Determination of the real poles of the Igusa zeta function for curves
Rev Mat Complut (2012) 25:581–597DOI 10.1007/s1316301100738
Determination of the real poles of the Igusazeta function for curves
Denis Ibadula
·
Dirk Segers
Received: 10 February 2011 / Accepted: 26 April 2011 / Published online: 18 May 2011© Revista Matemática Complutense 2011
Abstract
The numerical data of an embedded resolution determine the candidatepoles of Igusa’s
p
adic zeta function. We determine in complete generality whichreal candidate poles are actual poles in the curve case.
Keywords
Igusa’s
p
adic zeta function
Mathematics Subject Classiﬁcation (2010)
11D79
·
11S80
·
14B05
·
14E15
1 Introduction
Several mathematicians have already obtained partial results about the determinationofthepolesofIgusa’s
p
adiczetafunctionforcurves.Inthispaper,wewilldeterminethe real poles for an arbitrary polynomial
f
in two variables which is deﬁned over a
p
adic ﬁeld. People are interested in the poles of Igusa’s
p
adic zeta function
Z
f
(s)
because they determine the asymptotic behaviour of the number of solutions of polynomial congruences and because they are the subject of the monodromy conjecture(see for example [2]).
D. Segers is Postdoctoral Fellow of the Fund for Scientiﬁc Research – Flanders (Belgium).D. IbadulaInstitute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1764,014700 Bucharest, RomaniaD. IbadulaFaculty of Mathematics and Informatics, “Ovidius” University, Constanta, Mamaia Bd. 124,900527 Constanta, Romaniaemail: denis@univovidius.roD. Segers (
)Department of Mathematics, University of Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgiumemail: dirk.segers@wis.kuleuven.beurl: http://wis.kuleuven.be/algebra/segers/segers.htm
582 D. Ibadula, D. Segers
Historically, one considered ﬁrst only curves which are absolutely analytically irreducible. Partial results were obtained by Igusa [3] and Strauss [12]. Meuser [8]
determined the real poles, but she did not consider the candidate pole
−
1. In 1985Igusa [4] solved that problem completely. He proved that the candidate poles associated to the strict transform of
f
are poles when the domain of integration is smallenough. Moreover, another candidate pole of the minimal embedded resolution of
f
is a pole if and only if it is associated to an exceptional curve which is intersectedby three other irreducible components of the pullback of
f
. We have incorporated ageneralization of this result (Proposition 2).In the general case, Loeser [7] obtained that an exceptional curve
E
i
does not contribute to the poles of
Z
f
(s)
if
E
i
is intersected one or two times by other componentsof the pullback of
f
and if there are no other intersection points over an algebraicclosure. This was ﬁrst proved by Strauss in the absolutely analytically irreduciblecase, where the last condition is automatically satisﬁed.The next paper we want to mention is [13] of Veys. He considers a polynomial
f
in two variables over a number ﬁeld
F
and takes the minimal embedded resolutionof
f
over an algebraic closure of
F
. This setup allowed him to use a formula [1] of Denef for
Z
f
(s)
, which is valid for almost all
p
adic completions of
F
. He supposesthat all intersection points of irreducible components of the pullback of
f
are deﬁned over
F
. Under this condition, he proves the converse of the result of Loeser forreal candidate poles and for almost all
p
adic completions of
F
. Moreover, he dealswith the problem of a possible cancellation of several contributions to the same realcandidate pole.In the proofs of the mentioned vanishing and nonvanishing results, one neededcertain relations between the various numerical data of the embedded resolution.They were systematically derived in [4, 8, 12] for absolutely analytically irreducible
curves and ﬁnally, Loeser [7] obtained the necessary relations in the general case.Igusa [4] and Loeser [7] used a formula of Langlands [6] to calculate the contribution
of an exceptional curve to the residue of
Z
f
(s)
at a candidate pole of candidate orderone. We will use a slight variant of this formula which was obtained in [10]. Givenan embedded resolution written as a composition of blowingups, the second authorexplained there how to calculate this contribution to the residue at the stage where theexceptional curve is created. In Proposition 1, we determine when this contributionis zero and when not. For this, we need new ideas. It is not at all a straightforwardgeneralization of what was already known. Finally in Sect. 4, we will prove that contributions to the same candidate pole will not cancel out. For this, we use that thedual embedded resolution graph is an ordered tree. This was obtained in [14] whenthe base ﬁeld is algebraically closed.
2 Deﬁnitions and our tools
Let
K
be a
p
adic ﬁeld, i.e., an extension of
Q
p
of ﬁnite degree. Let
R
be the valuation ring of
K
,
P
the maximal ideal of
R
and
q
the cardinality of the residue ﬁeld
R/P
. For
z
∈
K
, let ord
z
∈
Z
∪{+∞}
denote the valuation of
z
and

z
=
q
−
ord
z
theabsolute value of
z
.
Determination of the real poles of the Igusa zeta function for curves 583
Let
f(x
1
,x
2
)
∈
K
[
x
1
,x
2
]
be a polynomial in two variables over
K
and put
x
=
(x
1
,x
2
)
. Let
X
be an open and compact subset of
K
2
. Igusa’s
p
adic zeta function of
f
is deﬁned by
Z
f
(s)
=
X

