Zeta functions and Alexander modules
Johannes Nicaise
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 $F:X\rightarrow \A^{r}_{k}$, a geometric point $\xi$ of $X$, and an object in the derived category $D^{b}_{c}(F^{-1}(\GG^{r}_{m,k}),\bar{\mathbb{Q}}_{l})$. These are sheaves of modules on the scheme $\mathcal{C}(\GG^{r}_{m,k})$ of continuous characters $\pi_{1}(\GG^{r}_{m,k})^{t}\rightarrow \bar{\mathbb{Q}}_l^{\times}$. 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.
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\begin{document}
\title[Zeta functions and Alexander modules]{Zeta functions and Alexander modules}
\author{Johannes Nicaise}
\address{KULeuven, Dept. of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium}
\email{johannes.nicaise@wis.kuleuven.ac.be}


\begin{abstract}
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 $F:X\rightarrow
\A^{r}_{k}$, a geometric point $\xi$ of $X$, and an object in the
derived category
$D^{b}_{c}(F^{-1}(\GG^{r}_{m,k}),\bar{\mathbb{Q}}_{l})$. These are
sheaves of modules on the scheme $\mathcal{C}(\GG^{r}_{m,k})$ of
continuous characters $\pi_{1}(\GG^{r}_{m,k})^{t}\rightarrow
\bar{\mathbb{Q}}_l^{\times}$. 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.
\end{abstract}

\maketitle

\section{Introduction}
Let us recall the classical definition of a (univariate) local
zeta function, as it was given by Igusa
\cite{Igusa1}\cite{Igusa2}\cite{Igusa3}\cite{Igusa:intro}. A
survey of the univariate theory can be found in Denef's Bourbaki
report \cite{DenefBour}.

In its simplest form, a local zeta function is defined by the
$p$-adic integral
$$Z(f,s)=\int_{\Z_p^{n}}|f(x)|^{s}|dx|\,, $$
where $s$ is a complex variable, $f$ is a polynomial over $\Q_p$
in $n$ variables, $|f|$ is its $p$-adic norm, and $|dx|$ denotes
the Haar measure on the compact group $\Z_p^{n}$, normalized
to give $\Z_p^{n}$ measure $1$. A priori,
$Z(f,s)$ is only defined when $\Re(s)>0$. However, Igusa proved,
using resolution of singularities, that it has a meromorphic
continuation to the complex plane. Moreover, it is a rational
function in $p^{-s}$. The numerical data of an embedded resolution
for $f$ yield a set of candidate poles of $Z(f,s)$, but, since
this set depends on the chosen resolution, a lot of these
candidate poles will not be actual poles of $Z(f,s)$.

When $f$ is defined over $\Q$, Igusa's Monodromy Conjecture
\ref{monoconjecture} predicts an interesting connection between
the eigenvalues of the monodromy at some point of
$f^{-1}(0)\subset \C^{n}$, and the poles of $Z(f,s)$, for almost
all primes $p$, where $f$ is considered as a complex, respectively
$p$-adic polynomial. When the zeta function is twisted by
introducing a multiplicative character
$\chi:\Z_p^{\times}\rightarrow \C^{\times}$, Denef's Holomorphy
Conjecture \ref{holoconjecture} states that poles only occur when
the order of $\chi$ divides the order of an eigenvalue of the
monodromy at some point of $f^{-1}(0)\subset \C^{n}$.

In this paper, we consider multivariate zeta functions, associated
to an $r$-tuple of polynomials $F=(f_1,\ldots,f_r)$. They are
defined in Section \ref{def}. Multivariate local zeta functions
were already studied by Loeser in \cite{ENS}. We state his
generalizations of the Monodromy Conjecture \ref{gmonoconjecture}
and the Holomorphy Conjecture \ref{gholoconjecture}. Since in
general, $F$ does not induce a locally trivial fibration near a
singular point of the complex zero locus, we no longer have a
geometric Milnor fiber at our disposal. However, the formalism of
nearby cycles still makes sense in the multivariate setting, as
was shown by Sabbah in \cite{Sabbah}, such that we do have
something like a cohomological Milnor fiber: the Alexander complex
(actually, it already exists as a motivic object, see
\cite{Guibert}). The monodromy action makes these cohomological
objects into $\C[\Z^{r}]$-modules, and the support of the induced
coherent sheaves on $Spec\,\C[\Z^{r}]$ generalizes the eigenvalues
of the monodromy transformation. Sabbah also proved a generalized
A'Campo formula, expressing the monodromy zeta function of the
Alexander complex at a singular point in terms of a simultaneous
embedded resolution for the polynomials $f_i$.

Using ideas from \cite{GabLoe}, we introduce the \'etale
counterpart of this Alexander complex. The Alexander modules are
associated to a morphism of varieties $F:X\rightarrow \A^{r}_{k}$,
a geometric point $\xi$ of $X$, and an object in the derived
category $D^{b}_{c}(F^{-1}(\GG^{r}_{m,k}),\bar{\mathbb{Q}}_{l})$.
They are sheaves of modules on the scheme
$\mathcal{C}(\GG^{r}_{m,k})$ of continuous characters
$\pi_{1}(\GG^{r}_{m,k})^{t}\rightarrow
\bar{\mathbb{Q}}_l^{\times}$.
 We prove a comparison
theorem, stating that our definition is equivalent to the
 definition of the \'etale nearby cycles if $r=1$, and to
the transcendent one if our base field is the complex field $\C$.
In the latter case, the stalk of the $q$-th Alexander module at a
continuous character $\chi:\pi_{1}(\GG^{r}_{m,k})^{t}\rightarrow
\bar{\mathbb{Q}}_l^{\times}$ corresponds to the subspace of the
$q$-th transcendent Alexander module on which the semi-simple part
of the monodromy acts as multiplication by $\chi$.

Let us give an overview of the structure of the paper. In Section
\ref{def}, we define the multivariate local zeta functions, and we
recall a result of Igusa's concerning their rationality, and the
set of candidate poles associated to a simultaneous embedded
resolution (Theorem \ref{rat}). In Section \ref{alex}, we define
the Alexander modules, and we prove that their support is
contained in a finite union of translated cotori of codimension
one in the scheme of continuous characters
$\mathcal{C}(\G^{r}_{m,k})$ (Proposition \ref{support}). The
generalized Monodromy and Holomorphy Conjectures are stated in
Section \ref{conj}, while Section \ref{trans} contains the
comparison theorems mentioned above (Proposition \ref{comp1} and
\ref{monoco}), making use of a local computation in the normal
crossings case. In Section \ref{form}, we study a generalization
of Denef's formula for the local zeta function in terms of the
geometry of the exceptional locus of a simultaneous embedded
resolution (Theorem \ref{formdenef}). This allows us to prove a
criterion to unmask certain candidate poles that do not contribute
to the polar locus of the local zeta function (Theorem
\ref{unmask}). We prove a result concerning the limit of the zeta
function at $-\infty$ in Section \ref{degree}, Theorem
\ref{theoremdegree}. To conclude, in Section \ref{proofs}, we
prove the generalized Monodromy and Holomorphy Conjecture in the
case where $dim(X)=2$.
\section{Several variables Igusa local zeta functions}\label{def}

\subsection{}Let us fix some notation.
We choose a prime number $p$.
We denote by $K$ a finite extension of $\QQ_p$,
with ring of integers $R$, and residue field $k$, a finite extension
of $\FF_p$. We choose
an uniformizing parameter $\varpi$ and denote by $q$ the cardinality of $k$.
We denote by ${\rm val}$ the valuation on $K$ and
we normalize the norm on
$K$ by setting $|x| = q^{-{\rm val} (x)}$.
For $x$ non zero in $K$, we define $\ac (x)$ as the image in
$k$ of $x \varpi^{-{\rm val} (x)}$ and we set
$\ac (x) = 0$ if $x = 0$.


Let $X$ be a smooth compact $p$-adic (locally) analytic
variety (hence $X$ is a finite disjoint union of balls) of pure dimension
$d$ and let $\omega$ be a gauge form on
$X$, i.e. a nowhere vanishing analytic $n$-form on
$X$. In a standard way (cf. \cite{serre}),
we may attach to $\omega$ a volume form
$|\omega|$ on $X$.







\subsection{}We denote by $\cC (\GG_m^r (k), \CC)$
the group
of characters
$$\chi = (\chi_1, \dots, \chi_r) :
\GG_m^r (k) = (k^*)^r \longrightarrow \CC^*.$$
For any analytic function
$F = (f_1, \dots, f_r) : X \rightarrow K^r$,
for
$\phi$ locally constant on $X$, for $\chi$ in
$\cC (\GG_m^r (k), \CC)$
and for $s = (s_1,\dots, s_r) $ in
$\CC^r$,
we consider Igusa's local zeta function
$$
Z_{\phi} (F, \chi ; s)
:=
\int_X \phi \prod_{i = 1}^r
\chi_i (\ac (f_i (x)))
\prod_{i = 1}^r
|f_i (x)|^{s_i} |\omega|,
$$
which is defined when $\Re (s_i) > 0$, for each $i$, where $\Re
(s_i)$ denotes the real part of $s_i$. We will often omit
$\phi,F,\chi$ from the notation, and denote the zeta function
simply by $Z(s)$. Furthermore, when $X=R^{n}$, and $\phi$ is the
characteristic function of $P^{n}$, with $P$ the maximal ideal in
$R$, we write $Z_0(s)$ instead of $Z_{\phi}(F,\chi;s)$.

\subsection{}Consider
an embedded resolution $\pi : Y \rightarrow X$ of the $f_j$'s. By
this we mean that $\pi : Y \rightarrow X$ is a proper morphism
between compact locally analytic $p$-adic varieties, that $\pi$
induces an isomorphism between $Y \setminus \pi^{-1} (D)$ and $X
\setminus D$, with $D = \cup_{j} f_j^{-1} (0)$, and that $\pi^{-1}
(D)$ is a divisor $E$ with (global) normal crossings. We denote by
$E_i$, $i \in I$, the set of irreducible components of $E$. In
particular, the $E_i$ are smooth. We denote by $N_i^j$ the order
of vanishing of $f_j \circ \pi$ at the generic point of $E_i$, and
by $\nu_i - 1$ the order of vanishing of $\pi^* \omega$ at the
generic point of $E_i$. We shall write $N_i s$ for $\sum_j N_i^j
s_j$, and $\chi^{N_i}$ for $\prod_j \chi_j^{N_i^j}$.


\begin{theorem}[Igusa]\label{rat}
The function
$Z_{\phi} (F, \chi ; s)$ is a rational function
in the variables $q^{-s_j}$. Furthermore,
given
an embedded resolution $\pi : Y \rightarrow X$  of the $f_j$'s,
a necessary condition for
$Z_{\phi} (F, \chi ; s)$ to have a pole for a given value
of $\chi$ and $s$ is that
$$
\prod_{i \in I} (1 -q^{- \nu_i} q^{ -N_i s}) = 0
$$
and
$$
\prod_{i \in I} (\chi^{N_i} - 1) =0.
$$
\end{theorem}


\subsection{}By Theorem \ref{rat},
the set of real parts of the poles of $Z_{\phi} (F, \chi ; s)$ are
located on a finite set of affine hyperplanes
$$
H : \quad \sum_j N^j s_j + \nu = 0
$$
with $N^j$ in $\NN$ and $\nu > 0$. We denote this finite set of
hyperplanes by $\cP (Z_{\phi} (F, \chi ; s))$.


