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.
We show that coefficients in the Laurent series of Igusa Zeta functions are periods. This will be used in a subsequent paper (by P. Brosnan) to show that certain numbers occurring in study of Feynman amplitudes (upto gamma factors) are periods.
The numerical data of an embedded resolution determine the candidate poles of Igusa's p-adic zeta function. We determine in complete generality which real candidate poles are actual poles in the curve case.
The aim of this paper is to describe explicitly the poles of the meromorphic continuation of the Igusa local zeta function associated to several polynomials. Using resolution of singularities is possible to express the Igusa's local zeta function as a finite sum of p-adic monomial integrals. We compute these monomial integrals using techniques of toroidal geometry. In this way, we obtain an explicit list for the candidates to poles of the local zeta function associated to several polynomials.
Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of K^n. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than -1/2 for n=2 and less than -1 for n=3 which occur as the real part of a pole.
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.
In this paper, we study the Igusa's local zeta functions of a class of hybrid polynomials with coefficients in a non-archimedean local field of positive characteristic. Such class of hybrid polynomial was first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We prove the rationality of these local zeta functions and describe explicitly their poles. The proof is based on Igusa's stationary phase formula.
For p prime, we give an explicit formula for Igusa's local zeta function associated to a polynomial mapping f=(f_1,...,f_t): Q_p^n -> Q_p^t, with f_1,...,f_t in Z_p[x_1,...,x_n], and an integration measure on Z_p^n of the form |g(x)||dx|, with g another polynomial in Z_p[x_1,...,x_n]. We treat the special cases of a single polynomial and a monomial ideal separately. The formula is in terms of Newton polyhedra and will be valid for f and g sufficiently non-degenerated over F_p with respect to their Newton polyhedra. The formula is based on, and is a generalization of results of Denef - Hoornaer…
This paper provides specific results on the Igusa local zeta function for the curves $x^n+y^m$. In addition to specific results, we give an introduction to $p$-adic analysis and a discussion of various methods which have been used to compute these zeta functions. The paper was written by the 1992 REU group in $p$-adic analysis supervised by Margaret Robinson. It has been available on the Mount Holyoke REU website.
Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n>1 variables and let χbe a character of R^{\times}. Let M_i(u) be the number of solutions of f=u in (R/P^i)^n for i \in \mathbb{Z}_{\geq 0} and u \in R/P^i. These numbers are related with Igusa's p-adic zeta function Z_{f,χ}(s) of f. We explain the connection between the M_i(u) and the smallest real part of a pole of Z_{f,χ}(s). We also prove that M_i(u) is divisible by q^{\ulcorner(n/2)(i-1)\urcorner}, where the corners indicate that we …
We investigate the relationship between the poles of Igusa zeta integrals and the unextendability of semi-invariant distributions. Under some algebraic conditions, we obtain an upper bound for the order of the poles of Igusa zeta integral, and by using the order of the poles we give a criterion on the unextendability of semi-invariant distributions. A key ingredient of our method is the idea of generalized semi-invariant distributions.
Let $K$ be a local field and $f(x)\in K[x]$ be a non-constant polynomial. When ${\rm char}K=0$, Igusa showed the local zeta function is a rational function. However, when ${\rm char}K>0$, the rationality of the local zeta function is unknown in general. In this paper, we study the local zeta functions for the so-called hybrid polynomials in three variables with coefficients in a non-archimedean local field of positive characteristic. These hybrid polynomials were first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We establish the rationality…
Let $K$ be a local field and $f(x)\in K[x]$ be a non-constant polynomial. The local zeta function $Z_f(s, χ)$ was first introduced by Weil, then studied in detail by Igusa. When ${\rm char}(K)=0$, Igusa proved that $Z_f(s, χ)$ is a rational function of $q^{-s}$ by using the resolution of singularities. Later on, Denef gave another proof of this remarkable result. However, if ${\rm char}(K)>0$, the question of rationality of $Z_f(s, χ)$ is still kept open. Actually, there are only a few known results so far. In this paper, we investigate the local zeta functions of two-variable polynomial $g(x,…
In this paper we present a polynomial time algorithm to compute the local zeta function Z(s,f) attached to a polynomial f(x) in Z[x] (in one variable, with splitting field Q) and a prime p. The algorithm reduces in polynomial time the computation of Z(s,f) to the computation of a factorization of f(x) over Q. This reduction is accomplished by constructing a weighted tree from the p-adic expansion of the roots of f(x) modulo a certain power of p, and then associating a generating function to this tree. The generating function constructed in this way coincides with the local zeta function of f(x…
In this paper we provide a geometric description of the possible poles of the Igusa local zeta function associated to an analytic mapping and a locally constant function, in terms of a log-principalizaton of an ideal naturally attached to the mapping. Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in terms of the log-canonic…
The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function $Z_{\text{DL}}(h,T)$ of a quasi-ordinary power series $h$ of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent $Z_{\text{DL}}(h,T)=P(T)/Q(T)$ such that almost all the candidate poles g…
We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called flag Hilbert-Poincaré series. These series are intimately connected with Igusa local zeta functions of products of linear polynomials, and their motivic and topological relatives. Our main results include a self-reciprocity result for central arrangements defined over fields of characteristic zero. We also prove combinatorial formulae for a specialization of the flag Hilbert-Poincaré series for irreducible Coxeter arrangements of types $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{D}$ i…
The local zeta functions (also called Igusa's zeta functions) over p-adic fields are connected with the number of solutions of congruences and exponential sums mod p^{m}. These zeta functions are defined as integrals over open and compact subsets with respect to the Haar measure. In this paper, we introduce new integrals defined over submanifolds, or more generally, over non-degenerate complete intersection varieties, and study their connections with some arithmetical problems such as estimation of exponential sums mod p^{m}. In particular we extend Igusa's method for estimating exponential su…
On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities
KU Leuven（荷语鲁汶大学）；Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学技术院数学研究所）；Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo（东京大学宇宙物理数学研究所（Kavli））；Department of Mathematics, Graduate School of Science, the University of Osaka（大阪大学理学研究科数学系）
A new method is devised for calculating the Igusa local zeta function $Z_f$ of a polynomial $f(x_1,\dots,x_n)$ over a $p$-adic field. This involves a new kind of generating function $G_f$ that is the projective limit of a family of generating functions, and contains more data than $Z_f$. This $G_f$ 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 explicitl…
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 …
We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric formula for a class of $p$-adic integrals under non-degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.
