We introduce the etale framework to study Igusa zeta functions in several variables, generalizing the machinery of vanishing cycles in the univariate case. We define the etale Alexander modules, associated to a morphism of varieties F from X to affine r-space, a geometric point of X, and an object in the derived category of constructible l-adic sheaves on the inverse image of the r-torus under F. The Alexander modules are sheaves of modules on the scheme of continuous characters of the tame fundamental group of the r-torus. We formulate Loeser's Monodromy and Holomorphy Conjectures for multivariate p-adic zeta functions, and prove them in the case where dim(X)=2, generalizing results from the univariate case. Furthermore, we prove a comparison theorem with the transcendent case, we study a formula of Denef's for the zeta function in terms of a simultaneous embedded resolution, and we generalize a result concerning the degree of the zeta function to our setting.
We discuss some formulae which express the Alexander polynomial (and thus the zeta-function of the classical monodromy transformation) of a plane curve singularity in terms of the ring of functions on the curve. One of them describes the coefficients of the Alexander polynomial or of zeta-function of the monodromy transformation as Euler characteristics of some explicitly constructed spaces. For the Alexander polynomial these spaces are complements to arrangements of projective hyperplanes in projective spaces. For the zeta-function they are disjoint unions of such spaces. Under the influence …
We describe the Alexander modules and Alexander polynomials (both over $\Q$ and over finite fields $\FF{p}$) of generalized trigonal curves. The rational case is closed completely; in the case of characteristic $p>0$, a few points remain open.
We consider an arbitrary polynomial map $f:{\mathbb C}^{n+1}\to {\mathbb C} $ and we study the Alexander invariants of ${\mathbb C}^{n+1}\setminus X$ for any fiber $X$ of $f$. The article has two major messages. First, the most important qualitative properties of the Alexander modules are completely independent of the behaviour of $f$ at infinity, or about the special fibers. Second, all the Alexander invariants of all the fibers of the polynomial $f$ are closely related to the monodromy representation of $f$. In fact, all the torsion parts of the Alexander modules (associated with all the pos…
We complete the enumeration of the possible roots of the Alexander polynomial (both conventional and over finite fields) of a trigonal curve. The curves are not assumed proper or irreducible.
We discuss a notion of integration with respect to the Euler characteristic in the projectivization $¶{\cal O}_{\C^n,0}$ of the ring ${\cal O}_{\C^n,0}$ of germs of functions on $C^n$ and show that the Alexander polynomial and the zeta-function of a plane curve singularity can be expressed as certain integrals over $¶{\cal O}_{\C^2,0}$ with respect to the Euler characteristic.
To any complex algebraic variety endowed with a morphism to a complex affine torus we associate multivariable cohomological Alexander modules, and define natural mixed Hodge structures on their maximal Artinian submodules. The key ingredients of our construction are Gabber-Loeser's Mellin transformation and Hain-Zucker's work on unipotent variations of mixed Hodge structures. As applications, we prove the quasi-unipotence of monodromy, we obtain upper bounds on the sizes of the Jordan blocks of monodromy, and we explore the change in the Alexander modules after removing fibers of the map. We a…
We show that the degree of the Alexander polynomial of an irreducible plane algebraic curve with nodes and cusps as the only singularities does not exceed ${5 \over 3}d-2$ where $d$ is the degree of the curve. We also show that the Alexander polynomial $Δ_C(t)$ of an irreducible curve $C=\{F=0\}\subset \mathbb P^2$ whose singularities are nodes and cusps is non-trivial if and only if there exist homogeneous polynomials $f$, $g$, and $h$ such that $f^3+g^2+Fh^6=0$. This is obtained as a consequence of the correspondence, described here, between Alexander polynomials and ranks of Mordell-Weil gr…
In previous work jointly with Geske, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this paper, we give a positive answer to that question, which implies that the direct sum decomposition of the torsion part of Alexander modules into ge…
The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally we give examples motivated from the theory of Galois coverings of curves.
Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let $U$ be a smooth connected complex algebraic variety and let $f\colon U\to \mathbb{C}^*$ be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of $\mathbb{C}^*$ by $f$ gives rise to an infinite cyclic cover $U^f$ of $U$. The action of the deck group $\mathbb{Z}$ on $U^f$ induces a $\mathbb{Q…
Let $V$ be a degree $d$, reduced hypersurface in $\mathbb{CP}^{n+1}$, $n \geq 1$, and fix a generic hyperplane, $H$. Denote by $\mathcal{U}$ the (affine) hypersurface complement, $\mathbb{CP}^{n+1}- V \cup H$, and let $\mathcal{U}^c$ be the infinite cyclic covering of $\mathcal{U}$ corresponding to the kernel of the linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules $H_i(\mathcal{U}^c;\mathbb{Q})$ of the hypersurface complement and show that, if $i \leq n$, these are torsion over the ring of rational Laurent polynomials. We also…
We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric formula for a class of $p$-adic integrals under non-degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.
We define new higher-order Alexander modules $\mathcal{A}_n(C)$ and higher-order degrees $δ_n(C)$ which are invariants of the algebraic planar curve $C$. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve $C$. These invariants are in the spirit of those developed by T. Cochran in \cite{C} and S. Harvey in \cite{H} and \cite{Har}, which were used to study knots, 3-manifolds, and finitely presented groups, respectively. We show that for curves in general position at infinity, the higher-order degrees are finite. This provides …
Building on our previous work (arXiv:1405.5711), we develop the first practical algorithm for computing topological zeta functions of nilpotent groups, non-associative algebras, and modules. While we previously depended upon non-degeneracy assumptions, the theory developed here allows us to overcome these restrictions in various interesting cases.
The integral $t$-motivic cohomology and the class module of a (rigid analytically trivial) Anderson $t$-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor powers of the Carlitz $t$-motive, namely, the function field counterpart $\underline{A}(n)$ of Tate twists $\mathbb{Z}(n)$. We find out that these modules are in relation with fundamental objects of function field arithmetic: integral $t$-motivic cohomology governs linear relations among Carlitz polylogarithms, its torsion is expressed in terms of the denomina…
The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for p-cohomology in characteristic p.
We overview numerous algorithms in computational $D$-module theory together with the theoretical background as well as the implementation in the computer algebra system \textsc{Singular}. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein-Sato polynomials and also algorithms, recovering any kind of Bernstein-Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein-Sato …