\section{Alexander modules}\label{alex}


\subsection{Notations and conventions}
We fix an algebraically closed field $k$ of characteristic
exponent $p$ (i.e. $p = 1$ when $k$ is of characteristic $0$, and
$p$ is equal to the characteristic of $k$ when $k$ is of
characteristic $> 0$). We fix a prime number $l \not= p$. We
denote by $\bar \QQ_{l}$ a fixed algebraic closure of $\QQ_{l}$.
We set $\hat \ZZ (1) (k) := \limproj_{(p, n) = 1} \mu_n (k)$ and
$\ZZ_{l} (1) (k) := \limproj_{n} \mu_{l^n} (k)$. Let $T$ be a
$k$-torus, $T \simeq \GG_{m, k}^r$. We set $X_{\ast} (T) := {\rm
Hom}_{k-{\rm gr}} (\GG_{m, k}, T)$, $\hat X_{\ast} (T) := X_{\ast}
(T) \otimes_{\ZZ} \hat \ZZ (1) (k)$ and $X_{\ast l} (T):= X_{\ast}
(T)\otimes_{\ZZ} \ZZ_{l} (1) (k)$. We denote by $\pi_1 (T)$ the
fundamental group of $T$ pointed at 1, by $\pi_1 (T)^t$ the tame
fundamental group of $T$, and by $\pi_1 (T)_{l}$ the maximal
pro-$l$ quotient of $\pi_1 (T)^t$. By Kummer theory, there are
canonical isomorphisms $\hat X_{\ast} (T) \simeq \pi_1 (T)^t$ and
$X_{\ast l} (T) \simeq \pi_1 (T)_{l}$.




\subsection{Torsion by the generic character}
Let $R$ be a complete regular local ring with residual
characteristic $l$. For every $k$-torus $T$, we set
$$
\Omega^R_T := R [[\pi_1 (T)_{l}]].
$$
This is a complete regular local ring of residual characteristic
$l$. By Kummer theory, $\Omega^R_T$ may be identified with $R
[[X_{\ast l} (T)]]$. For every choice of a generator $\gamma$ of
$\ZZ_{l} (1) (k)$, we have, by Iwasawa theory, an isomorphism
$$
R [[\ZZ_{l} (1) (k)]] \simeq R[[t]]
$$
sending $\gamma$ to $1 +t$.
Hence, for every isomorphism $T \simeq (\GG_{m, k})^r$,
we have, $\gamma$ being given,
an isomorphism
$$
\Omega^R_T \simeq R[[t_1, \ldots, t_r]].
$$
There is a canonical continuous character
$$
{\rm Can}_T : \pi_1 (T) \rightarrow ({\Omega^R_T})^{\times}
$$
obtained by composition of the tautological character $\pi_1
(T)_{l} \rightarrow ({\Omega^R_T})^{\times}$, sending $\gamma$ to
$\gamma$, with the projection $\pi_1 (T) \rightarrow \pi_1
(T)_{l}$. We shall denote by $L_T^R$ the corresponding
$\Omega_T^R$-sheaf on $T$. It is a tamely ramified lisse
constructible free twisted rank 1 $\Omega_T^R$-sheaf on $T$.




\subsection{The scheme $\cC (T)$}
 In \cite{GabLoe}, Gabber and Loeser defined a $\bar \QQ_{l}$-scheme $\cC (T)$
whose set of closed points may be canonically identified with the
group of continuous characters $\pi_1 (T)^t \rightarrow \bar
\QQ_{l}^{\times}$. It is the disjoint union of the translates of
the connected component $\cC (T)_{l}$ of the trivial character by
the group of characters $\chi : \pi_1 (T)^t \rightarrow \bar
\QQ_{l}^{\times}$ of finite order prime to $l$. By its very
definition,
$$
\cC (T)_{l} := \Spec \bar \QQ_{l} \otimes \ZZ_{l} [[\pi_1
(T)_{l}]].
$$
By \cite{GabLoe} 3.2.2, $\cC (T)_{l}$ is a noetherian regular
scheme.

\subsection{}\label{cot}
If $\pi : T \rightarrow T'$
is a morphism of tori, we denote
by $\pi^{\vee} : \cC (T') \rightarrow \cC (T)$
the morphism which one deduces by
functoriality.
We say that
$\pi : T \rightarrow T'$
is a quotient of $T$, if $\pi$ is surjective with kernel a torus.
We call a subscheme of
$\cC (T)$
of the form $\pi^{\vee} (\cC (T'))$, with $\pi : T \rightarrow T'$
a quotient of $T$,
a cotorus of $\cC (T)$. A subscheme of the form
$ \chi \cdot \pi^{\vee} (\cC (T'))$,
with $\chi$ a closed point of $\cC (T)$, is called a
translated cotorus.
If $Z$ is cotorus translated by a character of order $n$
(i.e. $Z = \chi \cdot \pi^{\vee} (\cC (T'))$
with $\chi$ of order $n$),
we
set
$$
\overline Z := \bigcup_{1 \leq i \leq n} \chi^i \cdot \pi^{\vee} (\cC (T')).
$$
More generally, if $Z$ is a finite union $\cup Z_i$ of cotori
translated by a character of finite order, we set $\overline Z :=
\cup_i \overline Z_i$.


\subsection{}\label{fo}Assume $k$ is the
algebraic closure of a finite field $k_0$ with $q$ elements. Fix
an isomorphism $\bar \QQ_{l} \simeq \CC$. Then we may identify
$\cC (\GG_m^r (k_0), \CC)$ with the set of closed points of $\cC
(\GG_{m, k}^r)$ of order dividing $q - 1$, since we have a
canonical morphism $\hat \ZZ (1) (k) \rightarrow \mu_{q - 1} (k)
\simeq k_0^{\ast}$.




\subsection{}\label{rest}Assume $K$ is the algebraic closure of some local field
having $k$ as residue field.
The canonical injection $\hat \ZZ (1) (k) \hookrightarrow
\hat \ZZ (1) (K)$
induces an epimorphism
$r : \cC (\GG_{m, K}^r) \rightarrow \cC (\GG_{m, k}^r)$.

\subsection{Alexander modules}
Now we consider the affine space $$\A^r_k := \Spec k [T_1, \ldots,
T_r]$$ and the torus $j : T = \Spec k [T_1, T_1^{-1},\ldots, T_r,
T_r^{-1}] \hookrightarrow \A^r_k$. Let $F : X \rightarrow \A^r_k$
be a morphism of $k$-varieties and set $j : X_T := F^{-1} (T)
\hookrightarrow X$.


Let $R$ be the ring of integers of a finite extension of
$\QQ_{l}$, and let $\cF$ be an object in $D^b_c (X_T, R)$. As
explained in  \cite{GabLoe} A.1, one can define in a natural way
objects $\cF \otimes_R F^{\ast} L_T^R$ and $Rj_{\ast}  (\cF
\otimes_R F^{\ast} L_T^R)$ in $D^b_c (X_T, \Omega^R_T)$ and $D^b_c
(X, \Omega^R_T)$,  respectively. Hence, for every geometric point
$\xi$ of $X \setminus X_T$, and every integer $q \geq 0$, one may
consider
$$
\cA^{q, R}_{F, \xi} (\cF) :=
R^qj_{\ast}  (\cF \otimes_R F^{\ast} L_T^R)_{\xi}
$$
and view it as a coherent $\Omega^R_T$-module.




Now, if $\cF$ is an object in $D^b_c (X_T, \bar \QQ_{l})$, we
define a coherent $\cC (T)$-module $\cA^{q}_{F, \xi} (\cF)$, the
$q$-th Alexander module  of $\cF$ at $\xi$, as follows. We choose
a model $\cF_R$ in $D^b_c (X_T, R)$ for some $R$, and ask for  the
restriction of $\cA^{q}_{F, \xi} (\cF)$ to the connected component
$\cC (T)_{l}$ to be equal to
$$\cA^{q}_{F, \xi} (\cF)_{|\cC (T)_{l}}
:= \cA^{q, R}_{F, \xi} (\cF_R) \otimes \cC (T)_{l},$$ which is
clearly independent of the chosen model $\cF_R$. For  the
restriction of $\cA^{q}_{F, \xi} (\cF)$ to a general connected
component of the form $\{\chi\} \times \cC (T)_{l}$, with $\chi$ a
character of finite order prime to $l$, we set
$$\cA^{q}_{F, \xi} (\cF)_{|\{\chi\} \times \cC (T)_{l}}
:= \cA^{q}_{F, \xi} (\cF \otimes \cL_{\chi})_{|\cC (T)_{l}},
$$
with $\cL_{\chi}$ the Kummer sheaf attached to $\chi$ (its
definition is similar to the construction of
$\mathcal{L}_{\psi,\alpha}$ in Section \ref{form}).



If $\cF$ belongs to $D^b_c (X, \bar \QQ_{l})$, we shall write
$\cA^{q}_{F, \xi} (\cF)$ for $\cA^{q}_{F, \xi} (j^{\ast}\cF)$.
Also, if $k$ is not algebraically closed, we shall still write
$\cA^{q}_{F, \xi} (\cF)$ for the Alexander module attached to $F
\otimes \bar k  : X \otimes \bar k \rightarrow \A^n_{\bar{k}}$.


\subsection{Support of Alexander modules}

\begin{prop}\label{support}
Let $F : X \rightarrow \A^n_k$ be a morphism of $k$-varieties, and
let $\cF$ be an object in $D^b_c (X_T, \bar \QQ_{l})$. Then, for
every geometric point $\xi$ of $X \setminus X_T$ and every integer
$q$, the support of the module $\cA^{q}_{F, \xi} (\cF)$ is
contained in a finite set, not depending on $\xi$ and $q$, of
translated cotori of codimension 1 of $\cC (T)$. Furthermore, when
$\cF = \bar \QQ_{l}$, the support of the module $\cA^{q}_{F, \xi}
(\bar \QQ_{l})$ is contained in a finite set, not depending on
$\xi$ and $q$, of cotori of codimension 1 of $\cC (T)$ translated
by characters of finite order.
\end{prop}
\begin{proof}
We may suppose that $\cF$ is the extension by zero of a lisse
sheaf on a smooth irreducible locally closed subvariety $U$ of
$X_T$. We may furthermore assume that $U$ is dense in $X_T$, and
that the monodromy of $\cF$ is pro-$l$; see \cite{GabLoe}, Prop.
4.3.1'.

By de Jong's theorem \cite{dJ}, there exists a generically \'etale
alteration $\pi:X'\rightarrow X$, where $X'$ is regular, and the
inverse image $\pi^{-1}(Z)$ is a normal crossing divisor, with
$Z=X-U$. Shrinking $X$ if necessary, we may suppose that $\pi$ is
\'etale over $U$.
%Furthermore, there exists a regular
%compactification $\bar{X}$ of $X'$, such that the union of
%$\bar{X}\setminus X$ with $p^{-1}(Z)$ is still a normal crossing
%divisor.

\begin{picture}(100,100)
\put(120,70){$\pi^{-1}(U)$} \put(160,74){\vector(1,0){40}}
\put(170,76){$u'$} \put(205,70){$\pi^{-1}(X_T)$}
\put(250,74){\vector(1,0){40}} \put(260,77){$j'$}
\put(295,70){$X'$}

\put(135,65){\vector(0,-1){40}} \put(220,65){\vector(0,-1){40}}
\put(300,65){\vector(0,-1){40}} \put(303,45){$\pi$}

\put(130,10){$U$} \put(150,14){\vector(1,0){50}}\put(170,16){$u$}
\put(215,10){$X_T$}
\put(240,14){\vector(1,0){50}}\put(260,17){$j$} \put(295,10){$X$}
\end{picture}

For any object $\mathcal{G}$ in $D^{b}_{c}(X_T,R)$, and any
character $\chi$ of finite order prime to $l$, we have a short
exact sequence
$$0\rightarrow u_{!}u^{*}\mathcal{G}\otimes \mathcal{L}_{\chi}\rightarrow \mathcal{G}\otimes \mathcal{L}_{\chi}\rightarrow i_{*}i^{*}\mathcal{G}\otimes \mathcal{L}_{\chi}\rightarrow 0 \,,$$
where $i$ is the inclusion of $Z\cap X_T$ in $X_T$, and
$\mathcal{L}_{\chi}$ is the Kummer sheaf of $\chi$ on $X_T$.
Applying the derived functor $R^{q}j_{*}$ yields a long exact
sequence, hence, it suffices to investigate the support of the
Alexander complex of $u_{!}u^{*}\mathcal{G}$ and
$i_{*}i^{*}\mathcal{G}$.

