Local Zeta Functions Research Articles

𝐾𝑈-local zeta-functions of finite CW-complexes

Begin with the Hasse-Weil zeta-function of a smooth projective variety over Q \mathbb {Q} . Replace the variety with a finite CW-complex, replace étale cohomology with complex K K -theory K U ∗ KU^* , and replace the p p -Frobenius operator with the p p th Adams operation on K K -theory. This simple idea yields a kind of “ K U KU -local zeta-function” of a finite CW-complex. For a wide range of finite CW-complexes X X with torsion-free K K -theory, we show that this zeta-function admits analytic continuation to a meromorphic function on the complex plane, with a nice functional equation, and whose special values in the left half-plane recover the K U KU -local stable homotopy groups of X X away from 2 2 . We then consider a more general and sophisticated version of the K U KU -local zeta-function, one which is suited to finite CW-complexes X X with nontrivial torsion in their K K -theory. This more sophisticated K U KU -local zeta-function involves a product of L L -functions of complex representations of the torsion subgroup of K U 0 ( X ) KU^0(X) , similar to how the Dedekind zeta-function of a number field factors as a product of Artin L L -functions of complex representations of the Galois group. For a wide range of such finite CW-complexes X X , we prove analytic continuation of the zeta-function, and we show that the special values in the left half-plane recover the K U KU -local stable homotopy groups of X X away from 2 2 if and only if the skeletal filtration on the torsion subgroup of K U 0 ( X ) KU^0(X) splits completely.

Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL2

We introduce a local zeta-function for an irreducible admissible supercuspidal representation π of the metaplectic double cover of SL2 over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of π. We obtain an expression of the gamma factor, which shows its entireness on C. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on PGL2≅SO3 associated with the prehomogenous vector space of symmetric matrices of degree 2.

Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces

We explore the effective gravitational action for two-dimensional massive Euclidean Majorana fermions in a small mass expansion, continuing and completing the study initiated in a previous paper [1]. We perform a detailed analysis of local zeta functions, heat kernels, and Green’s functions of the Dirac operator on arbitrary Riemann surfaces. We obtain the full expansion of the effective gravitational action to all orders in m2. For genus one and larger, this requires the understanding of the role of the zero-modes of the (massless) Dirac operator which is worked out.Besides the Liouville action, at order m0, which only involves the background metric and the conformal factor σ, the various contributions to the effective gravitational action at higher orders in m2 can be expressed in terms of integrals of the renormalized Green’s function at coinciding points of the squared (massless) Dirac operator, as well as of higher Green’s functions. In particular, at order m2, these contributions can be re-written as a term ∫ e2σσ characteristic of the Mabuchi action, much as for 2D massive scalars, as well as several other terms that are multi-local in the conformal factor σ and involve the Green’s functions of the massless Dirac operator and the renormalized Green’s function, but for the background metric only, and certain area-like parameters related to the zero-modes.

Flux vacua and modularity for $\mathbb{Z}_2$ symmetric Calabi-Yau manifolds

We find continuous families of supersymmetric flux vacua in IIB Calabi-Yau compactifications for multiparameter manifolds with an appropriate \mathbb{Z}_{2}ℤ2 symmetry. We argue, supported by extensive computational evidence, that the numerators of the local zeta functions of these compactification manifolds have quadratic factors. These factors are associated with weight-two modular forms, these manifolds being said to be weight-two modular. Our evidence supports the flux modularity conjecture of Kachru, Nally, and Yang. The modular forms are related to a continuous family of elliptic curves. The flux vacua can be lifted to F-theory on elliptically fibred Calabi-Yau fourfolds. If conjectural expressions for Deligne’s periods are true, then these imply that the F-theory fibre is complex-isomorphic to the modular curve. In three examples, we compute the local zeta function of the internal geometry using an extension of known methods, which we discuss here and in more detail in a companion paper. With these techniques, we are able to compare the zeta function coefficients to modular form Fourier coefficients for hundreds of manifolds in three distinct families, finding agreement in all cases. Our techniques enable us to study not only parameters valued in \mathbb{Q}ℚ but also in algebraic extensions of \mathbb{Q}ℚ, so exhibiting relations to Hilbert and Bianchi modular forms. We present in appendices the zeta function numerators of these manifolds, together with the corresponding modular forms.

Resolution of singularities for C∞ functions and meromorphy of local zeta functions

In this paper, we attempt to resolve the singularities of the zero variety of a C∞ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the “almost” normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with C∞ functions of two variables. As is well known, the desingularization theorem of Hironaka implies that the local zeta functions associated with real analytic functions admit the meromorphic continuation to the whole complex plane. On the other hand, it is recently observed that the local zeta function associated with a specific (non-real analytic) C∞ function has a singularity different from the pole. From this observation, the following questions are naturally raised in the C∞ case: how wide the meromorphically extendible region can be and what kinds of information essentially determine this region? This paper shows that this region can be described in terms of some kind of multiplicity of the zero variety of each C∞ function. By using our blowings up algorithm, it suffices to investigate local zeta functions in the almost normal crossings case. This case can be effectively analyzed by using real analysis methods; in particular, a van der Corput-type lemma plays a crucial role in the determination of the above region.

