For a divisor representing a function and another divisor representing a differential form on a normal surface singularity, there is a notion of motivic and topological zeta function. In this paper, given a finite morphism between two normal surfaces, we prove that the set of poles of the motivic zeta function associated with the target is contained in the one associated with the source. We illustrate by examples that this inclusion is strict in general, and that on the topological level there are in general no inclusions between the sets of poles on source and target. On the other hand, when the morphism is the quotient map induced by an action of a finite abelian group on ℂ^2, and the divisor associated with the differential form on the source is trivial, we do show equality between the corresponding sets of poles, both on motivic and topological level. In addition, again for the quotient map induced by an action of a finite abelian group on ℂ^2, but now with a general divisor associated with a differential form, we provide a criterion when the topological zeta function on the target is just a multiple of the one on the source. Finally, we compare log canonical models on source and target with a view on zeta functions.

We fix a complex analytic normal singularity germ (X,o) of dimension ≥ 2 and a (not necessarily irreducible) reduced Weil divisor (S,o)⊂ (X,o). The embedded resolution of the pair determines a multi-index filtration of the local ring 𝒪_X,o, which measures the embedded geometry of the pair. Furthermore, from the (induced) resolution of (S,o) we also consider a multi-index filtration associated with (S,o). This latter one can be lifted to a filtration of 𝒪_X,o too. The main result proves that the second filtration of 𝒪_X,o can be realized as a `limit' filtration of the first one (if we blow up certain centers sufficiently many times).

We consider the class of all homogeneous, possibly non-reduced, polynomials f whose associated reduced projective divisor D_red⊂ℙ^n-1 has (at worst) quasi-homogeneous isolated singularities. In an arbitrary number of variables n and with d denoting the degree of f, we characterize when -n/d is a root of the Bernstein–Sato polynomial of f in terms of elementary data involving logarithmic derivations. When we restrict to three variables, we prove the resulting class of polynomials satisfies the Strong Monodromy Conjecture, in the motivic sense.

Motivic and topological zeta functions are singularity invariants, mainly associated to a function f and a top differential form ω on a smooth variety. When ω is the standard form dx_1∧…∧ dx_n on affine n-space, the monodromy conjecture states that poles of these zeta functions should induce monodromy eigenvalues of f. We study natural generalized statements of the monodromy conjecture for functions f on complex surface germs; more precisely on singular surfaces for forms ω that generalize the standard form, and on the affine plane for forms ω that are intrinsically associated to f. For all cases, we provide counterexamples to the statement. In addition, when the intrinsically associated ω is given by the generic polar of f, we discover a relation between the poles of the zeta functions and the intersection behaviour of the polar curve.

The Koba-Nielsen local zeta functions are integrals depending on several complex parameters, used to regularize the Koba-Nielsen string amplitudes. These integrals are convergent and admit meromorphic continuations in the complex parameters. In the original case, the integration is carried out on the n-dimensional Euclidean space. In this work, the integration is over a variety of (bounded or unbounded) convex subsets; the resulting integrals also admit meromorphic continuations in the complex parameters. We describe the meromorphic continuation's polar locus explicitly, using the technique of embedded resolution. This result can be reinterpreted as saying that the meromorphic continuations are weighted sums of Gamma functions, evaluated at linear combinations of the complex parameters, where the weights are holomorphic functions. The integrals announced in the title of this paper occur as a particular case of these new Koba-Nielsen local zeta functions, or of a further generalization to arbitrary hyperplane arrangements.

We define a birational analog of the motivic zeta function of a reduced polynomial in terms of minimal models. It admits an intrinsic meaning in terms of contact loci of arcs, an analog of a result of Denef and Loeser in the motivic case. We show that for local plane curve singularities the poles of the birational zeta function essentially coincide with the poles of the motivic zeta function.

In arXiv:1408.4708, Xu defines the dlt motivic zeta function associated to a regular function $f$ on a smooth variety $X$ over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser. The dlt motivic zeta function is defined on a dlt modification via a Denef-Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. We provide explicit examples that show that the dlt motivic zeta function depends on the choice of dlt modification, contrary to what is claimed in arXiv:1408.4708, and that it is therefore not well-defined.

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-Mustata-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 $-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-Mustata-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions. (c) 2021 Elsevier Masson SAS. All rights reserved.

These notes constitute the content of the series of lectures Introduction to local zeta functions by the second author, in the Lluís Santaló Research Summer School 2019: p p -Adic Analysis, Arithmetic and Singularities, June 24-28, at the Palacio de la Magdalena in Santander. We want to thank the organizers of the school for their excellent work. The lectures were intended as an elementary introduction to p p -adic Igusa zeta functions and related topics. We hope that these notes reflect that goal. Our text is complementary to the one of León-Cardenal and Zúñiga-Galindo [Rev. Integr. Temas Mat., 37, 2019, 45–76]. The notes of Nicaise [MSJ Mem., 21, 2010, 141–166] are an introduction to p p -adic and motivic zeta functions. Substantial survey articles are the \lq old\rq Bourbaki report of Denef [Séminaire Bourbaki, 1990/91, 1992, 359–386] and the more recent paper of Meuser [Amer. J. Math., 138, 2016, 149–179].

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.

Roughly speaking, the monodromy conjecture for a singularity states that every pole of its motivic Igusa zeta function induces an eigenvalue of its monodromy. In this note, we determine both the motivic Igusa zeta function and the eigenvalues of monodromy for a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a plane branch. In particular, this yields a proof of the monodromy conjecture for such a curve.

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.

We study motivic zeta functions for Q-divisors in a Q-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the action of a nonabelian group, from which we compute some invariants of motivic nature after constructing a Q-resolution.

We show various properties of numerical data of an embedded resolution of singularities for plane curves, which are inspired by a conjecture of Nguyen and motivated by a conjecture of Igusa on exponential sums.

We provide a gentle introduction to arc spaces, motivic integration and stringy invariants. We explain the basic concepts and first results, including the p-adic number theoretic pre-history, and we provide concrete examples. The text is a slightly adapted version of the 'extended abstract' of the author's talks at the 12th MSJ-IRI "Singularity Theory and Its Applications" (2003) in Sapporo. At the end we included a list of various recent results.

Let V be a finite set of divisorial valuations coming from a modification of \(K^d\), where K is a field. We present results on the semigroup of values and the Poincaré series associated to V, assuming that V has a finite generating sequence. First, if K is infinite, this semigroup is finitely generated. Secondly, for any K, the Poincaré series associated to V is a rational function whose denominator can be expressed in terms of the valuation vectors of the elements in the generating sequence.

We investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension, inspired by the results for curves on surfaces by Smith and Thompson and Tucker. In particular, we construct an example of an exceptional divisor that is not contracted in the log canonical model, and does not contribute any jumping numbers.

The monodromy conjecture predicts that the poles of the topological zeta function and related zeta functions associated to a polynomial f induce monodromy eigenvalues of f. However, not every monodromy eigenvalue can be recovered from a pole. More generally, one also considers zeta functions associated to a polynomial and a differential form. We attach to f a suitable class of differential forms, such that each pole of the topological zeta function of f and such a form induces a monodromy eigenvalue, and moreover such that all monodromy eigenvalues are obtained this way.

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.

We investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension, inspired by the results for curves on surfaces by Smith and Thompson and Tucker. In particular, we construct an example of an exceptional divisor that is not contracted in the log canonical model, and does not contribute any jumping numbers.