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In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to the number of poles of the motivic zeta function associated with a generic curve in the family., Comment: v3: 33 pages; final version to appear in Advances in Geometry

This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding surface, we construct an embedded $\mathbb Q$-resolution of this pair and use an A'Campo formula in terms of this resolution to compute the zeta function of monodromy. Combining all results, we prove the monodromy conjecture for this class of monomial curves., 50 pages; final version to appear in Publicacions Matem\`atiques

We study endomorphisms of complete Noetherian local rings in the context of motivic integration. Using the notion of an auto-arc space, we introduce the (reduced) auto-Igusa zeta series at a point, which appears to measure the degree to which a variety is not smooth that point. We conjecture a closed formula in the case of curves with one singular point, and we provide explicit formulas for this series in the case of the cusp and the node. Using the work of Denef and Loeser, one can show that this series will often be rational. These ideas were obtained through extensive calculations in Sage. Thus, we include a Sage script which was used in these calculations. It computes the affine arc spaces ź n X provided that X is affine, n is a fat point, and the ground field is of characteristic zero. Finally, we show that the auto Poincare series will often be rational as well and connect this to questions concerning new types of motivic integrals.

We investigate an arithmetic function representing a generalization of the gcd-sum function, considered by Kurokawa and Ochiai in 2009 in connection with the multivariable global Igusa zeta function for a finite cyclic group. We show that the asymptotic properties of this function are closely connected to the Piltz divisor function. A generalization of Menon's identity is also considered., Comment: 9 pages

In this paper we provide a geometric description of the possible poles of the Igusa local zeta function Z Φ ( s , f ) Z_{\Phi }(s,\mathbf {f}) associated to an analytic mapping f = \mathbf {f}= ( f 1 , … , f l ) : U ( ⊆ K n ) → K l \left (f_{1},\ldots ,f_{l}\right ) :U(\subseteq K^{n})\rightarrow K^{l} , and a locally constant function Φ \Phi , with support in U U , in terms of a log-principalizaton of the K [ x ] − K\left [x \right ] - ideal I f = ( f 1 , … , f l ) \mathcal {I}_{\mathbf {f}}=\left (f_{1},\ldots ,f_{l}\right ) . 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-canonical threshold of the subscheme given by I f \mathcal {I}_{\mathbf {f}} . We associate to an analytic mapping f \boldsymbol {f} = = ( f 1 , … , f l ) \left (f_{1},\ldots ,f_{l}\right ) a Newton polyhedron Γ ( f ) \Gamma \left (\boldsymbol {f}\right ) and a new notion of non-degeneracy with respect to Γ ( f ) \Gamma \left (\boldsymbol {f}\right ) . The novelty of this notion resides in the fact that it depends on one Newton polyhedron, and Khovanskii’s non-degeneracy notion depends on the Newton polyhedra of f 1 , … , f l f_{1},\ldots ,f_{l} . By constructing a log-principalization, we give an explicit list for the possible poles of Z Φ ( s , f ) Z_{\Phi }(s,\mathbf {f}) , l ≥ 1 l\geq 1 , in the case in which f \mathbf {f} is non-degenerate with respect to Γ ( f ) \Gamma \left (\boldsymbol {f}\right ) .

We show that the real parts of the poles of the Igusa zeta function of a monomial ideal can be computed from the torus-invariant divisors on the normalized blow-up of the affine space along the ideal. Moreover, we show that every such number is a root of the Bernstein-Sato polynomial associated to the monomial ideal.

We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if $\Gamma$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than one, let $r_n(\Gamma)$ be the number of irreducible $n$-dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (for example, $C=746$ suffices) such that $r_n(\Gamma)=O(n^C)$ for every such $\Gamma$. This answers a question of Larsen and Lubotzky., Comment: version 2: The prove of the main theorem was simplified and a result on arithmetic algebraic geometry was added. 29 pages

On the space of real rectangular n × m matrices, we introduce a composite power function and study the zeta integral associated with it. We describe the properties of the Igusa zeta function on the basis of the properties of a generalized composite power function and establish a functional relation for the zeta integral. As a result, the Fourier transform of a generalized composite power function is found in explicit form.