The theorem holds for the latter, by Noetherian induction, since
$i_{*}$ is exact, hence, by Leray's spectral sequence,
$$R^{q}j_{*}(i_{*}i^{*}\mathcal{G}\otimes \mathcal{L}_{\chi})\cong R^{q}(j\circ
i)_{*}(i^{*}\mathcal{G}\otimes i^{*}\mathcal{L}_{\chi})\,.$$

As for the former, by proper base change, one sees that
$$u_{!}u^{*}\mathcal{G}\otimes \mathcal{L}_{\chi}\cong \pi_{*}u'_{!}\pi^{*}u^{*}\mathcal{G}\otimes \mathcal{L}_{\chi} \,.$$
Leray's spectral sequence reads
$$E_{2}^{p,q}= (R^{p}j_{*})(R^{q}\pi_{*})(u'_{!}\pi^{*}u^{*}\mathcal{G}\otimes \pi^{*}\mathcal{L}_{\chi})
\Rightarrow R^{p+q}(j\circ
\pi)_{*}(u'_{!}\pi^{*}u^{*}\mathcal{G}\otimes
\pi^{*}\mathcal{L}_{\chi})\,.$$ We use the proper base change
theorem to compute the stalks of
$(R^{q}\pi_{*})(u'_{!}\pi^{*}u^{*}\mathcal{G}\otimes
\pi^{*}\mathcal{L}_{\chi})$. These are zero outside of $U$, and
since $\pi$ is \'etale over $U$, they vanish in all other points
as well, if $q>0$. Therefore, our spectral sequence yields
isomorphisms
$$R^{p}j_{*}(u_{!}u^{*}\mathcal{G}\otimes \mathcal{L}_{\chi})\cong R^{p}(\pi\circ j')_{*}(u'_{!}\pi^{*}u^{*}\mathcal{G}\otimes \pi^{*}\mathcal{L}_{\chi})\,.$$
After applying Leray's spectral sequence and proper base change
once more, one sees that it suffices to consider the support of
$R^{p}j'_{*}(u'_{!}\pi^{*}u^{*}\mathcal{G}\otimes
\pi^{*}\mathcal{L}_{\chi})_{\xi'}$, where $\xi'$ is a geometric
point of $\pi^{-1}(X\setminus X_T)$.

This means we may assume that $W$ is smooth over $k$, that the
complement of $U$ in $X$ is a normal crossing divisor, and that
the monodromy of $\mathcal{F}$ on $U$ is pro-$l$. Now one can
proceed as in the proof of \cite{GabLoe}, Proposition 4.3.1'.
%Just plug de Jong's result \cite{dJ} in the proof of Proposition
%4.3.1$'$ of \cite{GabLoe}. [To be detailed.]
\end{proof}



\section{Conjectures}\label{conj}
\subsection{}In this section, we shall consider
a number field $K$, and a smooth algebraic variety $X$ of
dimension $d$ over $K$, together with a gauge form $\omega$ on
$X$. Let $F := (f_1, \dots, f_r)$ be a morphism $X \rightarrow
\A^r_K$. For every finite place $\gP$ of $K$, we denote by
$K_{\gP}$ the completion of $K$ and by $R_{\gP}$ its ring of
integers. Taking an appropriate model $\cX$ of $X$ over the
integers of $K$ imposes, at least for almost all $\gP$, the
structure of a smooth compact locally analytic $K_{\gP}$-variety
on $X_{\gP} := \cX (R_{\gP})$. Also, taking appropriate models of
$\omega$ and $F$, we obtain a gauge form $\omega_{\gP}$ on
$X_{\gP}$ and an analytic function $F_{\gP} = (f_{1 \gP}, \dots,
f_{r \gP}) : X_{\gP} \rightarrow K_{\gP}^r$. All these data are
independent of choices, for almost all $\gP$. Hence, for suitable
$\chi$ and $\phi$, we can consider Igusa's zeta function $Z_{\phi}
(F_{\gP}, \chi ; s)$.




\subsection{}Let us first review
the $r = 1$ case, i.e. the case when there is only one function $f$.
We write $f$ for $F = (f)$.


We shall say a locally constant function $\phi$ on $X_{\gP} := \cX
(R_{\gP})$ is residual if its value at a point $x$ depends only of
the image in $\cX (\FF_{\gP})$, where $\FF_{\gP}$ denotes the
residue field.

\begin{monoconjecture}[Igusa, \cite{DenefBour}]\label{monoconjecture}
For almost all finite places $\gP$, if $s_0$ is a pole of
$Z_{\phi} (f_{\gP}, \chi ; s)$, with $\phi$ residual on $X_{\gP}$,
then $\exp (2 \pi i {\rm Re} (s_0))$ is an eigenvalue of the local
monodromy at some point of $f^{-1} (0)$.
\end{monoconjecture}


\begin{holoconjecture}[Denef, \cite{Denef5}]\label{holoconjecture}
For almost all finite places $\gP$, if $Z_{\phi} (f_{\gP}, \chi_0
; s)$, with $\phi$ residual on $X_{\gP}$, is not a holomorphic
function of $s$, then the order of $\chi_0$ divides the order of
some eigenvalue of the local monodromy at some point of $f^{-1}
(0)$.
\end{holoconjecture}


Here we should make more precise what is meant by ``some
eigenvalue of the local monodromy at some point of $f^{-1} (0)$''.
The easiest way is to fix a complex embedding of $K$ in $\CC$ and
to stick to  the complex points of $X$ with complex topology. We
can consider the usual complex of nearby cycles $R \psi \CC$ on
$f^{-1} (0)$, and then ``some eigenvalue of the local monodromy at
some point of $f^{-1} (0)$'' means ``some eigenvalue of the
monodromy action on  some cohomology object of the stalk $(R \psi
\CC)_x$ at some complex point $x$ of $f^{-1} (0)$''. More
algebraically, and this amounts to the same by the classical
comparison theorems, one could consider the complex of nearby
cycles $R \psi \QQ_{l}$ on $f^{-1} (0)$ for the \'etale topology
and $l$-adic cohomology, and
 interprete it as
``some eigenvalue of the geometric monodromy action on  some
cohomology object of the stalk $(R \psi \QQ_{l})_x$ at some
geometric point $x$ of $f^{-1} (0)$''.


\subsection{}We fix an isomorphism $\bar \QQ_{l} \simeq \CC$.
In particular, we have an embedding of $\bar \QQ$ in $\CC$. We
denote by $\gamma$ the topological generator $\gamma = (\zeta_n)$,
$\zeta_n = \exp (2 \pi i / n)$, of $\hat \ZZ (\QQ)$. To an affine
hyperplane $H$ with equation $\sum_j N^j s_j + \nu = 0$ in
$\CC^r$, with $N^j$ and $\nu$ in $\NN$, $\nu > 0$, we associate
$\cH$, a finite union of cotori translated by a character of
finite order in $\CC (\GG_{m, \bar K})$, as follows. We deduce
from $\gamma$ a topological basis $(\gamma_1, \dots, \gamma_r)$ of
$\pi_1 (\GG_{m, \bar K}^r)^t = \hat \ZZ (1)^r (\bar K)$. For any
$\alpha$ in $(\QQ/ \ZZ)^r$, we denote by $\varphi (\alpha)$ the
character which sends $(\gamma_1, \dots, \gamma_r)$ to $\exp (2
\pi i \alpha)$. We consider the image $[H (\QQ)]$ of $H (\QQ)$ in
$(\QQ/ \ZZ)^r$, and we define $\cH$ as the Zariski closure of
$\{\varphi (\alpha)|\alpha \in [H (\QQ)]\}$ in $\CC (\GG_{m, \bar
K}^r)$. It is clearly a finite union of cotori translated by a
character of finite order.



\subsection{}Fran\c cois Loeser proposes the following generalisation of the monodromy
conjecture:

\begin{gmonoconjecture}\label{gmonoconjecture}\
For almost all finite place $\gP$, if $H$ is a polar hyperplane in
$\cP (Z_{\phi} (F_{\gP}, \chi ; s))$, with $\phi$ residual on
$X_{\gP}$, then, for every irreducible component $Z$ of $\cH$,
there exists an integer $q$ and a geometric point $\xi$ such that
$Z$ is contained in the support of $\cA^{q}_{F,  \xi} (\bar
\QQ_{l})$.
\end{gmonoconjecture}




Loeser also generalizes the holomorphy conjecture as follows
(using notations and conventions from \ref{fo} and \ref{rest}):


\begin{gholoconjecture}\label{gholoconjecture}
For almost all finite place $\gP$, if $Z_{\phi} (F_{\gP}, \chi_0 ;
s)$ with $\phi$ residual on $X_{\gP}$ is not a holomorphic
function of $s$, then there exists an integer $q$ and a geometric
point $\xi$, such that $\chi_0$ belongs to $r (\overline{{\rm
Supp} \, \cA^{q}_{F,  \xi} (\bar \QQ_{l})})$.
\end{gholoconjecture}


\begin{remark}By Proposition \ref{monoco}, when $r = 1$,
Conjectures \ref{gmonoconjecture} and \ref{gholoconjecture}
are indeed equivalent to
Conjectures \ref{monoconjecture} and \ref{holoconjecture}.
\end{remark}

\section{Comparison with the transcendent theory}\label{trans}
In this section, we compare our definition of the Alexander
complex with the classical definition \cite{Sabbah} in the complex
case, following Deligne's treatment of the nearby cycles
\cite{SGA7b}. Let $X$ be a complex manifold, and let
$F=(f_1,\ldots,f_{r}):X\rightarrow \C^{r}$ be an analytic map. We
denote the complex analytic torus $\mathbb{G}_{m,\C}^{r}(\C)$ by
$T$, the inverse image of $T$ under $F$ by $X_{T}$, and the
complement of $X_{T}$ in $X$ by $X_0$. Let
$\pi:\tilde{T}\rightarrow T$ be a universal cover of $T$. Consider
the following diagram:

\begin{picture}(200,80)(-30,20)
\put(30,80){$X_0$} \put(45,83){\vector(1,0){30}} \put(82,80){$X$}
\put(125,83){\vector(-1,0){30}} \put(130,80){$X_T$}
\put(175,83){\vector(-1,0){28}} \put(180,80){$X_{T}\times_T
\tilde{T}=X_{\tilde{T}}$}

%\put(37,77){\vector(0,-1){30}}
 \put(86,77){\vector(0,-1){30}}
\put(137,77){\vector(0,-1){30}} \put(196,77){\vector(0,-1){30}}

\put(90,58){$F$} \put(162,42){$\pi$} \put(162,85){$\tilde{\pi}$}
\put(57,85){$i$} \put(107,86){$j$}

%\put(35,35){$0$} \put(45,40){\vector(1,0){30}}
\put(82,35){$\C^{r}$} \put(127,40){\vector(-1,0){32}}
\put(135,35){$T$} \put(185,40){\vector(-1,0){40}} \put(193,35){$
\tilde{T}$}

\end{picture}

\noindent In the univariate case ($r=1$), after reducing $X$ to a
sufficiently small neighbourhood of a point $x\in F^{-1}(0)$ (i.e.
the intersection of a small ball centered at $x$ with the inverse
image of a small ball around the origin in $\C$), the space
$X_{T}\times_T \tilde{T}$ is called the canonical Milnor fiber of
$F$ at $x$, since it is homotopy-equivalent to the fiber of the
Milnor fibration at $x$ \cite{Milnor}\cite{Kuli}. However, in the
multivariate case, $F$ will, in general, not be a fibration in any
neighbourhood of $x$.

