Local Zeta Functions Research Articles

Nonpolar singularities of local zeta functions in some smooth case

Nonpolar singularities of local zeta functions in some smooth case

New bounds for exponential sums with a non-degenerate phase polynomial

We prove a recent conjecture due to Cluckers and Veys on exponential sums modulo pm for m≥2 in the special case where the phase polynomial f is sufficiently non-degenerate with respect to its Newton polyhedron at the origin. Our main auxiliary result is an improved bound for certain related exponential sums over finite fields. This bound can also be used to settle a conjecture of Denef and Hoornaert on the candidate-leading Taylor coefficient of Igusa's local zeta function associated with a non-degenerate polynomial, at its largest non-trivial real candidate pole.

On Archimedean zeta functions and Newton polyhedra

Let $f$ be a polynomial function over the complex numbers and let $\phi$ be a smooth function over $\mathbb{C}$ with compact support. When $f$ is non-degenerate with respect to its Newton polyhedron, we give an explicit list of candidate poles for the complex local zeta function attached to $f$ and $\phi$. The provided list is given just in terms of the normal vectors to the supporting hyperplanes of the Newton polyhedron attached to $f$. More precisely, our list does not contain the candidate poles coming from the additional vectors required in the regular conical subdivision of the first orthant, and necessary in the study of local zeta functions through resolution of singularities. Our results refine the corresponding results of Varchenko and generalize the results of Denef and Sargos in the real case, to the complex setting.

Regularization of p-adic string amplitudes, and multivariate local zeta functions

We prove that the p-adic Koba-Nielsen type string amplitudes are bona fide integrals. We attach to these amplitudes Igusa-type integrals depending on several complex parameters and show that these integrals admit meromorphic continuations as rational functions. Then we use these functions to regularize the Koba-Nielsen amplitudes. As far as we know, there is no a similar result for the Archimedean Koba-Nielsen amplitudes. We also discuss the existence of divergencies and the connections with multivariate Igusa's local zeta functions.

On p-adic string amplitudes in the limit p approaches to one

In this article we discuss the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the p → 1 limit. Previously, we established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa’s local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit p → 1 of a Igusa’s local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser’s theory of topological zeta functions, we show that limit p → 1 of tree-level p-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (reproducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. We show that the Feynman amplitudes of this last Lagrangian are precisely the amplitudes introduced here. Finally, the amplitudes for four and five points are computed explicitly.

Enumerating traceless matrices over compact discrete valuation rings

We enumerate traceless square matrices over finite quotients of compact discrete valuation rings by their image sizes. We express the associated rational generating functions in terms of statistics on symmetric and hyperoctahedral groups, viz. Coxeter groups of types A and B, respectively. These rational functions may also be interpreted as local representation zeta functions associated to the members of an infinite family of finitely generated class-2-nilpotent groups. As a byproduct of our work, we obtain descriptions of the numbers of traceless square matrices over a finite field of fixed rank in terms of statistics on the hyperoctahedral groups.

Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbb{F}_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the $\ell$-torsion by structured polynomial systems. Our main result improves on previously known complexity bounds by showing that there exists a constant $c > 0$ such that, for any fixed $g$, this algorithm has expected time and space complexity $O((\log q)^{cg})$ as $q$ grows and the characteristic is large enough.

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 $$6-\frac{15}{m+3}$$ .

Igusa’s Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials

We study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. Under the hypothesis that the polynomial is arithmetically non degenerate, we obtain an explicit list of candidates for the poles in terms of geometric data obtained from a family of arithmetic Newton polygons attached to the polynomial. The notion of arithmetical non degeneracy due to Saia and Zúñiga-Galindo is weaker than the usual notion of non degeneracy due to Kouchnirenko. As an application we obtain asymptotic expansions for certain exponential sums attached to these polynomials.

Computing local zeta functions of groups, algebras, and modules

We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six. We also determine the generic local subalgebra zeta functions associated with $\mathfrak {gl}_2(\mathbf {Q})$. Finally, we introduce and compute examples of graded subobject zeta functions.

Monodromy conjecture and the Hessian differential form

In this paper it is shown that the local topological zeta function of a germ associated with its Hessian differential form does not satisfy the monodromy conjecture, i.e., that the Hessian form is not an allowed differential 2-form.

Local zeta functions, pseudodifferential operators and Sobolev-type spaces over non-Archimedean local fields

In this articlewe introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers of norms of polynomials multiplied by infinitely pseudo-differentiable functions. In characteristic zero, the new local zeta functions admit meromorphic continuations to the whole complex plane, but they are not rational functions. The real parts of the possible poles have a description similar to the poles of Archimedean zeta functions. But they can be irrational real numbers while in the classical case are rational numbers. We also study, in arbitrary characteristic, certain connections between local zeta functions and the existence of fundamental solutions for pseudodifferential equations.

