Holomorphy of local zeta functions for curves (original) (raw)

Monodromy conjecture for some surface singularities

Annales Scientifiques de l’École Normale Supérieure, 2002

In this work we give a formula for the local Denef-Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef-Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surfaces singularities. These results are applied to the study of rational arrangements of plane curves whose Euler-Poincaré characteristic is three.  2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Dans ce travail, nous donnons une formule pour la fonction zêta locale de Denef-Loeser d'une singularité superisolée d'hypersurface, en termes des fonctions zêta locales des singularités de son cône tangent. Nous démontrons la conjecture de la monodromie pour certaines singularités de surfaces. Nous appliquons ces résultats à l'étude d'arrangements, de caractéristique d'Euler trois, de courbes rationnelles.

Monodromy eigenvalues are induced by poles of zeta functions: the irreducible curve case

Bulletin of the London Mathematical Society, 2010

The 'monodromy conjecture' for a hypersurface singularity f predicts that a pole of its topological (or related) zeta function induces one of its monodromy eigenvalues. However, in general only a few eigenvalues are obtained this way. The second author proposed to consider zeta functions associated with the hypersurface and with a differential form and raised the following question. Can one find a list of differential forms ω i such that any pole of the zeta function of f and an ω i induces a monodromy eigenvalue of f , and such that all monodromy eigenvalues of f are obtained this way? Here we provide an affirmative answer for an arbitrary irreducible curve singularity f .

On atypical values and local monodromies of meromorphic functions

Arxiv preprint math/ …, 1998

A meromorphic function on a compact complex analytic manifold defines a C ∞ locally trivial fibration over the complement of a finite set in the projective line CP 1. We describe zeta-functions of local monodromies of this fibration around atypical values. Some applications to polynomial functions on C n are described. §1.-Introduction We want to consider fibrations defined by meromorphic functions. In order to have more general statements we prefer to use the notion of a meromorphic function slightly different from the standard one. Let M be an n-dimensional compact complex analytic manifold.

A non-vanishing theorem for zeta functions ofGL n

Inventiones Mathematicae, 1976

A and lI its ring of adeles and group of ideles. We regard the group G,= GL(n) as an F-group and, for n > 2, denote by Z, its center. Assume n => 2. A function ~ on the quotient *-G. (V) \ G. (A) G nis said to be cuspidal if the integral (1.1) q~v(g)= S (o(ug)du, U*=U(F)\U(A), U* vanishes for all g 6 G, (A) and all horicycles U of G,. (A horicycle is the unipotent radical of a proper F-parabolic subgroup of G,.) If co is a character (of modulus one) oflI/F • we denote by L2 (~, n) the space of all functions f on G,(A) such that f(Tag)=~o(a)f(g) for 7~G,(F), a~Z,(A)~II, ]f(g)]2 dg< +~. Z.(A) Gn(F)-. Gn(A) We denote by L2o (co, n) the closed subspace of cuspidal elements of L 2 (co, n). The representation of G,(A) on L2(co, n) (by right translations) decomposes into a discrete sum of irreducible representations, each occurring with multiplicity one [15, 2, 13]. To an irreducible component 7r of L2o (o0, n), one can attach as in [5] an L-function L(s, ~z) which satisfies a functional equation (1.2) L(s,n)=~(s, rr)L(1-s,~), where ~ is the representation contragredient to ft. (For other references see [1]). Of course ~ is a component of L 2 (co-~, n). Our main result is as follows:

Zeta Functions for Curves and Log Canonical Models

Proceedings of the London Mathematical Society, 1997

The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial f and an arithmetical invariant associated to a polynomial f over a p{adic eld. When f is a polynomial in two variables we prove a formula for both zeta functions in terms of the so{called log canonical model of f ?1 f0g in A 2 . This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non{symmetric`q{deformation' of the intersection matrix of the minimal resolution of a Hirzebruch{Jung singularity.

THE AUTO IGUSA-ZETA FUNCTION OF A PLANE CURVE SINGULARITY IS RATIONAL

The Auto Igusa-Zeta Function of a Plane Curve Singularity is Rationality, 2019

We show that the auto Igusa-zeta function ζ C,p (t) of a plane curve C over an algebraically closed field k is rational away from points p ∈ C of wild ramification-i.e., it is of the form f (t)/g(t) where f (t) ∈ Gr(Var k)[L −1 , t], where Gr(Var k) is Grothendieck ring of varieties, and g(t) = n i=1 (1 − L a i t b i) with a i ∈ Z and b i ∈ N \ {0}, where L := [A 1 k ] is the Leftshetz motive. As a consequence, we give a new characterization for a curve C on a smooth surface S to be smooth at a point p on C when the ground field is algebraically closed and of characteristic zero.

Arrangement of hyperplanes. I: Rational functions and Jeffrey-Kirwan residue

Annales Scientifiques De L Ecole Normale Superieure, 1999

Consider the space RΔ of rational functions of several variables with poles on a fixed arrangement Δ of hyperplanes. We obtain a decomposition of RΔ as a module over the ring of differential operators with constant coefficients. We generalize the notions of principal part and of residue to the space Δ, and we describe their relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of the work by L. Jeffrey and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions.Nous considérons l'espace RΔ des fonctions rationnelles en plusieurs variables, dont les pôles sont dans un arrangement d'hyperplans Δ fixé. Nous obtenons une décomposition de RΔ comme module sur l'anneau des opérateurs différentiels à coefficients constants. Nous généralisons à l'espace RΔ les notions de partie principale et de résidu, et nous décrivons ses relations avec les transformées de Laplace des fonctions localement polynomiales. Ceci explique des aspects algébriques des travaux de L. Jeffrey et F. Kirwan sur les intégrales de classes de cohomologie équivariantes dans les variétés hamiltoniennes. Comme autres applications, nous construirons, dans un autre article, des versions multidimensionnelles des séries d'Eisenstein, et nous obtiendrons une autre démonstration d'une formule de résidus pour les fonctions zêta de Witten, due à A. Szenes.