Igusa's Zeta Function Research Articles

Motivic Classes of Curvilinear Hilbert Schemes and Igusa Zeta Functions

Motivic Classes of Curvilinear Hilbert Schemes and Igusa Zeta Functions

Flag Hilbert–Poincaré series and Igusa zeta functions of hyperplane arrangements

Flag Hilbert–Poincaré series and Igusa zeta functions of hyperplane arrangements

Motivic zeta functions of hyperplane arrangements

AbstractFor each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef–Loeser motivic zeta function of $\mathcal{A}$ . We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which $Z_{\mathcal{A}}^{\hat\mu}(T)$ fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, we use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge–Deligne specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of the characteristic polynomials of related arrangements.

Bernstein-Sato polynomials for general ideals vs. principal ideals

We show that given an ideal a \mathfrak {a} generated by regular functions f 1 , … , f r f_1,\ldots ,f_r on X X , the Bernstein-Sato polynomial of a \mathfrak {a} is equal to the reduced Bernstein-Sato polynomial of the function g = ∑ i = 1 r f i y i g=\sum _{i=1}^rf_iy_i on X × A r X\times \mathbf {A}^r . By combining this with results from Budur, Mustaţă, and Saito [Compos. Math. 142 (2006), pp. 779–797], we relate invariants and properties of a \mathfrak {a} to those of g g . We also use the result on Bernstein-Sato polynomials to show that the Strong Monodromy Conjecture for Igusa zeta functions of principal ideals implies a similar statement for arbitrary ideals.

The monodromy conjecture for a space monomial curve with a plane semigroup

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 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.

The motivic zeta functions of a matroid

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements. We show that these motivic zeta functions satisfy a functional equation arising from matroid Poincar\'{e} duality in the sense of Adiprasito-Huh-Katz. In the process, we obtain a formula for the Hilbert series of the cohomology ring of a matroid, in the sense of Feichtner-Yuzvinsky. We then show that our motivic zeta functions specialize to the topological zeta functions for matroids introduced by van der Veer, and we compute the first two coefficients in the Taylor expansion of these topological zeta functions, providing affirmative answers to two questions posed by van der Veer.

Note on the monodromy conjecture for a space monomial curve with a plane semigroup

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.

The auto Igusa-zeta function of a plane curve singularity is rational

It is shown that the auto-Igusa zeta function of a plane curve singularity is rational. This gives a new criterion for a plane curve over an algebraically closed field of characteristic zero to be smooth at a point.

Natural boundaries for Euler products of Igusa zeta functions of elliptic curves

We study the analytic behavior 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.

Igusa Zeta Functions and the Non-archimedean SYZ Fibration

We explain the proof, obtained in collaboration with Chenyang Xu, of a 1999 conjecture of Veys about poles of maximal order of Igusa zeta functions. The proof technique is based on the Minimal Model Program in birational geometry, but the proof was heavily inspired by ideas coming from non-archimedean geometry and mirror symmetry; we will outline these relations at the end of the paper. This text is intended to be a low-tech introduction to these topics; we only assume that the reader has a basic knowledge of algebraic geometry.

On the auto Igusa-zeta function of an algebraic curve

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 ∇nX provided that X is affine, n is a fat point, and the ground field is of characteristic zero. Finally, we show that the auto Poincaré series will often be rational as well and connect this to questions concerning new types of motivic integrals.

Power moments of Kloosterman sums

In this paper we give an essential treatment for power moments of Kloosterman sums over the residue class ring Z/qZ of q elements. When q is a large enough power of a prime, we prove concrete formulas using computations with Igusa zeta functions. As a consequence, we show that there are explicit formulas for power moments of Kloosterman sums whenever q is powerful.

Another generalization of the gcd-sum function

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.

Integration in algebraically closed valued fields with sections

We construct Hrushovski–Kazhdan style motivic integration in certain expansions of ACVF. Such an expansion is typically obtained by adding a full section or a cross-section from the RV-sort into the VF-sort and some (arbitrary) extra structure in the RV-sort. The construction of integration, that is, the inverse of the lifting map L, is rather straightforward. What is a bit surprising is that the kernel of L is still generated by one element, exactly as in the case of integration in ACVF. The overall construction is more or less parallel to the main construction of Hrushovski and Kazhdan (2006) [10], as presented in Yin (2010) [19] and Yin (2011) [20]. As an application, we show uniform rationality of Igusa zeta functions for non-archimedean local fields with unbounded ramification degrees.

Motivic Milnor fiber of a quasi-ordinary hypersurface

Abstract. 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 ) $(S, 0) \subset (\mathbf {C}^{d+1}, 0)$ .

Determination of the real poles of the Igusa zeta function for curves

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.

Local Zeta Functions Supported on Analytic Submanifolds and Newton Polyhedra

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 pm. These zeta functions are defined as integrals over open and compact subsets with respect to the Haar measure. In this article, we introduce new integrals defined over submanifolds, or more generally, over nondegenerate complete intersection varieties, and study their connections with some arithmetical problems such as estimation of exponential sums mod pm. In particular, we extend Igusa's method for estimating exponential sums mod pm to the case of exponential sums mod pm along nondegenerate smooth varieties.

A multivariable Euler product of Igusa type and its applications

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.

Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra

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 ) .

On Igusa zeta functions of monomial ideals

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.