Let $\mathcal{F}$ be an object in $D_{c}^{b}(X_T,\C)$. We define
the Alexander complex to be
$$R\psi_{F}(\mathcal{F})=i^{*}R(j\circ \tilde{\pi})_{*}
\tilde{\pi}^{*}\mathcal{F}\,.$$ The deck transformations of
$X\times_{T}\tilde{T}$ make $R\psi_{F}(\mathcal{F})$ into a
constructible $\C[\Z^{r}]$-module.

Now let $A$ be a regular Noetherian ring, $\mathcal{F}$ an
$A$-constructible sheaf on $X_0$, and $\mathcal{G}$ an object in
$D_{c}^{b}(X_0,A)$. The support Supp$_x \mathcal{F}$ of
$\mathcal{F}$ at a point $x$ of $X_0$ is defined to be the
collection of prime ideals $\mathcal{P}$ of $A$ such that
$\mathcal{F}_{x}\otimes_{A}A_{\mathcal{P}}\neq 0$; the support of
$\mathcal{G}$ at $x$ is the union of Supp$_{x}h^{i}(\mathcal{G})$.
If $d$ is the dimension of the support of $\mathcal{G}$ at $x$, we
denote by Supp$_{x,d}\,\mathcal{G}$ the collection of prime ideals
in Supp$_{x}\mathcal{G}$ of coheight $d$. The cycle associated to
$\mathcal{G}$ at $x$ is the cycle
$$\zeta_{x,d}(\mathcal{G})=\sum_{\mathcal{P}\in\,
\mathrm{Supp}_{x,d}\mathcal{G}}\chi_x(\mathcal{G}_{\mathcal{P}})V(\mathcal{P}),$$
where $V(\mathcal{P})$ is the subvariety of Spec\,$A$ associated
to $\mathcal{P}$, $\mathcal{G}_{\mathcal{P}}$ is the localization
$\mathcal{G} \otimes_A A_{\mathcal{P}}$, and
$\chi_x(\mathcal{G}_{\mathcal{P}})$ equals $\sum
(-1)^{i}\mathrm{length}\,h^{i}(\mathcal{G_P})_x$. In our setting,
$A$ equals $\C[t_i,t_i^{-1}]_{i=1}^{r}$, and $d$ will be equal to
$r-1$, so we can replace the cycle by a defining function. This
zeta function $\zeta_{x,d}(\mathcal{G})$ is a rational function in
$t_1,\ldots,t_r$, defined modulo multiplication with a monomial.
We get a duality formula
$$\zeta_{x,d}(D_A\mathcal{G})(t_1,\ldots,t_r)=\zeta_{x,d}(\mathcal{G})(t_1,\ldots,t_r)^{-1},$$
where $D_A\mathcal{G}$ is the dual complex of $\mathcal{G}$.

In his article \cite{Sabbah}, Sabbah formulates a generalized
A'Campo formula \cite{A'C}: let
$F=(f_1,\ldots,f_r):(\C^{n},0)\rightarrow (\C^{r},0)$ be an
analytic germ, and let $h:E\C^{n}\rightarrow \C^{n}$ be a proper
modification, where $E\C^{n}$ is smooth. Assume that there exists
a divisor $E=\cup E_i$ on $E\C^{n}$ with normal crossings, such
that $(f_i\circ h)^{-1}(0)$, as well as $h^{-1}(0)$ are unions of
components of $E$. Then
$$\zeta_0(R\psi_F\C)(t_1,\ldots,t_r)=\prod_{\{i\in I\,|\,E_i\in
\pi^{-1}(0)\}}(t^{N_i}-1)^{-\chi(E^{o}_i)}\,,$$ where
$t^{N}=t_1^{N^{1}}\ldots t_r^{N^r}$, the $N_i$ are multiplicity
vectors of $F$ along the $E_i$, and $E_i^{o}$ is equal to the set
of points on $E_i$ which do not belong to any other divisor $E_j$.

It is well-known \cite{Sabbah} that $Rh_*\circ R\psi_{F\circ
h}=R\psi_{F}\circ Rh_*$. The comparison of the analytic with the
algebraic definition is based upon an explicit computation of the
Alexander complex associated to $F\circ h$, which reduces to the
computation of the Alexander complex associated to
$$G:Y=\C^{n}\rightarrow \C^{r}:x=(x_1,\ldots,x_n)\mapsto
(x^{N^{1}},\ldots,x^{N^{r}})\ ,$$ with $N^{j}\in \N^{n}$. We may
assume that for each index $j=1\ldots n$, there exists a $k$ such
that $N^{k}_j\neq 0$. The map
$$p_1:\C^{n}\rightarrow Y_T=(\C_0)^{n}:(w_1,\ldots,w_n)\mapsto
(e^{2i\pi w_1},\ldots, e^{2i\pi w_n})$$ is a universal covering
space. The space $Y_{\tilde{T}}$ falls apart in $\kappa$ connected
components, where $\kappa$ equals $\prod_i
gcd\,\{N^{j}_{i}\}_{j=1}^{r}$. These components are transitively
permutated by the monodromy actions. For each of these components
$Z$, we get an induced covering projection $p_2:\C^{n}\rightarrow
Z$ with
$$M=\{(z_1,\ldots,z_n)\in \Z^{n}\,|\,\sum z_i \,N^{j}_i=0\
\mathrm{for\  each}\ j\}\subset \Z^{n}=\pi_1(Y_T)$$ as
transformation group. This means that, for each constant sheaf
$\Lambda$ on $Y$ and each $q$,
$$R^{q}\psi_{G}(\Lambda)_0=H^{q}(Y_{\tilde{T}},\Lambda)=\wedge^{q}(M\otimes \Lambda)\otimes \psi^{0}_{G}(\Lambda)_0\,,$$ which
follows from the fact that, for each sheaf $\mathcal{F}$ on $Z$,
the global sections of $\mathcal{F}$ are the global sections of
$p_2^{*}\mathcal{F}$ which are invariant under $M$, and the
resulting spectral sequence.

Let us try to reformulate this definitions in a more algebraic
way. For each $i=1,\ldots,r$, let $S_i$ be the spectrum of a
strictly Henselian ring with residue field $k$, closed point $s_i$
and generic point $\eta_i$. Consider a geometric point
$\bar{\eta_i}$, localized at $\eta_i$. The residue field of the
closed point of the normalization $\bar{S}_i$ of $S_i$ in
$k(\bar{\eta_i})$ is an algebraic separably closed extension
$k(\bar{s}_i)$ of $k$, and thus defines a geometric point
$\bar{s}_i$ localized at $s_i$. Let $F:X\rightarrow
S$ be a scheme of finite type over the fiber
product $S$ of the $S_i$ over Spec\,$k$. Fix a subset $J$ of
$\{1,\ldots,r\}$. Let $\Lambda$ a Noetherian torsion ring in which
the characteristic exponent of $k$ is invertible, and let
$\mathcal{F}$ be an object in $D_{c}^{b}(X_T,\Lambda)$.

Consider the diagram

\begin{picture}(200,80)(-30,20)
%\put(30,80){$X_{s_J}$} \put(45,83){\vector(1,0){30}}
\put(69,80){$X_{s_J}$} \put(85,83){\vector(1,0){40}}
\put(130,80){$X$} \put(185,83){\vector(-1,0){42}}
\put(190,80){$X_{\bar{\eta}}$}

%\put(37,77){\vector(0,-1){30}}
 \put(76,76){\vector(0,-1){30}}
\put(134,77){\vector(0,-1){31}} \put(198,76){\vector(0,-1){30}}

\put(139,58){$F$} \put(162,85){$\tilde{\pi}$}
 \put(107,85){$i$}

\put(73,35){$s_{J}$}% \put(45,40){\vector(1,0){30}}
 \put(85,38){\vector(1,0){42}} \put(131,35){$S$}
\put(193,38){\vector(-1,0){20}} \put(197,35){$ \bar{\eta}$}
\put(163,35){$\eta$} \put(159,38){\vector(-1,0){20}}

\end{picture}

\noindent where $\eta=(\eta_i)$, and $s_J=(\alpha_i)$ with
$\alpha_i=s_i$ if $i\in J$ and $\alpha_i=\eta_i$ else. We define
the $J$-th Alexander complex to be
$$R\psi_{F,J}(\mathcal{F})=i^{*}R\tilde{\pi}_{*}\mathcal{F}_{\bar{\eta}}\,.$$
It is endowed with an action of $\prod
\mathrm{Gal}(\bar{\eta_i}/\eta_i)$; the $i$-part acts trivially if
$i\notin J$. If we replace, for each $i$, $\bar{\eta_i}$ by the
point $\eta_{i,t}$ associated to the maximal tamely ramified
extension of $k(\eta_i)$ in $k(\bar{\eta_i})$, we get the tame
Alexander complex $R\psi_{F,J}^{(t)}$.

 It follows from proper base change that $Rh_*\circ R\psi_{F\circ h,J}^{(t)}=R\psi_{F,J}^{(t)}\circ
 Rh_*$ if $h:Y\rightarrow X$ is a proper morphism. In our
 comparison theorem, we will apply this property to the case where
 $h$ is a simultaneous embedded resolution for the $f_i$; so let
 us investigate what happens if, for each $i\in J$, $X_{\eta_i}$
 is smooth over $\eta_i$, and $D=\cup_{i\in J}X_{s_i}$ is a normal
 crossing divisor. The calculation follows the same lines as in
 the analytic case; see \cite{SGA7a} for the univariate case $r=1$.
  %We suppose that $(X,D_j)$ is a smooth pair for each component $D_j$ of $D$.

  Let $\bar{x}$ be a
 geometric point of $X$, localized at a point $x$ mapped to $s_J$
 by $F$. Consider the strict localization $X_{(\bar{x})}$ of $X$
 at $\bar{x}$, let $U$ be equal to the complement of the inverse image of $D$ in
 $X_{(\bar{x})}$, and let $t_j$ be a local equation for $D_j$. Now
 put $X_n=X_{(\bar{x})}[t_j^{\frac{1}{n}}]$, and denote by $U_n$ the
 inverse image of $U$ in $X_n$. By purity and the long exact
 sequence for relative cohomology,
 $H^{q}(U_n,\Lambda)=\wedge^{q}\Lambda(-1)^{C}$, where $C$ is the
 number of components $D_j$ containing $x$. For $m=nd$, the
 morphism from $H^{q}(U_n,\Lambda)$ to $H^{q}(U_m,\Lambda)$,
 associated to the obvious morphism $U_m\rightarrow U_n$, is
 induced by multiplication by $d^{q}$. If we define $\tilde{U}$ to
 be the projective limit of the $U_n$, taken over the indices $n$
 prime to the characteristic exponent of $k$, we get
 $$\left\{\begin{array}{ll}H^{0}(\tilde{U},\Lambda)=\Lambda &
 \\ H^{q}(\tilde{U},\Lambda)=\lim_{\to}H^{q}(U_n,\Lambda)=0 &
 \mbox{if $q\neq 0$.}\end{array}\right.$$
 The scheme $\tilde{U}$
 is a procover of $U$ with group $\hat{\Z}(1)(k)^{C}$.
 Furthermore, it is a procover with group $M\otimes
 \hat{\Z}(1)(k)$ of each of the connected components of
 $U_{\eta_t}$, which are permutated transitively by the tame
 inertia group $I_t=\hat{\Z}(1)(k)^r$. Since, by definition,
 $R\psi^{(t),q}_{F,J}(\Lambda)_{\bar{x}}=H^{q}(U_{\eta_t},\Lambda)$,
 the Hochschild-Serre spectral sequence yields
 $$R^{q}\psi^{(t)}_{F,J}(\Lambda)_{\bar{x}}\cong \wedge^{q}(M\otimes
 \Lambda(-1))\otimes_{\Lambda}R^{0}\psi^{(t)}_{F,J}(\Lambda)_{\bar{x}}\,,$$
 and $R^{0}\psi^{(t)}_{F,J}(\Lambda)_{\bar{x}}$ is a
 $\Lambda$-algebra $\Lambda^{E}$, with $E$ a set on which the tame
 inertia group acts transitively, and $|E|$ equal to the greatest
 divisor of $\kappa$ prime to $p$. Of course, $M$ and $\kappa$ are
defined in the same way as in the analytic case.