In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial f is smaller than expected. We carry out this study in the case that f is sufficiently non-degenerate with respect to its Newton polyhedron Γ(f), and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial f to the same question about polynomials f μ , where μ are faces of Γ(f) depending on the examined pole and f μ is obtained from f by throwing away all monomials of f whose exponents do not belong to μ. Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial f μ , with μ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than expected.

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 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 where $Q$ 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., Comment: 54 pages

We give a very explicit formula for Igusa's local zeta function Z(f)(s) associated to a polynomial f in several variables over the p-adic numbers, when f is sufficiently non-degenerated with respect to its Newton polyhedron Gamma (f). Using this Formula and the estimates of Adolphson and Sperber on exponential sums over finite fields, we study the largest real pole different from -1 of Z(f)(s) (C) 2001 Academic Press. ispartof: Journal of number theory vol:89 issue:1 pages:31-64 status: published

Let f : ( 𝐂 d + 1 , 0 ) → ( 𝐂 , 0 ) $f \hspace{-1.0pt}:\hspace{-1.0pt}(\mathbf {C}^{d+1}, 0) \hspace{-1.0pt}\rightarrow \hspace{-1.0pt}(\mathbf {C},0)$ be a germ of complex analytic function such that its zero level defines an irreducible germ of quasi-ordinary hypersurface ( S , 0 ) ⊂ ( 𝐂 d + 1 , 0 ) $(S, 0) \subset (\mathbf {C}^{d+1}, 0)$ . We describe the motivic Igusa zeta function, the motivic Milnor fibre and the Hodge–Steenbrink spectrum of f at 0 in terms of topological invariants of ( S , 0 ) ⊂ ( 𝐂 d + 1 , 0 ) $(S, 0) \subset (\mathbf {C}^{d+1}, 0)$ .

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., Comment: 18 pages

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 given by $Q(T)$ are poles. Anyway, these candidate poles give eigenvalues of the monodromy action of the complex of nearby cycles on $h^{-1}(0).$ In particular we prove in this case the monodromy conjecture made by Denef-Loeser for the local motivic zeta function and the local topological zeta function. As a consequence, if $h$ is a quasi-ordinary polynomial defined over a number field we prove the Igusa monodromy conjecture for its local Igusa zeta function., Comment: 74 pages

Nous calculons la fonction zeta d'Igusa motivique de Denef-Loeser associee a une serie irreductible de deux variables et retrouvons a l'aide de ce resultat la formule donnant le spectre de Hodge-Steenbrink d'une courbe plane irreductible en termes des donnees de Puiseux. Nous etudions ensuite une generalisation de la fonction d'Igusa a une famille de fonctions et montrons que cette fonction d'Igusa permet de retrouver les invariants d'Alexander de la famille. Nous appliquons ce resultat en dimension deux pour obtenir une expression du polynome d'Alexander d'une courbe plane. ¶¶Abstract. We compute the motivic Igusa zeta function of Denef-Loeser associated with a two variables irreducible serie and use this result to give a new proof of the formula expressing the Hodge-Steenbrink spectrum in terms of the Puiseux data. We study a generalisation of the motivic Igusa function to a family of functions and show that this Igusa function is related with the Alexander invariants of the family. Using this result, we obtain a formula for the Alexander plolynomial of a plane curve.

ispartof: Bulletin of the Belgian Mathematical Society - Simon Stevin vol:9 issue:4 pages:589-606 status: published

Let X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain 'motivic integral', living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization to singular varieties. Batyrev already considered such a 'Kontsevich invariant' for log terminal varieties (on the level of Hodge polynomials instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any Q-Gorenstein variety X we associate a motivic zeta function and a 'Kontsevich invariant' to effective Q-Cartier divisors on X whose support contains the singular locus of X., AMS-TeX (using PicTeX), 27 pages; some minor improvements

Knowing the number of solutions for a Diophantine equation is an important step to study various arithmetic problems. Igusa originated the study of Igusa zeta functions associated to local Diophantine problems. Multiplying all these local Igusa zeta functions we obtain the global version in the natural way. Unfortunately, investigations on global Igusa zeta functions are rare up to now. Reformulating the global Igusa zeta function via the number of morphisms between algebraic systems we discover a new aspect: the multivariable Euler product of Igusa type and its applications. A purpose of this paper is to encourage experts for further studies on global Igusa zeta functions by treating a simple interesting example.