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 (2008) and Cluckers-Mustaţǎ-Nguyen (2019). 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 functions of $f$ has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with b…
We prove the modulo $p$ and modulo $p^2$ cases of Igusa's conjecture on exponential sums. This conjecture predicts specific uniform bounds in the homogeneous polynomial case of exponential sums modulo $p^m$ when $p$ and $m$ vary. We introduce the motivic oscillation index of a polynomial $f$ and prove the stronger, analogue bounds for $m=1,2$ using this index instead of the original bounds. The modulo $p^2$ case of our bounds holds for all polynomials; the modulo $p$ case holds for homogeneous polynomials and under extra conditions also for nonhomogeneous polynomials. We obtain natural lower b…
In this article we initiate the study of the heat traces and spectral zeta functions for certain p-adic Laplacians. We show that the heat traces are given by p-adic integrals of Laplace type, and that the spectral zeta functions are p-adic integrals of Igusa-type. We find good estimates for the behaviour of the heat traces when the time tends to infinity, and for the asymptotics of the function counting the eigenvalues less than or equal to a given quantity.
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 paramet…
For a nonzero ideal I of C[x_1,...,x_n], with 0 in supp I, a generalization of a conjecture of Igusa - Denef - Loeser predicts that every pole of its topological zeta function is a root of its Bernstein-Sato polynomial. However, typically only a few roots are obtained this way. Following ideas of Veys, we study the following question. Is it possible to find a collection G of polynomials g in C[x_1,...,x_n], such that, for all g in G, every pole of the topological zeta function associated to I and the volume form gdx on the affine n-space, is a root of the Bernstein-Sato polynomial of I, and su…
One mentions in a lot of papers that the poles of Igusa's p-adic zeta function determine the asymptotic behavior of the number of solutions of polynomial congruences. However, no publication clarifies this connection precisely. We try to get rid of this gap.
Motivated by recent work of Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, Connes-Consani, and the author, we define some multivariable deformed zeta functions of Hurwitz-Igusa type for a Noetherian $\F_1$-scheme $X$ in the sense of Connes-Consani. Our zeta functions generalize both the zeta functions studied by Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, and the log derivative of the modified Soulé type zeta function Connes-Consani. We give an explicit presentation for these zeta functions using the Hurwitz zeta functions, and so, we can derive its meromorphicity. When restricted to the log derivati…
We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions - the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa - match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations …
This paper is dedicated to the description of the poles of the Igusa local zeta functions $Z(s,f,v)$ when $f(x,y)$ satisfies a new non degeneracy condition, that we have called arithmetic non degeneracy. More precisely, we attach to each polynomial $f(x,y)$, a collection of convex sets $Γ^{A}$ called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non degeneracy with respect to $Γ^{A}(f)$. The set of degenerate polynomials, with respect to a fixed geometric Newton polygon, contains an open subset, for the Zariski topology, formed by non degenerate polynomials …
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 $p \to 1$ 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 …
The holomorphy conjecture states roughly that Igusa's zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article we prove the holomorphy conjecture for surface singularities which are nondegenerate over $\mathbb{C}$ with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volume (which appears in the formula of Varchenko for the zeta functi…
Two number fields are called arithmetically equivalent if they have the same Dedekind zeta function. In the 1970's Perlis showed that this is equivalent to the condition that for almost every rational prime $\ell$ the arithmetic type of $\ell$ is the same in each field. In the 1990's Perlis and Stuart gave an unexpected characterization for arithmetic equivalence; they showed that to be arithmetically equivalent it is enough for almost every prime $\ell$ to have the same number of prime factors in each field. Here, using an $\ell-p$ switch trick, we provide an elementary proof of that fact bas…
It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime $p$ is the same in both fields for almost all prime $p$. Here, almost all means with the possible exception of a set of Dirichlet density zero. One of the results of this paper shows that the condition density zero can be improved to a specific positive density that depends solely in the degree of the fields. More specifically, for every positive $n$ we exhibit a positive constan $c_{n}$ such that any two degree $n$ number fields $K$ and $L$ are arithmetically equivale…
We prove a recent conjecture due to Cluckers and Veys on exponential sums modulo $p^m$ for $m \geq 2$ 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 on 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 to a non-degenerate polynomial, at its largest non-trivial real candidate pole.
The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s,F) is a product of L-series of a pair of elliptic cusp forms. Two key examples are the forms F defined by the symmetric cube of a non-CM elliptic curve E over the rationals, and those defined by anticyclotomic twists (of even non-zero weight) of the base change of such an E to an imaginary qu…
On a uniform bound for exponential sums modulo $p^m$ for Deligne polynomials
KU Leuven（荷语鲁汶大学）；Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学技术研究院数学研究所）；Normandie Université, Université de Caen Normandie - CNRS, Laboratoire de Mathématiques Nicolas Oresme (LMNO)（诺曼底大学，卡昂诺曼底大学-法国国家科学研究中心，尼古拉·奥雷数学实验室）