 Now we compare both
definitions. For each $i=1,\ldots,r$, let $f_i:X\rightarrow
\A^{1}_{\C}$ be a separated reduced $\A^{1}_{\C}$-scheme of finite
type. Consider the strict localization $S$ of $\A_{\C}^{1}$ at
$0$, with closed point $s$, and generic point $\eta$. By abuse of
notation, we will also write $f_i:X\rightarrow S$ for the
restriction of $f_i$ to $X\times_{\A^{1}_{\C}}S$. Put $F=(f_i)$,
and let $R\psi^{an}_{F,I}(\mathcal{F})$ be the restriction of the
analytically defined complex of Alexander modules to $X_I=\{x\in
X_0\,|\,f_i(x)=0\  \mathrm{iff}\  i\in J\}$.

We can make use of an \'etale dictionary, allowing us to travel
from the algebraic to the analytic world, associating to a scheme
$X$ of finite type over $\C$ the analytic space $X(\C)$ of its
complex points, and to an \'etale sheaf $\mathcal{F}$ on $X_T$ a
sheaf $\mathcal{F}^{an}$ on $X_T(\C)$, with pleasant compatibility
properties - see, e.g. \cite{Frei}\cite{Milne}. Our Henselian ring
$S$ corresponds to the germ of a disc around the origin. The
universal cover of the one-dimensional torus, however, is not
algebraic in nature - this is exactly why we have to use limits in
the definition of the algebraic fundamental group \cite{Murre}. We
now explain how it is linked to a geometric generic point
$\bar{\eta}$ of $S$, see also \cite{SGA7b}.

Let $T_1$ be the torus $\A^{1}_{\C}\setminus \{0\}$, and
$\tilde{T}_1$ a universal cover. We can add a point $0$ to
$\tilde{T}_1$ to obtain a map $p:\tilde{\A}\rightarrow
\A^{1}_{\C}$ such that $p(0)=0$, $\tilde{T}_1$ is open in
$\tilde{\A}$, and the sets $p^{-1}(U)$, where $U$ runs over the
neighbourhoods of $0$ in $\A_{\C}^{1}$, form a fundamental system
of neighbourhoods of $0$ in $\tilde{\A}$. The cover $\tilde{T}_1$
determines an algebraic closure of the residue field $k(\eta)$ as
follows: $S$ is isomorphic to the ring of germs of holomorphic
functions on $\A^1_{\C}$ at $0$, that are algebraic over the field
of rational functions $K(\A_{\C}^{1})$. The field of germs at
$0\in \tilde{\A}$ of meromorphic functions on $\tilde{T}_1$ that
are algebraic over $K(\A^{1}_{\C})$, is an algebraic closure
$k(\bar{\eta})$ of $K(\A^{1}_{\C})$, hence of $k(\eta)$. The group
of deck transformations $\pi_1(T_1)$ acts on $k(\bar{\eta})$,
which yields a monomorphism
$$\Z=\pi_1(T_1)\rightarrow
\mathrm{Gal}(k(\bar{\eta})/k(\eta))=\hat{\Z}(1)(\C)\ (*).$$ Let
$\bar{\eta}$ be the geometric point corresponding to the inclusion
$k(\eta)\subset k(\bar{\eta})$.

 Our first comparison statement is the following:
\begin{prop}\label{comp1}
Let $\mathcal{G}$ be an object in $D_{c}^{b}(X_T,\Lambda)$, and
let $\mathcal{G}^{an}$ be the corresponding object in
$D_{c}^{b}(X_T(\C),\Lambda)$. Consider the constructible complex
$R\psi_{F,J}^{(t)}(\mathcal{G})^{an}$ on the stratum $X_J$ (or its
intersection with the inverse image of a small disc in $\C^{r}$).
It inherits a $\hat{\Z}(1)(\C)^{r}$ action, which induces, via the
morphism (*), a monodromy action of $\Z^{r}$. This latter object
is quasi-isomorphic to $R\psi_{F,J}^{an}(\mathcal{G}^{an})$.
\end{prop}
The proof is completely analogous to the proof in the univariate
case \cite{SGA7b}, and consists of a reduction to the normal
crossing-case for which we established formulas above.

Our next task is to compare the algebraic definition of
$R\psi_{F,I}^{(t)}$ with our definition of the Alexander complex.
 For each $i=1,\ldots,r$, let $f_i:X\rightarrow \A^{1}_{k}$ be
a morphism of $k$-varieties. Consider the strict
localization $S$ of $\A_{k}^{1}$ at $0$, with closed point $s$,
and generic point $\eta$. By abuse of notation, we will also write
$f_i:X\rightarrow S$ for the restriction of $f_i$ to
$X\times_{\A^{1}_{k}}S$. Put $F=(f_1,\ldots,f_r)$. Let $\xi$ be a
closed point of $X$ that is mapped to $s_J$; we might as well
assume that $J=\{1,\ldots,r\}$, and we drop the subscript $J$. Let
$\mu$ be the generic point of a strict localization of $X$ at
$\xi$, and let $\bar{\mu}$ be a geometric point localized at
$\mu$. We fix an isomorphism of fiber functors
$V_{F(\bar{\mu})}\cong V_{(1,\ldots,1)}$ on the category of tame
\'etale covers of $\G_{m,k}^{r}$. Consider an object $\mathcal{F}$
in $D_{c}^{b}(X_{\mathbb{G}^{r}_{m,k}},\bar{\Q}_l)$. We denote by
$R\psi^{(t),op}_{\mu,F}(\mathcal{F})$ the tame Alexander complex
at $\mu$, with opposite Gal$(\bar{\mu}/\mu)$-action.

Now we give $R\psi^{(t),op}_{\mu,F}(\mathcal{F})$ the structure of
a sheaf of modules on $\mathcal{C}(\G_{m,k}^{r})$, or rather: we
split up $R\psi^{(t),op}_{\mu,F}(\mathcal{F})$ in different direct
summands, in analogy with the Jordan decomposition of a linear
transformation, and each of these summands will get the structure
of a sheaf of modules on $\{\chi\}\times
\mathcal{C}(\G_{m,k}^{r})_l$, for some character $\chi$. For each
$\chi$ in $\mathcal{C}(\G_{m,k}^{r},\bar{\Q}_l^{\times})_f$ (see
\cite{GabLoe} for a definition), we denote by
$R\psi^{(t),op}_{\mu,F}(\mathcal{F})^{(\chi)}$ the biggest
subobject of $R\psi^{(t),op}_{\mu,F}(\mathcal{F})$ such that the
monodromy action, translated by $\chi^{-1}$, factors through
$\pi_{1}(\G_{m,k}^{r})_l$. Now it is clear how to give
$R\psi^{(t),op}_{\mu,F}(\mathcal{F})^{(\chi)}$ the structure of a
sheaf of modules on $\{\chi\}\times \mathcal{C}(\G_{m,k}^{r})$.

 For each closed point $\varphi$ of
$\{\chi\}\times \mathcal{C}(\mathbb{G}_{m,k}^{r})_l$, the stalk
$R\psi^{(t),op}_{\mu,F}(\mathcal{F})^{(\chi)}_{\varphi}$ is the
largest subobject of $R\psi^{(t),op}_{\mu,F}(\mathcal{F})$ on
which the action of each member $\gamma_j$ of a topological basis
$(\gamma_1,\ldots,\gamma_r)$ for $\pi_{1}(\GG_{m,k}^{r})^{t}$ is
of the form $\varphi(\gamma_j)+N_{\gamma_j}$, with $N_{\gamma_j}$
nilpotent.

\begin{prop}\label{monoco}
For each character $\chi$ in
$\mathcal{C}(\G_{m,k}^{r},\bar{\Q}_l^{\times})_f$, we get a
canonical isomorphism of sheaves of modules
$$\mathcal{A}^{q}_{F,\xi}|_{\{\chi\}\times
\mathcal{C}(\G_{m,k}^{r})_l}(\mathcal{F})\cong
R^{q-r}\psi^{(t),op}_{\mu,F}(\mathcal{F})^{(\chi)}\,.$$
\end{prop}
\begin{proof}
It suffices to prove the theorem when $\chi$ is trivial. We
replace $\mathcal{F}$ by a model in
$D_{c}^{b}(X_{\G_{m,k}^{r}},R)$, where $R$ is the ring of integers
of a finite extension of $\Q_l$. We denote this model again by
$\mathcal{F}$. Then the equality we want to prove becomes
$$\mathcal{A}^{q,R}_{F,\xi}(\mathcal{F})\cong
R^{q-r}\psi^{pro-l,op}_{\mu,F}(\mathcal{F})\otimes_{\Z_l}(\wedge_{\Z_l}^{r}\pi_1(\G_{m,k}^{r})_l)^{\vee}\,,$$
 where $^{\vee}$ denotes the dual $\Z_l$-module, and the
isomorphism is a canonical isomorphism of
$R[[\pi_1(\G_{m,k}^{r})_l]]$-modules.

