2,133 research outputs found
On the smallest poles of Igusa's p-adic zeta functions
Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is
associated to a K-analytic function on an open and compact subset of K^n. First
we deduce a formula for an important coefficient in the Laurent series of this
meromorphic function at a candidate pole. Afterwards we use this formula to
determine all values less than -1/2 for n=2 and less than -1 for n=3 which
occur as the real part of a pole.Comment: 27 page
Monodromy eigenvalues and zeta functions with differential forms
For a complex polynomial or analytic function f, one has been studying
intensively its so-called local zeta functions or complex powers; these are
integrals of |f|^{2s}w considered as functions in s, where the w are
differential forms with compact support. There is a strong correspondence
between their poles and the eigenvalues of the local monodromy of f. In
particular Barlet showed that each monodromy eigenvalue of f is of the form
exp(a2i\pi), where a 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.Comment: To appear in Advances in Mathematics. 17 page
Zeta functions and monodromy for surfaces that are general for a toric idealistic cluster
In this article we consider surfaces that are general with respect to a 3-
dimensional toric idealistic cluster. In particular, this means that blowing up
a toric constellation provides an embedded resolution of singularities for
these surfaces. First we give a formula for the topological zeta function
directly in terms of the cluster. Then we study the eigenvalues of monodromy.
In particular, we derive a useful criterion to be an eigenvalue. In a third
part we prove the monodromy and the holomorphy conjecture for these surfaces
Stringy E-functions of hypersurfaces and of Brieskorn singularities
We show that for a hypersurface Batyrev's stringy E-function can be seen as a
residue of the Hodge zeta function, a specialization of the motivic zeta
function of Denef and Loeser. This is a nice application of inversion of
adjunction. If an affine hypersurface is given by a polynomial that is
non-degenerate with respect to its Newton polyhedron, then the motivic zeta
function and thus the stringy E-function can be computed from this Newton
polyhedron (by work of Artal, Cassou-Nogues, Luengo and Melle based on an
algorithm of Denef and Hoornaert). We use this procedure to obtain an easy way
to compute the contribution of a Brieskorn singularity to the stringy
E-function. As a corollary, we prove that stringy Hodge numbers of varieties
with a certain class of strictly canonical Brieskorn singularities are
nonnegative. We conclude by computing an interesting 6-dimensional example. It
shows that a result, implying nonnegativity of stringy Hodge numbers in lower
dimensional cases, obtained in our previous paper, is not true in higher
dimension.Comment: 21 page
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