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.

On the degree of polynomial subgroup growth of nilpotent groups

Let N be a finitely generated nilpotent group. The subgroup zeta function \(\zeta _N^{\scriptscriptstyle \leqslant }(s)\) and the normal zeta function \(\zeta _N^{\scriptscriptstyle \lhd }(s)\) of N are Dirichlet series enumerating the finite index subgroups or the finite index normal subgroups of N. We present results about their abscissae of convergence \(\alpha _N^{\scriptscriptstyle \leqslant }\) and \(\alpha _N^{\scriptscriptstyle \lhd }\), also known as the degrees of polynomial subgroup growth and polynomial normal subgroup growth of N, respectively. We first prove some upper bounds for the functions \(N\mapsto \alpha _N^{\scriptscriptstyle \leqslant }\) and \(N\mapsto \alpha _N^{\scriptscriptstyle \lhd }\) when restricted to the class of torsion-free nilpotent groups of a fixed Hirsch length. We then show that if two finitely generated nilpotent groups have isomorphic \(\mathbb {C}\)-Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. This follows, via the Mal’cev correspondence, from a similar result that we establish for zeta functions of rings. This result is obtained by proving that the abscissa of convergence of an Euler product of certain Igusa-type local zeta functions introduced by du Sautoy and Grunewald remains invariant under base change. We also apply this methodology to formulate and prove a version of our result about nilpotent groups for virtually nilpotent groups. As a side application of our result about zeta functions of rings, we present a result concerning the distribution of orders in number fields.

Zeta functions enumerating normal subgroups of [formula omitted]-groups and their behavior on residue classes

Let G be a finitely generated torsion-free nilpotent class-2 group. We investigate how ζG,p◃(s), the local zeta function enumerating normal subgroups of p-power indices in G, varies on residue classes. We first show how the behaviors of local normal subgroup zeta functions ζG,p◃(s) and ζG×Zr,p◃(s) on residue classes are closely related for certain G. Then we show that for G of Hirsch length less than or equal to 7, under certain assumptions the normal subgroup zeta function of G is always a rational function on residue classes. We also show that there are examples of groups with Hirsch length 8 whose normal subgroup zeta function is not a rational function on residue classes.

Fields over Fields

We investigate certain arithmetic properties of field theories. In particular, we study the vacuum structure of supersymmetric gauge theories as algebraic varieties over number fields of finite characteristic. Parallel to the Plethystic Programme of counting the spectrum of operators from the syzygies of the complex geometry, we construct, based on the zeros of the vacuum moduli space over finite fields, the local and global Hasse-Weil zeta functions, as well as develop the associated Dirichlet expansions. We find curious dualities wherein the geometrical properties and asymptotic behaviour of one gauge theory is governed by the number theoretic nature of another.

Euclidean quantum field formulation of p-adic open string amplitudes

We study in a rigorous mathematical way p-adic quantum field theories whose N-point amplitudes are the expectation of products of vertex operators. We show that this type of amplitudes admit a series expansion where each term is an Igusa's local zeta function. The lowest term in this series is a regularized version of the p-adic open Koba-Nielsen string amplitude.

Graphs, local zeta functions, log-Coulomb gases, and phase transitions at finite temperature

We study a log-gas on a network (a finite, simple graph) confined in a bounded subset of a local field (i.e., R, C, and Qp being the field of p-adic numbers). In this gas, a log-Coulomb interaction between two charged particles occurs only when the sites of the particles are connected by an edge of the network. The partition functions of such gases turn out to be a particular class of multivariate local zeta functions attached to the network and a positive test function, which is determined by the confining potential. The methods and results of the theory of local zeta functions allow us to establish that the partition functions admit meromorphic continuations in the parameter β (the inverse of the absolute temperature). We give conditions on the charge distributions and the confining potential such that the meromorphic continuations of the partition functions have a pole at a positive value βUV, which implies the existence of phase transitions at finite temperature. In the case of p-adic fields, the meromorphic continuations of the partition functions are rational functions in the variable p−β. We give an algorithm for computing such rational functions. For this reason, we can consider the p-adic log-Coulomb gases as exact solvable models. We expect that all these models for different local fields share common properties and that they can be described by a uniform theory.

Universal Equivalence and Majority of Probabilistic Programs over Finite Fields

We study decidability problems for equivalence of probabilistic programs for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication, and conditionals and thus is sufficiently expressive to encode Boolean and arithmetic circuits. We consider two variants of equivalence: The first one considers an interpretation over the finite field F q , while the second one, which we call universal equivalence, verifies equivalence over all extensions F q k of F q . The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of F 2 k for any k . While the first problem is obviously decidable, we establish its exact complexity, which lies in the counting hierarchy. To show decidability and a doubly exponential upper bound of the universal variant, we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. We then devise a general way to draw links between the universal probabilistic problems and widely studied problems on linear recurrence sequences. Finally, we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy. We also define and provide some insights about program indistinguishability, proving that it is decidable for programs always returning 0 or 1.