%By definition of $L^{R}_{\G_{m,k}^{r}}$, the complex $\mathcal{F}$
%is the Gal$(\bar{\mu}/\mu)$-invariant part of the complex
%$R\psi^{pro-l}_{\mu,F}(\mathcal{F}\otimes_R^{L}L^{R}_{\G_{m,k}^{r}})$,
%so
We have a canonical isomorphism
$$\mathcal{A}^{q,R}_{F,\xi}(\mathcal{F})=R^{q}j_{*}(\mathcal{F}\otimes_{R}^{L}F^{*}L^{R}_{\G^{r}_{m,k}})_{\xi}
\cong
H^{q}(\mathrm{Gal}(\bar{\mu}/\mu),R\psi^{pro-l}_{\mu,F}(\mathcal{F}\otimes_R^{L}F^{*}L^{R}_{\G_{m,k}^{r}}))\,,$$
which may be rewritten as an isomorphism
$$\mathcal{A}^{q,R}_{F,\xi}(\mathcal{F})\cong
H^{q}(\pi_{1}(\G_{m,k}^{r})_l,R\psi^{pro-l}_{\mu,F}(\mathcal{F})\otimes_R^{L}L^{R}_{\G_{m,k}^{r}})\,,$$
where we view $L^{R}_{\G_{m,k}^{r}}$ as an $R$-module with
continuous $\pi_{1}(\G_{m,k}^{r})_l$-action. Now let us take a
look at the hypercohomological spectral sequence
\begin{eqnarray*}
E^{i,j}_{2}&=&H^{i}(\pi_1(\G_{m,k}^{r})_l,R^{j}\psi^{pro-l}_{\mu,F}(\mathcal{F})\otimes_R^{L}L^{R}_{\G_{m,k}^{r}})
\\&&\Rightarrow
H^{i+j}(\pi_1(\G_{m,k}^{r})_l,R\psi^{pro-l}_{\mu,F}(\mathcal{F})\otimes_R^{L}L^{R}_{\G_{m,k}^{r}})\,.\end{eqnarray*}
By \cite{GabLoe}, proposition 4.2.2.1, the spectral sequence
degenerates at $E_2$, and yields the desired isomorphism.
\end{proof}

%\noindent Let $K\subset \C$ be a number field with ring of
%integers $R$, and let $\mathcal{P}$ be a prime ideal of $R$.
%Consider the $\mathcal{P}$-adic completion $K_{\mathcal{P}}$ of
%$K$, its ring of integers $R_{\mathcal{P}}$, and the residue field
%$k_{\mathcal{P}}$. Let $X$ be a smooth variety over $K$, and let
%$F=(f_1,\ldots,f_r)$ be a morphism $X\rightarrow \A^{r}_{K}$.
%Taking a appropriate models for $X$ and $F$ over $R$ yields, for
%almost all $\mathcal{P}$ a proper smooth $R_{\mathcal{P}}$-variety
%$X_{\mathcal{P}}$, and a morphism
%$F_{\mathcal{P}}:X_{\mathcal{P}}\rightarrow
%A^{r}_{R_{\mathcal{P}}}$.
%
%Now there are different objects we can study. To begin with, there
%are the analytical and algebraical Alexander complexes of $F$ over
%$\C$, which, as we've seen, are the same.
%%There's the algebraic
%%Alexander complex over $K$; this is again the same object because
%%of smooth base change.
%But we can also consider the reduction $\bar{f}:\bar{X}\rightarrow
%\A^{r}_{k}$, $k=k_{\mathcal{P}}$, of $F_{\mathcal{P}}$ modulo
%$\mathcal{P}$, and its Alexander complex with coefficients in
%$\bar{\Q}_l$ over the algebraic closure $k$ of $k_{\mathcal{P}}$.
%
What does this all mean for the support of the Alexander complex
$\mathcal{A}^{q}_{F,\xi}(\bar{\Q}_l)$ if $k=\C$? Consider the
universal cover
$$p:\C\rightarrow\C^{*}:z\mapsto e^{2\pi i z}\,,$$ and let
$\bar{\eta}$ be the corresponding geometric generic point of the
Henselization of $\A^{1}_{\C}$ at $0$. Fix an embedding of
$\bar{\Q}_l$ in $\C$. The choice of $\bar{\eta}$ yields a
distinguished topological basis $(\gamma_1,\ldots,\gamma_r)$ for
$\pi_1(\G_{m,\C}^{r})^t$, and the embedding of $\bar{\Q}_l$ in
$\C$ allows us to identify a closed point $\chi$ of
$\mathcal{C}(\G_{m,k}^{r})$ with a point $(e^{2\pi i\alpha_j})_j$
in Spec\,$\C[\Z]$, with $\alpha_j\in \Q$. This correspondence
identifies the support of $\mathcal{A}^{q}_{F,\xi}(\bar{\Q}_l)$
with the support of the stalk of the analytical complex
$\psi^{an}_{F}(\C)$ at $\xi$.
%%!!!!!!!!!!!!!!!!!!!!!KLOPT DIT WEL MET DIE OPPOSITE ACTION?????????????????????????????????????????????????
%%maakt niet veel uit, support is invariant onder complexe conjugatie (=nemen van invers)

 To conclude, let us consider the behaviour of the
Alexander complex with respect to resolution of singularities. Let
$h:Y\rightarrow X$ be a simultaneous resolution for the $f_i$, and
let $\mathcal{F}$ be an object in $D_{c}^{b}(Y_{T},R)$. It follows
from the proper base change theorem that
$$Rh_{*}Rj_{*}(\mathcal {F}\otimes h^{*}F^{*}L^{R}_{T})\cong
Rj_{*}(Rh_{*}\mathcal{F}\otimes F^{*}L^{R}_{T})\,.$$ Suppose that
the resolution is tame, i.e. all multiplicities $N^{i}_{j}$ are
prime to the characteristic exponent of $k$. In this case, the
complex $Rj_{*}(\bar{\Q}_l\otimes h^{*}F^{*}L^{R}_{T})$ is lisse
on the strata $E_{I}^{o}$, by \cite{GabLoe}, lemma 4.3.2, and on
each stratum $E_{I}^{o}$, we have an isomorphism
$$R^{q}j_{*}(\bar{\Q}_l\otimes h^{*}F^{*}L^{R}_{T})\cong R^{0}j_{*}(\bar{\Q}_l\otimes h^{*}F^{*}L^{R}_{T})\otimes \wedge^{q}(M_{I}\otimes
\bar{\Q}_l(-1))\,$$ where $M_{I}$ is defined in the obvious way.
\section{A formula for the local zeta function}\label{form}
\noindent We recall, and introduce, some notation. Let $K$ be a
finite field extension of $\Q_p$, with ring of integers $R$, and
residue field $k\cong F_{q}$. We denote by $P$ the maximal ideal
of $R$. Let $\Phi$ be a residual, locally constant function on
$R^{n}$, and $F=(f_{1},\ldots,f_{r})$, with $f_i$ a polynomial in
$x=(x_1,\ldots,x_n)$ over $R$. Let
$\chi=(\chi_{1},\ldots,\chi_r)$, where $\chi_j$ is a
multiplicative character on $R^{\times}$ of order $d_{j}$, trivial
on $1+P$. Put $d=(d_1,\ldots,d_r)$. Denote by $N_i$ the vector
$(N_i^1,\ldots,N_i^r)$, and by $s$ the $r$-tuple of complex
variables $(s_1,\ldots,s_r)$. Let $\bar{.}$ denote reduction
modulo $P$. Let $(Y,h)=h:Y\rightarrow \A^{n}_K$ be a simultaneous
embedded resolution for $F$, and always assume tame good reduction
\cite{D}\cite{Denef7}\cite{Denef6}. Let $T$ be the index set of
the exceptional components, where $T_s$ indexes the components of strict
transform, and $T_e$ the exceptional divisors. The symbol $E_{I}$,
with $I\subset T$, means $\cap_{i\in I}E_i$, and $E_{I}^{o}$
stands for $E_{I}\setminus \cup_{j\in T\setminus I}E_j$.
Analogously, we write $\bar{E}_{I}$ instead of $\cap_{i\in
I}\bar{E}_i$, and $\bar{E}_{I}^{o}$ instead of
$\bar{E}_{I}\setminus \cup_{j\in T\setminus I}\bar{E}_j$.
%We can use additional
%blow-ups to ensure that in each point $y$ of $Y$ the following
%property holds: if $i_1,\ldots,i_t$ are the indices $i$ in
%$\{1,\ldots,j\}$ for which $f_i\circ h(y)=0$, then there exists a
%divisor through $y$ for which all these $i$-th multiplicity
%entries are nonzero.

We say that a multiplicity vector $N_i$ satisfies condition
$\gamma(N_i)$ if the character $\prod_{j=1}^{r}\chi_{j}^{N_i^{j}}$
is trivial. The condition $\gamma(N_i)$ replaces the condition
$d|N_i$ in the univariate case.
% Of course, if $d_j$ divides
%$N_i^j$ for all indices $j$, $\gamma(N_i)$ will be satisfied; but
%there are other possibilities, due to the interplay of the
%characters $\chi_j$.
% We say that a resolution for $f^{-1}(0)$ has
%$\gamma$-tame good reduction if it has good reduction and,
%furthermore, the.
%This definition guarantees that, in case of
%$\gamma$-tame reduction
\begin{theorem}\label{formdenef}%veralgemening formule denef
Let $f_i\in R[x]$, $\bar{F}\neq 0$, and let $(Y,h)$ have good
reduction modulo $P$. Then
$$Z_{\Phi}(F,\chi;s)=q^{-n}\sum_{I\subset T, \gamma(N_i)\,if\, i\in
I}c_{I,\chi,\Phi}\prod_{i\in I}\frac{q-1}{q^{N_i s+\nu_i}-1}$$
where $c_{I,\chi,\Phi}=\sum_{a\in
\bar{E}_I^{o}(k)}\bar{\Phi}(a)\Omega_{\chi}(a)$ and
$\Omega_{\chi}(a)=\prod_{j=1}^{r}\chi_j(u_j)$, with $u_j$ as in
the univariate case \cite{Denef6} ($\chi_j(u_j)$ is ill-defined,
but their product isn't).
% In the other case, the function $Z(s)$ is
%constant, and, moreover, it will be zero if in addition $\bar{f}$
%is smooth outside $\bar{f}^{-1}(0)$.
\end{theorem}

\begin{proof}
See the proof of Denef in \cite{Denef6}. His proof can be read as
if $s$, $\psi$ and the $N_i$ were vectors, replacing the condition
$d|N_i$ by $\gamma(N_i)$ to obtain the generalized formula.
%The second part of the assertion should be dealt with more
%carefully. Assume that the product $\prod \chi_{j}$ is not trivial
%on $1+P$.
%te bewijzen: \prod \chi_j^{N_j} is niet triviaal.
%reduction to the case \chi_j(u_j) constant voor T_{a,j} niet leeg; kan omwille van de veronderstelling over
%de resolutie.
\end{proof}
%
%cohomologische interpretatie
%

\noindent We write $c_{I,\chi}$ for $c_{I,\chi,\Phi}$ when $\Phi$
is the characteristic function of $R^{n}$, and $c_{I,\chi,0}$ when
$\Phi$ is the characteristic function of $P^{n}$.

These coefficients have a cohomological interpretation. Choose a
prime number $l$, not dividing $q$, such that all characters
$\chi_j$ take values in $\Q_l$, identifying $\Q_l$ with a subfield
of $\C$. Let $C$ be the torus $(\A_{F_{q}}\setminus 0)^{r}$,
define $\bar{Y}_C$ as $(\bar{F}\circ\bar{h})^{-1}(C)$ and let
$\alpha:\bar{Y}_C\rightarrow C$ be the induced map.