Geodesic bulk diagrams on the Bruhat–Tits tree

Geodesic bulk diagrams were recently shown to be the geometric objects which compute global conformal blocks. We show that this duality continues to hold in $p$-adic AdS/CFT, where the bulk is replaced by the Bruhat-Tits tree, an infinite regular graph with no cycles, and the boundary is described by $p$-adic numbers, rather than reals. We apply the duality to evaluate the four-point function of scalar operators of generic dimensions using tree-level bulk diagrams. Relative to standard results from the literature, we find intriguing similarities as well as significant simplifications. Notably, all derivatives disappear in the conformal block decomposition of the four-point function. On the other hand, numerical coefficients in the four-point function as well as the structure constants take surprisingly universal forms, applicable to both the reals and the $p$-adics when expressed in terms of local zeta functions. Finally, we present a minimal bulk action with nearest neighbor interactions on the Bruhat-Tits tree, which reproduces the two-, three-, and four-point functions of a free boundary theory.

Stability results for local zeta functions of groups algebras, and modules

AbstractVarious types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

Asymptotic expansions and conformal covariance of the mass of conformal differential operators

We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of m-Laplace type operators L on compact Riemannian manifolds in terms of Riesz distributions. The constant term in this asymptotic expansion turns out to be given by the local zeta function of L. In particular, the constant term in the asymptotic expansion of the Green’s function L^{-1} is often called the mass of L, which (in case that L is the Yamabe operator) is an important invariant, namely a positive multiple of the ADM mass of a certain asymptotically flat manifold constructed out of the given data. We show that for general conformally invariant m-Laplace operators L (including the GJMS operators), this mass is a conformal invariant in the case that the dimension of M is odd and that ker L = 0, and we give a precise description of the failure of the conformal invariance in the case that these conditions are not satisfied.

Zeta functions and oscillatory integrals for meromorphic functions

In the 70's Igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero. In the present article this theory is extended to the case of rational functions, or, more generally, meromorphic functions f/g, with coefficients in a local field of characteristic zero. This generalization is far from being straightforward due to the fact that several new geometric phenomena appear. Also, the oscillatory integrals have two different asymptotic expansions: the ‘usual’ one when the norm of the parameter tends to infinity, and another one when the norm of the parameter tends to zero. The first asymptotic expansion is controlled by the poles (with negative real parts) of all the twisted local zeta functions associated to the meromorphic functions f/g−c, for certain special values c. The second expansion is controlled by the poles (with positive real parts) of all the twisted local zeta functions associated to f/g.

P-Adic AdS/CFT

We construct a $p$-adic analog to AdS/CFT, where an unramified extension of the $p$-adic numbers replaces Euclidean space as the boundary and a version of the Bruhat-Tits tree replaces the bulk. Correlation functions are computed in the simple case of a single massive scalar in the bulk, with results that are strikingly similar to ordinary holographic correlation functions when expressed in terms of local zeta functions. We give some brief discussion of the geometry of $p$-adic chordal distance and of Wilson loops. Our presentation includes an introduction to $p$-adic numbers.

An explicit formula for the local zeta function of a Laurent polynomial

In a recent paper Zúñiga-Galindo and the author begun the study of the local zeta functions for Laurent polynomials. In this work we continue this study by giving a very explicit formula for the local zeta function associated to a Laurent polynomial f over a p-adic field, when f is weakly non-degenerate with respect to the Newton polytope of f at infinity.

A new generating function for calculating the Igusa local zeta function

A new method is devised for calculating the Igusa local zeta function Zf of a polynomial f(x1,…,xn) over a p-adic field. This involves a new kind of generating function Gf that is the projective limit of a family of generating functions, and contains more data than Zf. This Gf resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, facilitating calculation of local zeta functions. This new technique is used to expand significantly the set of quadratic polynomials whose local zeta functions have been calculated explicitly. Local zeta functions for arbitrary quadratic polynomials over p-adic fields with p odd are presented, as well as for polynomials over unramified 2-adic fields of the form Q+L where Q is a quadratic form and L is a linear form such that Q and L have disjoint variables. For a quadratic form over an arbitrary p-adic field with odd p, this new technique makes clear precisely which of the three candidate poles are actual poles.

A survey of Igusa’s local zeta function

The origin and development of the Igusa local zeta function, by Igusa and others, is presented. In particular, we discuss the various conjectures Igusa made and the notable results that have so far been obtained. We also explain how topological and motivic zeta functions arose from the Igusa local zeta function and present the current status of the analogous conjectures. Igusa's conjecture on exponential sums that are related to his zeta function is described, and along with the progress made towards its resolution.