the zeta function of the associated nilpotent group. This example demonstrates that nilpotent groups have a rich arithmetic beyond the connection with quadratic forms.

Zeta functions of groups and rings: Uniformity (page 1)

There are various natural local zeta functions associated with groups and rings for each primep. We consider the question of how these functions behave as we vary the primep and the groups (or rings) range over a specific class of groups (or rings), e.g. finitely generated torsion-free nilpotent groups of a fixed Hirsch length orp-adic analytic groups of a fixed dimension. Using a result of Macintyre’s on the uniformity of parameterizedp-adic integrals, together with various natural parameter spaces we define for these classes of groups, we prove a strong finiteness theorem on the possible poles of these local zeta functions.

Zeta functions of classical groups and their friendly ghosts

Abstract The zeta functions of the classical reductive groups in general have natural boundaries. We introduce in this Note the concept of the ghost zeta function which exists for any zeta function expressed as an Euler product of uniformly rational functions. We give an explicit description of the ghost zeta functions for the classical groups and prove that they are meromorphic functions.

Symmetry: A Bridge Between the Two Cultures

When I was a kid I didn’t want to be a mathematician at all. My dream was to become a spy. This dream was fueled by my mother who had been in the foreign office before she’d had children. But becoming a mother was apparently incompatible with being a diplomat and so she had to leave her job. But she told me and my sister that they had at least allowed her to keep the black gun every member of the foreign office was required to carry as a member of the diplomatic service. I became convinced that my mother must in fact have been a spy and that she might be recalled to active duty at any moment.

Burden of proof

A controversy behind the biggest prize in mathematics highlights a troubling crack in its foundations. Mathematician Marcus du Sautoy reveals all