The \'etale cover
$$\mathrm{Spec}F_q[w,w^{-1},z]/(z_1^{d_1}-w_1,\ldots,z_r^{d_r}-w_r)$$ of
$C$, $z=(z_1,\ldots,z_r)$, $w=(w_1,\ldots,w_r)$, has Galois group
$\mu_d=\prod_{i=1}^{r} \mu_{d_{i}}$. Through the isomorphism
$\mu_d\cong\prod_{i}F_{q}^{\times}/(F_{q}^{\times})^{d_{i}}$,
$\psi=\prod_{i}\chi_i$ induces a character $\psi'$ on $\mu_{d}$,
which at its turn induces a
 continuous
representation of $\pi_1(C)$ in $\Q_l^{\times}$. We denote the
 locally constant $\Q_l$-sheaf associated to its inverse by $\mathcal{L}'_{\psi}$.
Let $\mathcal{L}_{\psi,\alpha}$ be
$\alpha^{*}\mathcal{L}'_{\psi}$, and $\mathcal{F}_{\psi}$ be
$\nu_{*}\mathcal{L}_{\psi,\alpha}$, where
$\nu:\bar{Y}_{C}\rightarrow \bar{Y}$ is the open immersion. The
sheaf is constructed in such a way that the action of the
geometric Frobenius corresponds to the character we want to study,
so we get a convenient Grothendieck trace formula \cite{SGA41/2}:
\begin{theorem}
The sheaf $\mathcal{F}_{\psi}$ is locally constant of rank 1 on
$U_{d}=\bar{Y}\setminus \cup_{\neg\gamma(N_i)}^{i\in T}\bar{E}_i$.
If $I\subset T$ and $\gamma(N_i)$ for all $i\in I$, then
$$c_{I,\chi} =\sum_{i}(-1)^{i}Tr(Fr,H_{c}^{i}(\bar{E}^{o}_{I}\otimes
_{F_{q}} F_{q}^{alg},\mathcal{F}_{\psi})),$$ and
$$c_{I,\chi,0} =\sum_{i}(-1)^{i}Tr(Fr,H_{c}^{i}(\bar{E}^{o}_{I}\cap \bar{h}^{-1}(0)\otimes
_{F_{q}} F_{q}^{alg},\mathcal{F}_{\psi}))\,$$ where $Fr$ is the
Frobenius endomorphism, and $F_{q}^{alg}$ denotes the algebraic
closure of $F_{q}$.
\end{theorem}
\begin{proof}
Cover $U_{d}$ with affine opens $V\subset U_d$, such that on each
$V$, we can write $\bar{f_j}\circ \bar{h}$ as $u_j \prod
y_i^{M^{j}_{i}}$, with $u_j$ nonvanishing on $V$, $j=1,\ldots, r$,
and $(y_j)$ a system of local parameters on $V$. Let $u$ be the
mapping from $V$ to the torus $C$ induced by the $u_j$. The
multiplicity vectors $M_i$ satisfy condition $\gamma$ for each
$i$, so it follows from the bimultiplicativity of the construction
of $\mathcal{F}$ that the sheaf
$\mathcal{G}=u^{*}\mathcal{L}_{\psi}$ on $V$ coincides with
$\mathcal{L}_{\psi,\alpha}$ on $V\cap \bar{Y}_{C}$. Since
$\mathcal{G}$ is, by purity, isomorphic to the direct image of its
restriction to $V\cap\bar{Y}_{C}$, we see that
$\mathcal{G}=\mathcal{F}_{\psi}|_{V}$.

Now let $I$ be a subset of $T$ so that $\gamma(N_i)$ for each
$i\in I$; thus $\bar{E}^{o}_{I}\subset U_d$. Let $a$ be a point in
$V$. The action of the geometric Frobenius $\varphi$ of $a$ on the
stalk of $\mathcal{F}_{\psi}$ at $a$, is induced by the monodromy
action on the stalk by the element of $\mu_{d}$ corresponding to
$\varphi$, and is given by $\Omega_{\chi}(a)$. Grothendieck's
trace formula yields the expressions for $c_{I,\chi}$ and
$c_{I,\chi,0}$.
\end{proof}
%The sheaf $\mathcal{L}_{\psi}$ on the $r$-torus $C$ is the exterior tensor
%product $\boxtimes_{j=1}^{r}\mathcal{L}_{\chi_j}$ of the sheaves
%associated to the characters $\chi_j$ in the univariate case, while
%$\mathcal{F}_{\psi}$ coincides on $\cap U_{d_{i}}$ with the tensor product
%$\otimes_{j=1}^{r}\mathcal{F}_{\psi_j}$. we kunnen hier niet onmiddellijk gebruik
% van maken want onze schoof is op een groter domein gedefinieerd.
We now study the higher direct images of
$\mathcal{L}_{\psi,\alpha}$ under $\nu$.
\begin{lem}
(i) $R^{j}\nu_{*}\mathcal{L}_{\psi,\alpha}$ is zero outside
$U_{d}$ for all $j\geq 0$.

\noindent(ii) For $I\subset T$, let $\nu_{I}$ be the open
immersion $\nu_{I}:\bar{E}_{I}^{o}\rightarrow \bar{E}_{I}$. Then
$R^{j}\nu_{I*}(\mathcal{F}_{\psi}|_{\bar{E}_{I}^{o}})$ is zero
outside $\bar{E}_{I}\cap U_{d}$ for all $j\geq 0$.

\noindent(iii) Suppose $i\in T$ and $\gamma(N_i)$. Then
$(R^{j}\nu_{*}\mathcal{L}_{\psi,\alpha})|_{\bar{E}^{o}_{i}}$ is
zero for all $j\geq 2$, and canonically isomorhpic to
$\mathcal{F}_{\psi}(-1)|_{\bar{E}^{o}_{i}}$ for $j=1$.
\end{lem}
\begin{proof}
(i) Let $a$ be a closed point of $\bar{Y}\setminus U_d$. This
point belongs to $\bar{E}^{o}_{I}$ for some $I\subset T$, and $I$
contains an index $i_0$ such that $N_{i_{0}}$ does not satisfy
$\gamma$. We will prove that the stalk of
$R^{j}\nu_{*}\mathcal{L}_{\psi,\alpha}$ at $a$ is zero. This
assertion is local for the \'etale topology, which allows us to
replace $\bar{Y}$ by $A^{n}_{F_q}$, $\bar{Y}_{C}$ by $(\prod_{i\in
I}(A^{1}_{F_q}\setminus 0))\times A_{F_q}^{n-card\, I}$, $\alpha$
by $(\prod_{i\in I}y_{i}^{N^{j}_{i}})_j$, and $a$ by the origin.
Then we can write
$$\mathcal{L}_{\psi,\alpha}=\alpha^{*}\mathcal{L}_{\psi}=\boxtimes_{i\in
I}(\otimes_{j=1}^{r}\mathcal{L}_{\chi_{j}}^{\otimes N_i})\boxtimes
\mathcal{C}$$ where $\mathcal{C}$ denotes the constant sheaf
$\Q_l$ on $A_{F_q}^{n-card\, I}$, and $\mathcal{L}_{\chi_j}$
denotes the Kummer sheaf associated to $\chi_j$. Let $\mu$ be the
open immersion of $A^{1}_{F_q}\setminus 0$ in $A^{1}_{F_q}$. Since
$N_{i_0}$ does not satisfy $\gamma$, the stalk of
$\mu_{*}\otimes_{j=1}^{r}\mathcal{L}_{\chi_{j}}^{\otimes N_{i_0}}$
at the origin is zero (because we get a nontrivial monodromy
action). Local duality yields
$(R^{1}\mu_{*}\otimes_{j=1}^{r}\mathcal{L}_{\chi_{j}}^{\otimes
N_{i_0}})_0=0$. Hence
$(R^{j}\mu_{*}\otimes_{j=1}^{r}\mathcal{L}_{\chi_{j}}^{\otimes
N_{i_0}})_0=0$ for all $j\geq 0$. Now we can apply the K\"unneth
formula to the expression for $\mathcal{L}_{\psi,\alpha}$, to
obtain the desired result.

(ii) Because of (i), we may suppose that $\bar{E}^{o}_{I}\subset
U_d$. When $N_i$ satisfies $\gamma$, we get a trivial monodromy
action on $\otimes_{j=1}^{r}\mathcal{L}_{\chi_{j}}^{\otimes N_i}$,
so this sheaf is constant. Now one can use a local description
similar to the one in (i).

(iii) Apply purity results to the smooth pair consisting of
$\bar{E}_{i}^{o}$ and $\bar{Y}\setminus \cup_{k\neq i}\bar{E}_k$.
\end{proof}
The same arguments as in the univariate case \cite{Denef6} can be
used to prove the following result from the preceding lemma.
\begin{lem}%non-contributie van divisoren die geen divisoren snijden waarvoor enz...
Let $E_{i_{0}}$ be proper, $\gamma(N_{i_{0}})$, and suppose that
$E_{i_{0}}$ intersects no $E_j$ with $\gamma(N_j)$, $j\neq i_0$.
Then
$$H^{i}_c(\bar{E}^{o}_{i_{0}}\otimes_{F_{q}}F_q^{alg},\mathcal{F}_{\psi})=0\
\mathrm{for}\ i\neq n-1\,.$$
\end{lem}
 So if in addition
$\chi(E_{i_{0}}^{o})=0$ (i.e. the topological Euler characteristic
of $E_{i_{0}}^{o}(\C)$ vanishes), we have no contribution in the
formula for $Z_{\Phi}(F,\chi;s)$, since $\chi(E_{i_{0}}^{o})$ and
$\chi_{c}(\bar{E}_{i}^{o}\otimes_{F_{q}}F_{q}^{alg},\Q_l)$ differ
by a nonzero factor, and this latter Euler characteristic with
compact support at its turn equals
$\chi_{c}(\bar{E}_{i}^{o}\otimes_{F_{q}}F_{q}^{alg},\mathcal{F}_{\psi})$,
because $\mathcal{F}_{\psi}|_{\bar{E}_{i}^{o}}$ has tame
ramification.
%nog nagaan!
 This
allows us to prove:
\begin{theorem}\label{unmask}
Suppose $n=2$. If $H: \sum_k c_k s_k+1=0$ is a polar hyperplane of
$Z$, then there exists an index $j$ in $T$, with
$|E_j/E^{o}_j|\geq 3$ or $j\in T_s$, and $c_k=N_j^{k}/\nu_j$ for
each $k$.
\end{theorem}
\begin{proof}
This proof generalizes the one in \cite{DenefBour}. Some notation:
fix $j$ in $T_e$, and let $a_t$, $t\in J$, be the geometric points
of $E_j\setminus E_j^{o}$. For $t\in J$, let $N^{k}_t,\nu_t$
denote the numerical data associated to the unique divisor
intersecting $E_j$ transversally in $a_t$, and put
$\alpha_t^{(k)}=\nu_t-N^{k}_t\nu_j/N^k_j$ if $N_j^k$ differs from
zero. It is well-known \cite{Loepadic}\cite{Veysnum} that
$$\sum_{t\in J}(\alpha_t^{(k)}-1)=-2,\ \mathrm{and}\ \sum_{t\in
J}N_t^{k}\equiv 0\ mod\,N_j^{k}\,\quad(*)$$ for each $k$ for which
the identities are defined.

 Let $S$ be the set of all divisors $E_j$ inducing the polar
hyperplane $H$. Suppose that each of its members satisfies
$|E_j\setminus E^{o}_j|\leq 2$ and $j\notin T_s$. It follows from
the identities above that different elements of $S$ cannot meet,
since otherwise we could build an infinite tree of divisors in
$S$, which is absurd. Suppose that $E_j$ belongs to $S$ and
satisfies $\gamma(N_j)$. Suppose that $|E_j\setminus
E^{o}_j|=\{a_1,a_2\}$; the other case is similar. Either both
$N_1$ and $N_2$ satisfy $\gamma$, or none of them does; in the
latter case $E_j$ does not contribute to $Z$ because of the
previous lemma, in the former case the sheaf $\mathcal{F}_\psi$
will be geometrically constant on $\bar{E}_j$, and it follows from
the trace formula and the formula for $Z$, combined with the
information about the $\alpha_t^{(k)}$, that $E_j$ does not
contribute to the polar locus of $Z$.
\end{proof}
%

\section{Degree of local zeta functions and
monodromy}\label{degree} \noindent In this section, we establish a
formula for the limit of the zeta function for $s\to - \infty$,
and we prove the following theorem, which is a generalization of a
theorem in \cite{Denef5}:
\begin{theorem}\label{theoremdegree}
Let $F=(f_1,\ldots,f_r)$ be defined over a number field $K\subset
\C$. For almost all completions of $K$, the degree of the local
zeta function $Z_{0}(f,\chi;s)$ will be strictly less then zero if
$\chi\notin \mathrm{Supp}\mathcal{A}^{q}_{F,0}(\bar{\Q}_l)$ for
all $q$.
\end{theorem}
Here the statement that the degree of $Z_{0}(f,\chi;s)$ is
strictly negative, means that the well-defined limit $\lim_{s\to
-\infty}Z_{0}(f,\chi;s)$ vanishes.
\begin{proof}
\noindent Let $\bar{\eta}$ be a geometric generic point of
$\A^{r}_{F_{q}}$, defining an algebraic closure $F_{q}^{alg}$ of
$F_{q}$. Let $S_0$ be the Henselization at $0$ of
 $\A^{r}_{F_{q}}$, and denote by
 $\eta_0$ its generic point. Put
$G_0=\mathrm{Gal}(\bar{\eta}/\eta_0)$.