f(x)

s

dx

for
s
∈
C
, Re
(s)>
0, where

dx

denotes the Haar measure on
K
2
, normalised so that
R
2
hasmeasure 1.Igusaprovedthat
Z
f
(s)
isarationalfunctionof
q
−
s
bycalculatingthe integral on an embedded resolution of
f
. Therefore, it extends to a meromorphicfunction
Z
f
(s)
on
C
which is also called Igusa’s
p
adic zeta function of
f
.Let
g
:
Y
→
X
be an embedded resolution of
f
. Here,
Y
is a
K
analytic manifold.The meaning of embedded resolution in our context is explained in [5, Sect. 3.2].Write
g
=
g
1
◦···◦
g
t
:
Y
=
Y
t
→
X
=
Y
0
as a composition of blowingups
g
i
:
Y
i
→
Y
i
−
1
,
i
∈
T
e
:={
1
,...,t
}
. The exceptional curve of
g
i
and also the strict transformsof this curve are denoted by
E
i
. The closed submanifolds of
Y
of codimension onewhich are the zero locusof the strict transform of an irreducible factor of
f
in
K
[
x,y
]
are denoted by
E
j
,
j
∈
T
s
. The corresponding transforms in
Y
i
,
i
∈{
0
,...,t
−
1
}
, aredenotedinthe same way.Notethatwe hadto be careful withthe notionof irreducible,because
X
is totally disconnected as a topological space. Put
T
=
T
e
∪
T
s
. For
i
∈
T
,let
N
i
and
ν
i
−
1 be the multiplicities of respectively
f
◦
g
and
g
∗
dx
along
E
i
. The
(N
i
,ν
i
)
are called the numerical data of
E
i
.Let us recall Igusa’s proof of the rationality of
Z
f
(s)
. As we already said, wecalculate the deﬁning integral on
Y
:
Z
f
(s)
=
X

f(x)

s

dx
=
Y

f
◦
g

s

g
∗
dx

.
Let
b
be an arbitrary point of
Y
. There are three cases. In the ﬁrst case, there are twovarieties
E
i
and
E
j
, with
i,j
∈
T
, that pass through
b
. We take a neighborhood
V
of
b
and analytic coordinates
(y
1
,y
2
)
on
V
such that
y
1
is an equation of
E
i
,
y
2
is anequation of
E
j
,
f
◦
g
=
εy
N
i
1
y
N
j
2
and
g
∗
dx
=
ηy
ν
i
−
11
y
ν
j
−
12
dy
on
V
for nonvanishing
K
analytic functions
ε
and
η
on
V
. We may suppose that
y(V)
=
P
k
1
×
P
k
2
, with
k
1
,k
2
∈
Z
≥
0
, and that

ε

and

η

are constant on
V
. We get
V

f
◦
g

s

g
∗
dx
=
P
k
1
×
P
k
2

ε

s

η

y
1

N
i
s
+
ν
i
−
1

y
2

N
j
s
+
ν
j
−
1

dy
=
ε

s

η

q
−
1
q
2
q
−
k
1
(N
i
s
+
ν
i
)
1
−
q
−
(N
i
s
+
ν
i
)
q
−
k
2
(N
j
s
+
ν
j
)
1
−
q
−
(N
j
s
+
ν
j
)
.
Note that this is a rational function of
q
−
s
. In the second case, there is one variety
E
i
,
i
∈
T
, that passes through
b
. We take a neighborhood
V
of
b
and analytic coordinates
(y
1
,y
2
)
on
V
such that
y
1
is an equation of
E
i
,
f
◦
g
=
εy
N
i
1
and
g
∗
dx
=
ηy
ν
i
−
11
dy
584 D. Ibadula, D. Segers
on
V
for nonvanishing
K
analytic functions
ε
and
η
on
V
. We may suppose that
y(V)
=
P
k
1
×
P
k
2
, with
k
1
,k
2
∈
Z
≥
0
, and that

ε

and

η

are constant on
V
. We get
V

f
◦
g

s

g
∗
dx
=
P
k
1
×
P
k
2

ε

s

η

y
1

N
i
s
+
ν
i
−
1

dy
=
ε

s

η

q
−
k
2
q
−
1
qq
−
k
1
(N
i
s
+
ν
i
)
1
−
q
−
(N
i
s
+
ν
i
)
.
In the third case, there is no variety
E
i
,
i
∈
T
, that passes through
b
. We take aneighborhood
V
of
b
and analytic coordinates
(y
1
,y
2
)
on
V
such that
f
◦
g
=
ε
and
g
∗
dx
=
ηdy
on
V
for nonvanishing
K
analytic functions
ε
and
η
on
V
. We maysuppose that
y(V)
=
P
k
1
×
P
k
2
, with
k
1
,k
2
∈
Z
≥
0
, and that

