arXiv subjects

Marcus du Sautoy

Publications and source records attributed to Marcus du Sautoy.

4 recordsLinked to original sources

arXiv2022

Uniformity in Higher class Free Lie algebras

Let $\mathfrak{f}_{c,2}$ denote a free class-$c$ Lie rings on $2$ generators. We investigate the zeta functions enumerating graded ideals in $\mathfrak{f}_{c,2}(\mathbb{F}_p)$ for $c\leq6$, prove that they are uniformly given by polynomials in $p$ for $c\leq5$ and not uniformly given by a polynomial in $p$ for $c=6$. We also show that the zeta functions enumerating one-step graded ideals $\mathfrak{f}_{c,2}(\mathbb{F}_p)$ is always given by a polynomial in $p$ for all $c$.

math.RA↗

arXiv2015

Natural boundaries for Euler products of Igusa zeta functions of elliptic curves

We study the analytic behaviour of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato-Tate conjectures we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.

math.NT↗

arXiv2011

Non-PORC behaviour of a class of descendant $p$-groups

We prove that the number of immediate descendants of order $p^10$ of $G_p$ is not PORC (Polynomial On Residue Classes) where $G_p$ is the $p$-group of order $p^9$ defined by du Sautoy's nilpotent group encoding the elliptic curve $y^2=x^3-x$. This has important implications for Higman's PORC conjecture.

math.GR↗

arXiv2000

Analytic properties of zeta functions and subgroup growth

In this paper we introduce some new methods to understand the analytic behaviour of the zeta function of a group. We can then combine this knowledge with suitable Tauberian theorems to deduce results about the growth of subgroups in a nilpotent group. In order to state our results we introduce the following notation. For αa real number and N a nonnegative integer, define s_N^α(G) = sum_{n=1}^N a_n(G)/n^α. Main Theorem: Let G be a finitely generated nilpotent infinite group. (1) The abscissa of convergence α(G) of ζ_G(s) is a rational number and ζ_G(s) can be meromorphically continued to Re(s)>α(G)-δfor some δ>0. The continued function is holomorphic on the line \Re(s) = (α)G except for a pole at s=α(G). (2) There exist a nonnegative integer b(G) and some real numbers c,c' such that s_{N}(G) ~ c N^{α(G)}(\log N)^{b(G)} s_{N}^{α(G)}(G) ~ c' (\log N)^{b(G)+1} for N\rightarrow \infty .

math.GR↗