%In the section about the comparison theorem, we established a
%formula for the stalks of $R^{j}\psi^{(t)}_{\bar{f}\circ
%\bar{h},J}(\bar{\Q}_l)$. If $h$ has tame good reduction, the
%action of $I_0$ on $R^{j}\psi^{(t)}_{\bar{f}\circ
%\bar{h},J}(\bar{\Q}_l)$ is tame. A local computation shows that
%the $R^{j}\psi^{(t)}_{\bar{f}\circ \bar{h},J}(\bar{\Q}_l)$ are
%lisse on $\bar{E}^{o}_{I}\otimes F^{alg}_{q}$, and that locally on
%$\bar{E}^{o}_{I}\otimes F^{alg}_{q}$, the expression for the
%stalks glue together to a canonical isomorphism
%$$R^{j}\psi^{(t)}_{\bar{f}\circ \bar{h},J}(\bar{\Q}_l)=R^{0}\psi^{(t)}_{\bar{f}\circ \bar{h},J}(\bar{\Q}_l)\otimes \wedge^{j}(M_I\otimes
%\bar{\Q}_l(-1))$$ on $\bar{E}^{o}_{I}\otimes F^{alg}_{q}$, which
%is compatible with the action of $G_0$.

Let $\sigma$ be an element of $G_0$ which induces the geometric
Frobenius $\varphi$ on $F_{q}^{alg}$. Then, using the notation
from the previous section,

\begin{eqnarray*}
&&\sum_a(-1)^{a}Tr(\sigma,\mathcal{A}^{a}_{\bar{F},0}(\bar{\Q}_l)_{\chi})
\\&=&\sum_{a,b}(-1)^{a+b}Tr(\sigma,H^{a}(\bar{h}^{-1}(0)\otimes
F_{q}^{alg},R^{b}\nu_{*}((\bar{F}\circ
\bar{h})^{*}L^{\Z_l}_{T}\otimes (\bar{F}\circ
\bar{h})^{*}\mathcal{L}_{\chi}))
\end{eqnarray*}
(By proper base change and Leray's direct image spectral sequence,
we can compute the trace on the exceptional locus of the embedded
resolution $\bar{h}$.)
\begin{eqnarray*}
&=&\sum_{I}\sum_{a,b}(-1)^{a+b}Tr(\sigma,H^{a}_{c}((\bar{E}^{o}_{I}\cap\bar{h}^{-1}(0))\otimes
F_{q}^{alg}, \\[-10pt]&&\qquad \qquad \qquad R^{0}\nu_{*}(
(\bar{F}\circ \bar{h})^{*}L^{\Z_l}_{T}\otimes (\bar{F}\circ
\bar{h})^{*}\mathcal{L}_{\chi})\otimes \wedge^{b}(M_{I}\otimes
\bar{\Q}_l(-1)))
\end{eqnarray*}
(By the explicit computations in the normal crossing case in
Section \ref{trans})
\begin{eqnarray*}
&=&\sum_{I}\sum_{a,b}(-1)^{a+b}Tr(\varphi,H^{a}_{c}((\bar{E}^{o}_{I}\cap\bar{h}^{-1}(0))\otimes
F_{q}^{alg},\mathcal{F}_{\chi}))Tr(\varphi,\wedge^{b}(M_{I}\otimes
\bar{\Q}_l(-1)))
\\&=&q^{-n}(1-q)^{\lambda}\lim_{s\to -\infty}Z_{0}(F,\chi,s)
\end{eqnarray*}
where $\lambda$ is the number of linearly independent multiplicity
vectors $(N^{j}_{i})_{i\in I}$, $j=1,\ldots,r$.
\end{proof}

In the motivic setting, the limit of the multivariate zeta
function yields a motivic incarnation of the Alexander complex
\cite{Guibert}. In the case $r=1$, the motivic Milnor fiber has
the same mixed Hodge structure as the geometric Milnor fiber, and
 the Hodge spectrum of $F=f_1$ can be recovered from this motivic object
\cite{DL5}\cite{DL3}.
\section{Proof of the monodromy and holomorphy conjectures for
curves}\label{proofs}

Now we are ready to prove Conjectures \ref{gmonoconjecture} and
\ref{gholoconjecture} when $n=2$ (see \cite{Rod} and \cite{Veys1}
for the univariate case $r=1$).

\begin{proof}[Proof of the Generalized Monodromy Conjecture \ref{gmonoconjecture} for $n=2$]
Polar hyperplanes induced by components of the strict transform
satisfy the conjecture, as can be seen by looking at the Alexander
polynomial of the stalk of the Alexander complex in a smooth point
of the zero locus. Let $E_j$ be an exceptional divisor
intersecting at least three other divisors and lying above a point
$x$ of $X$. Let $N'$ be an integer $r$-tuple, and denote by
$\mathcal{I}$ the set of exceptional components lying over $x$,
with a multiplicity vector which is an integer multiple of $N'$.
We say that $E_i$ satisfies $\delta$ if it belongs to
$\mathcal{I}$. Because of Sabbah's A'Campo formula, and the
previous arguments, it suffices to prove that the expression
$w_{\mathcal{I}}:=\sum_{E_{i}\in \mathcal{I}} \chi(E_i^{o})$ is
strictly less than zero, for each $N'$ such that $E_j$ satisfies
$\delta$. We may suppose that none of the components of the strict
transform satisfies $\delta$, since in the other case, it again
suffices to look at the Alexander complex in a smooth point. To
prove the monodromy conjecture, we use the inequality
$w_{\mathcal{I}}<0$ in the case where $N'$ equals $N_j$, divided
by the greatest common divisor of $\nu_j$ and the entries of
$N_j$.

Let $\mathcal{N}$ be the union over all $E_i$ satisfying $\delta$.
It follows from the relations (*) in the proof of Theorem
\ref{unmask}, that each connected component of $\mathcal{N}$
contains a prime divisor, intersecting at least two divisors $E_k$
not satisfying $\delta$ (we suppose $f_j(x)=0$ for some $j$). Let
$\mathcal{M}$ be the set of divisors in the connected component of
$\mathcal{N}$ containing $E_j$; it suffices to prove that
$w_{\mathcal{M}}<0$. It is known that $w_\mathcal{M}=0$ implies
that $\mathcal{M}$ is a tree intersecting the union of the other
exceptional components exactly twice \cite{Rod}. These two
intersection points have to lie on the same divisor $E_1$ in
$\mathcal{M}$. Suppose that $E_1=E_j$. There is still another
divisor $E_2$ in $\mathcal{M}$ intersecting $E_j$. We claim that
we can find an index $k$ in $\{1,\ldots,r\}$ such that
$\alpha^{(k)}_t<1$ for all $E_t$ intersecting $E_1$. Since
$\alpha^{(k)}_2\in \Z$, (*) yields $\alpha^{(k)}_2=0$. But $E_2$
has to intersect another divisor in $\mathcal{M}$, and repeatedly
applying (*) produces an infinite tree. If $E_1\neq E_j$, there
are at least three divisors in $\mathcal{M}$ intersecting $E_j$;
since only one component intersects a divisor not belonging to
$\mathcal{M}$, the same arguments can be used.

We still have to prove our claim. We claim that we can always find
an appropriate index $k$, unless we have done some redundant
blow-ups during the resolution process. It is known that this
property holds in the univariate case $r=1$. But all the divisors
meeting $h^{-1}(x)$ appear as well in a minimal resolution for
$\prod_{f_j(x)=0}f_j$, so the claim follows from
$\sum_{i=1}^{m}x_i/\sum_{i=1}^{m}y_i\leq \max_i\{x_i/y_i\}$ for
$x_i, y_i\in \N_0$.
\end{proof}
%WIJ HEBBEN GEBRUIKT DAT T^{\Lambda}-1 een deler is van T^{\Mu}-1 asa \Mu een veelvoud is van \Lambda.
%Dit bewijs je door substitutie van T_i door y^{t_i} met t_i willekeurig in N.
%voor elke Q-rationale s in het polair HV geassocieerd aan E_j kies je N_j' minimaal zodat N_j'.s in Z zit.
%Dit klaart de klus: T^M-1 en T^N-1 zijn relatief priem als M en N LO zijn over Q (argument met substituties herhalen;
%stel gem factor; je
%kan de t_i zo kiezen dat je geen termen kan groeperen na substitutie (kies bvb s veel groter dan t en alle coeff in de gem factor)
% en dat \sum t_i M_i en \sum t_i N_i relatief priem zijn;
%resultaat volgt nu uit het univariate geval).
%als in het andere geval N.s en M.s beide in Z zitten kunnen we een ggd G nemen voor M en N en deze voldoet dan ook aan
%G.s in Z. dus omwille van minimaliteit moet N=G en N|M. Als M.s niet in Z zit is er geen probleem.
%Besluit: als we in a'campo alle factoren t^M-1 weglaten waarvoor N en M LO zijn over Q komen we opnieuw een
%polynoom uit (en geen noemers). wat overblijft wordt nul in e^{2\pi i \alpha}.

\begin{proof}[Proof of the Generalized Holomorphy Conjecture
\ref{gholoconjecture} for $n=2$] If $Z$ is not holomorphic, there
must exist an index $j$ in $T$, such that the corresponding
divisor $E_j$ induces a pole in the formula for the zeta function.
This means that we can choose $j$ such that $\gamma(N_j)$ is
satisfied, and $j\in T_s$ or $|E_j\setminus E_j^{o}|\geq 3$. If
$j\in T_s$, it is clear from Sabbah's A'Campo formula that the
holomorphy conjecture is satisfied; so let us assume that
$|E_j\setminus E_j^{o}|\geq 3$, and that $E_j$ is an exceptional
divisor above $x$. We've proven above that that the expression
$w_I:=\sum_{E_{i}\in I} \chi(E_i^{o})$ is strictly less than zero
when we take the sum over all exceptional divisors lying over $x$
for which the multiplicity vector is a multiple of some integer
vector $N'$, whenever $E_j$ belongs to $I$.

Taking $N'$ equal to $N_j$, defining $g$ to be the greatest common
divisor of the entries of $N_j$, and putting $N'_j=N_j/g$, it
follows from Sabbah's A'Campo formula that the locus of
$T^{N'_j}=\xi_{g}$ is contained in the support of the Alexander
complex at $x$, where $\xi_g$ is a primitive $g$-th root. Hence,
it suffices to find a character $\psi$ so that $\psi^{N'_j}$ is a
primitive $g$-th root, and $\chi=\psi^{k}\chi'$, with
$\chi'^{N'_j}=1$ and $k\in \N$. Since $\gamma(N_j)$ is satisfied,
we can take $\chi'$ to be $\chi^{g}$, and furthermore, if $\chi$
is non-trivial, we can find $k\in \N$ so that
$\chi^{N_j(1-g)/k}=1$, but $\chi^{g'N'_j(1-g)/k}\neq 1$ unless $g$
divides $g'$. Now put $\psi=\chi^{(1-g)/k}$.
\end{proof}
%choosing, once and for all, a basis for \pi_1(T), allows us to take
%rational powers of characters.
\section*{Acknowledgements}
The author would like to thank Fran\c{c}ois Loeser for his
clarifications and comments, and for the inspiring discussions.
 \nocite{Loe2} \nocite{prox1}
\nocite{prox2}

\bibliographystyle{amsplain}
\bibliography{wanbib,wanbib2}
\end{document}

monofinal.bbl
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
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