An equivariant version of the monodromy zeta function (original) (raw)

An equivariant Poincaré series of filtrations and monodromy zeta functions

Revista Matemática Complutense

We define a new equivariant (with respect to a finite group G action) version of the Poincaré series of a multi-index filtration as an element of the power series ring A(G)[[t 1 ,. .. , t r ]] for a certain modification A(G) of the Burnside ring of the group G. We give a formula for this Poincaré series of a collection of plane valuations in terms of a G-resolution of the collection. We show that, for filtrations on the ring of germs of functions in two variables defined by the curve valuations corresponding to the irreducible components of a plane curve singularity defined by a G-invariant function germ, in the majority of cases this equivariant Poincaré series determines the corresponding equivariant monodromy zeta functions defined earlier. * Math. Subject Class. 14B05, 13A18, 16W22, 16W70. Keywords: finite group actions, filtrations, Poincaré series, monodromy zeta functions, plane valuations. † Supported by the grants MTM2012-36917-C03-01 / 02 (both grants with the help of FEDER Program). ‡ Supported by the grants RFBR-13-01-00755, NSh-5138.2014.1.

Monodromy eigenvalues and zeta functions with differential forms

Advances in Mathematics, 2007

For a complex polynomial or analytic function f , there is a strong correspondence between poles of the so-called local zeta functions or complex powers |f | 2s ω, where the ω are C ∞ differential forms with compact support, and eigenvalues of the local monodromy of f . In particular Barlet showed that each monodromy eigenvalue of f is of the form exp(2π √ −1s 0 ), where s 0 is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.

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 .

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.

Elliptic Zeta functions and equivariant functions

Canadian Mathematical Bulletin, 2017

In this paper we establish a close connection between three notions attached to a modular subgroup, namely, the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup, and the set of elliptic zeta functions generalizing theWeierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

Euler reflexion formulas for motivic multiple zeta functions

Journal of Algebraic Geometry, 2017

We introduce a new notion of-product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability and commuting with the limit of rational series. In the same context, we define a motivic multiple zeta function with respect to an ordered family of regular functions, which is integrable and connects closely to Denef-Loeser's motivic zeta functions. We also show that the-product is associative in the class of motivic multiple zeta functions. Furthermore, a version of the Euler reflexion formula for motivic zeta functions is nicely formulated to deal with the-product and motivic multiple zeta functions, and it is proved for both univariate and multivariate cases by using the theory of arc spaces. As an application, taking the limit for the motivic Euler reflexion formula we recover the well known motivic Thom-Sebastiani theorem.

An equivariant Lefschetz formula for finite reductive groups

Mathematische Annalen, 1980

Let a linear algebraic group G act on a projective variety X. For any G-linearized coherent sheaf ~,~ on X one defines the G-Lefschetz trace of o~ as the element ZG,x(~) = ~(-1) / Tr(alHi(S, ~)) in the representation ring Rk(G) of G over the base field k. In this paper we shall mainly be concerned with the case where X is a smooth projective curve and G a finite reductive group. We shall then give various expressions for LG,x(O~), depending on the nature of ~, involving the numerical invariants of o ~, X, Y=X/G, and a local contribution from the ramification locus of the morphism X~ g These formulas may be viewed as equivariant versions of the Hirzebruch-Riemann-Roch formula, or as Lefschetz formulas. One version generalizes the Chevalley-Weil formula [C] for the representation of G on the m-differentials, in the complex case, cf. Theorem 3.8. The proof of the formula in [C] makes use of some transcendental constructions and a matriciel Riemann-Roch formula (see [W]). These techniques are here replaced by equivariant K-theory and the Hirzebruch-Riemann-Roch formula for locally free sheaves on a curve, much in the same spirit as Grothendieck's proof of Weil's characterization of flat vector bundles on a curve, by means of G-Chern classes [G]. The similar computations in the topological case by Atiyah-Bott-Segal-Singer, in relation with the index theorem, are our basic source for inspiration. However, they compute the trace at particular elements of G in a suitable localization of Rk(G), whereas we need to know the full trace function in Rk(G), in order to identify the corresponding representation. Different algebraic proofs for the action of cyclic groups on the 1-differentials may be found in [V], and on general coherent sheaves in I-N]. The latter expresses the result in a localization of Rk(G), as did Atiyah et al. With a little more care, many of the computations below can be carried through also in the case where G is non-reductive. However, since Rk(G) is * Tilegnet den snart forventede nye Ellingsrud

The zeta function of a quasi-ordinary singularity

Compositio Mathematica, 2004

We prove that the zeta function of an irreducible hypersurface quasi-ordinary singularity f equals the zeta function of a plane curve singularity g. If the local coordinates (x 1 , . . . , x d+1 ) of f are 'nice', then g = . Moreover, the Puiseux pairs of g can also be recovered from (any set of) distinguished tuples of f .

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.