Local Zeta Functions and Koba–Nielsen String Amplitudes

This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p→1. Gerasimov and Shatashvili studied the limit p→1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to the limit p→1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba–Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.

On the motivic oscillation index and bound of exponential sums modulo pm via analytic isomorphisms

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12]. In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of ℓ-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A−D−E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

Computing Igusa’s local zeta function of univariates in deterministic polynomial-time

Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}(p^s)$. We give an elementary proof of this fact for a univariate polynomial $f$. Our proof is constructive as it gives a closed-form expression for the number of roots $N_{k}(f)$. Our proof, when combined with the recent root-counting algorithm of (Dwivedi, Mittal, Saxena, CCC, 2019), yields the first deterministic poly($|f|, \log p$) time algorithm to compute $Z_{f,p}(s)$. Previously, an algorithm was known only in the case when $f$ completely splits over $\mathbb{Q}_p$; it required the rational roots to use the concept of generating function of a tree (Zuniga-Galindo, J.Int.Seq., 2003).

Meromorphic continuation of Koba-Nielsen string amplitudes

In this article, we establish in a rigorous mathematical way that Koba-Nielsen amplitudes defined on any local field of characteristic zero are bona fide integrals that admit meromorphic continuations in the kinematic parameters. Our approach allows us to study in a uniform way open and closed Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities. As an application we present the regularization of p-adic open string amplitudes with Chan-Paton factors and constant B-field. Finally, all the local zeta functions studied here are partition functions of certain 1D log-Coulomb gases, which shows an interesting connection between Koba-Nielsen amplitudes and statistical mechanics.

P-Adic open string amplitudes with Chan-Paton factors coupled to a constant B-field

We establish rigorously the regularization of the p-adic open string amplitudes, with Chan-Paton rules and a constant B-field, introduced by Ghoshal and Kawano. In this study we use techniques of multivariate local zeta functions depending on multiplicative characters and a phase factor which involves an antisymmetric bilinear form. These local zeta functions are new mathematical objects. We attach to each amplitude a multivariate local zeta function depending on the kinematic parameters, the B-field and the Chan-Paton factors. We show that these integrals admit meromorphic continuations in the kinematic parameters. This result allows us to regularize the Ghoshal-Kawano amplitudes. The regularized amplitudes do not have ultraviolet divergencies. Due to the need for a certain symmetry, the theory works only for prime numbers which are congruent to 3 modulo 4. We also discuss the limit p→1 in the noncommutative effective field theory and in the Ghoshal-Kawano amplitudes. We show that in the case of four points, the limit p→1 of the regularized Ghoshal-Kawano amplitudes coincides with the Feynman amplitudes attached to the limit p→1 of the noncommutative Gerasimov-Shatashvili Lagrangian.

Meromorphy of local zeta functions in smooth model cases

It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general (C∞) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as u(x,y)xayb+ flat function, where u(0,0)≠0 and a,b are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each case. Our results show new interesting phenomena in one of these cases. Actually, when a<b, local zeta functions can be meromorphically extended to the half-plane Re(s)>−1/a and their poles on the half-plane are contained in the set {−k/b:k∈Nwithk<b/a}.

Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus

We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-g hyperelliptic curve defined over Fq with explicit real multiplication (RM) by an order Z[η] in a degree-g totally real number field.It is based on the approaches by Schoof and Pila in a more favourable case where we can split the ℓ-torsion into g kernels of endomorphisms, as introduced by Gaudry, Kohel, and Smith in genus 2. To deal with these kernels in any genus, we adapt a technique that the author, Gaudry, and Spaenlehauer introduced to model the ℓ-torsion by structured polynomial systems. Applying this technique to the kernels, the systems we obtain are much smaller and so is the complexity of solving them.Our main result is that there exists a constant c>0 such that, for any fixed g, this algorithm has expected time and space complexity O((logq)c) as q grows and the characteristic is large enough. We prove that c≤9 and we also conjecture that the result still holds for c=7.

Derived ℓ-adic zeta functions

Let K0(Vk) be the Grothendieck group of k-varieties. Campbell and Zakharevich have constructed a higher algebraic K-theory spectrum K(Vk) such that π0K(Vk)=K0(Vk). In this paper we construct non-trivial classes in the higher homotopy groups of K(Vk) when k is finite or a subfield of C. To do this we give a recipe for lifting motivic measures K0(Vk)→K0(E) to maps of spectra K(Vk)→K(E). We consider two special cases: the classical local zeta function, thought of as a homomorphism K0(VFq)→K0(End(Qℓ)), and the compactly-supported Euler characteristic, thought of as a homomorphism K0(VC)→K0(Q). We use lifts of these motivic measures to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map S→K(Vk) is nontrivial in higher dimensions when k is finite or a subfield of C, and, moreover, that when k is finite this map is not surjective on higher homotopy groups.