Global Zeta Functions Research Articles

Meromorphic Continuation of Global Zeta Function for Number Fields

Meromorphic Continuation of Global Zeta Function for Number Fields

Función Zeta del anillo de Burnside para Cp3

The main objective of this paper is to determine the local and global Zeta function of B(C_{p^3}) the Burnside ring for cyclic groups of order p^3 and to study some relations that fulfill this Zeta function.

A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions

The pro-isomorphic zeta function $$\zeta ^\wedge _\Gamma (s)$$ of a finitely generated nilpotent group $$\Gamma $$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $$\Gamma $$ . Such zeta functions can be expressed as Euler products of p-adic integrals over the $$\mathbb {Q}_p$$ -points of an algebraic automorphism group associated to $$\Gamma $$ . In this way they are closely related to classical zeta functions of algebraic groups over local fields. We describe the algebraic automorphism groups for a natural family of class-2 nilpotent groups $$\Delta _{m,n}$$ of Hirsch length $$\left( {\begin{array}{c}m+n-2 n-1\end{array}}\right) +\left( {\begin{array}{c}m+n-1 n-1\end{array}}\right) + n$$ and central Hirsch length n; these groups can be viewed as generalisations of $$D^*$$ -groups of odd Hirsch length. General $$D^*$$ -groups, that is ‘indecomposable’ finitely generated, torsion-free class-2 nilpotent groups with central Hirsch length 2, were classified up to commensurability by Grunewald and Segal. We calculate the local pro-isomorphic zeta functions for the groups $$\Delta _{m,n}$$ and obtain, in particular, explicit formulae for the local pro-isomorphic zeta functions associated to $$D^*$$ -groups of odd Hirsch length. From these we deduce local functional equations; for the global zeta functions we describe the abscissae of convergence and find meromorphic continuations. We deduce that the spectrum of abscissae of convergence for pro-isomorphic zeta functions of class-2 nilpotent groups contains infinitely many cluster points. For instance, the global abscissa of convergence of the pro-isomorphic zeta function of a $$D^*$$ -group of Hirsch length $$2m+3$$ is shown to be $$6-\frac{15}{m+3}$$ .

The local and global zeta functions of Gauss's curve

The singular curve $C\subset \mathbb{P}^2$ defined over $\mathbb{F}_p$ for a prime $p$ by the equation $x^2t^2+y^2t^2+x^2y^2-t^4=0$ is known as Gauss's curve. For $p\equiv 3$ mod $4$, we give a proof that the zeta function of $C$ is \[ Z_{C}(u)=\frac{(1+pu^2)(1+u)^2}{(1-pu)(1-u)}. \] We define the (Hasse-Weil) global zeta function for any geometric genus~1 singular curve and, in particular, find that the global zeta function of $C$ is \[ \zeta_C(s)=\frac{\zeta(s)\zeta(s-1)}{L_E(s)L(s,\chi^{\prime})^{2}}, \] where $E$ is a projective nonsingular model for $C$, $L_E(s)$ is its $L$-function, and $L(s, \chi^{\prime})$ is a Dirichlet $L$-series for a character $\chi^{\prime}$ that we specify. We then consider more generally the ratio $R_{X}(s)$ of the Hasse-Weil global zeta function of a singular curve $X$ and that of its normalization $\widetilde{X}$. We finish with questions about the analytic properties of $R_{X}(s)$.

Local complex dimensions of a fractal string

We investigate in this work a local version of the theory of fractal strings and associated geometric zeta functions. Such a generalisation allows to describe the asymptotic behaviour of a ‘fractal’ set in the neighbourhood of any of its points. We give basic properties and several examples illustrating the possible range of situations concerning in particular the evolution of the local complex dimensions along the set and the relation between local and global zeta functions.

A mean value theorem for orders of degree zero divisor class groups of quadratic extensions over a function field

Let k be a function field of one variable over a finite field with the characteristic not equal to two. In this paper, we consider the prehomogeneous representation of the space of binary quadratic forms over k. We have two main results. The first result is on the principal part of the global zeta function associated with the prehomogeneous vector space. The second result is on a mean value theorem for degree zero divisor class groups of quadratic extensions over k, which is a consequence of the first one.

GLOBAL ZETA FUNCTIONS OF FREUDENTHAL QUARTICS

We give two applications of an explicit formula for global zeta functions of prehomogeneous vector spaces in Math. Ann.315 (1999), 587–615. One is concerned with an explicit form of global zeta functions associated with Freudenthal quartics, and the other the comparison of the zeta function of a unsaturated prehomogeneous vector space with that of the saturated one obtained from it.

Casimir Effect for a Perfectly Conducting Wedge in Terms of Local Zeta Function

The vacuum energy density of electromagnetic field inside a perfectly conducting wedge is calculated by making use of the local zeta function technique. This regularization completely eliminates divergent expressions in the course of calculations and gives rise to a finite expression for the energy density in question without any subtractions. Employment of the Hertz potentials for constructing the general solution to the Maxwell equations results in a considerable simplification of the calculations. Transition to the global zeta function is carried out by introducing a cutoff nearby the cusp at the origin. Proceeding from this the heat kernel coefficients are calculated and the high temperature asymptotics of the Helmholtz free energy and of the torque of the Casimir forces are found. The wedge singularity gives rise to a specific high temperature behavior ∼T2 of the quantities under consideration. The obtained results are directly applicable to the free energy of a scalar massless field and electromagnetic field on the background of a cosmic string.

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By definition, these zeta functions are rational functions in one variable, and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non-degenerate with respect to its Newton polyhedron, we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; we give a similar result concerning the global zeta function. Moreover, for any f we show that poles of maximal possible order are always of the form −1/N with N a positive integer. 1991 Mathematics Subject Classification 14B05, 14E15, 32S50.