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

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.

On the monodromy at infinity of a plane curve and the Poincare series of its coordinate ring

There exists the problem of expressing topological invariants of maps or of their germs (say, multiplicity, real index, and so on) in analytic terms. Here we discuss this problem for the zeta-function of a monodromy of a plane algebraic curve (or, in other terms, of a polynomial in two variables) with one branch at infinity. We show that this zeta-function coincides with the Poincaré series of an appropriate ring of functions (or of an appropriate semigroup).

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.

Topological zeta functions and the monodromy conjecture for complex plane curves

arXiv (Cornell University), 2020

We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.