ε

and

η

are constanton
V
. We get
V

f
◦
g

s

g
∗
dx
=
ε

s

η

q
−
k
1
−
k
2
.
It follows now that
Z
f
(s)
is a rational function of
q
−
s
because we can partition
Y
into sets
V
of the above form.We obtain also from this calculation that every pole of
Z
f
(s)
is of the form
−
ν
i
N
i
+
2
kπ
√ −
1
N
i
log
q,
with
k
∈
Z
and
i
∈
T
. These values are called the candidate poles of
Z
f
(s)
. If
i
∈
T
is ﬁxed, the values
−
ν
i
/N
i
+
(
2
kπ
√ −
1
)/(N
i
log
q)
,
k
∈
Z
, are called the candidatepoles of
Z
f
(s)
associated to
E
i
. Because the poles of 1
/(
1
−
q
−
N
i
s
−
ν
i
)
have orderone, we deﬁne the expected order of a candidate pole
s
0
as the highest number of
E
i
’swith candidate pole
s
0
and with nonempty intersection. The order of
s
0
is of courseless than or equal to its expected order and a candidate pole
s
0
of expected order oneis a pole if and only if the residue of
Z
f
(s)
at
s
0
is different from 0.Let us explain the formula for the residue that we will use. Let
s
0
be a candidatepole of
E
i
,
i
∈
T
, and suppose that
s
0
is not a candidate pole of any
E
j
, with
j
∈
T
and
j
=
i
, which intersects
E
i
in
Y
. Let
U
be an open and compact subset of
E
i
. Thecontribution of
U
to the residue of
Z
f
(s)
at
s
0
is by deﬁnition the contribution to theresidue of
Z
f
(s)
at
s
0
of an open and compact subset
V
of
Y
which satisﬁes
V
∩
E
i
=
U
and which is disjoint from every other
E
j
with candidate pole
s
0
. Suppose that
U
already existsin
Y
r
and if
i
∈
T
s
we also suppose that it is nonsingular in
Y
r
. Supposethat
W
is an open and compact subset of
Y
r
for which
W
∩
E
i
=
U
and that
(z
1
,z
2
)
are analytic coordinates on
W
such that
z
1
=
0 is an equation of
U
on
W
. Write
f
◦
g
1
◦···◦
g
r
=
γz
N
i
1
and
(g
1
◦···◦
g
r
)
∗
dx
=
δz
ν
i
−
11
dy
on
W
, for
K
analytic functions
γ
and
δ
on
W
. Then, the contribution of
U
to theresidue of
Z
f
(s)
at
s
0
is equal to
q
−
1
qN
i
log
q
U

γ

s

δ

dz
2

mc
s
=
s
0
,
(1)
Determination of the real poles of the Igusa zeta function for curves 585
where
[·]
mc
s
=
s
0
denotes the evaluation in
s
=
s
0
of the meromorphic continuation of the function between the brackets. This formula was obtained by Langlands [6] inthe case
r
=
t
and in general by the second author in [10, Sect. 2.6].We explain now the relations that we will need. Fix
r
∈
T
e
. The exceptional curve
E
r
is obtained by blowingup at a point
P
∈
Y
r
−
1
. Let
y
=
(y
1
,y
2
)
be local coordinates on
Y
r
−
1
centered at
P
. Write in these local coordinates
f
◦
g
1
◦···◦
g
r
−
1
=
d
i
∈
S
(a
i
2
y
1
−
a
i
1
y
2
)
M
i
i
∈
S
h
M
i
i
(y
1
,y
2
)
+
terms of higher degree
,
where all factors
a
i
2
y
1
−
a
i
1
y
2
and
h
i
are essentially different (i.e. no factor is equalto another multiplied by an element of
K
×
) polynomials over
K
, where the
h
i
areirreducible homogeneous polynomials of degree at least two, where
M
i
≥
1 for every
i
∈
S
∪
S
and where
d
∈
K
×
. Write also
(g
1
◦···◦
g
r
−
1
)
∗
dx
=
e
i
∈
S
(a
i
2
y
1
−
a
i
1
y
2
)
µ
i
−
1
+
terms of higher degree
dy,
where
µ
i
≥
1 for every
i
∈
S
and
e
∈
K
×
. Let
s
0
=−
ν
r
/N
r
+
(
2
kπ
√ −
1
)/(N
r
log
q)
be an arbitrary candidate pole of
Z
f
(s)
associated to
E
r
. We advise the reader tospecialize everything what follows in this section to the case
k
=
0. Put
α
i
:=
µ
i
+
s
0
M
i
for every
i
∈
S
. Because
N
r
=
i
∈
S
M
i
+
i
∈
S
(
deg
h
i
)M
i
and
ν
r
=
i
∈
S
(µ
i
−
1
)
+
2
,
it is straightforward to check that
i
∈
S
(α
i
−
1
)
+
i
∈
S
s
0
(
deg
h
i
)M
i
=−
2
+
2
kπ
√ −
1log
q.
(2)We now give another description of the
α
i
. Let
F
i
be the point on
E
r
which hascoordinates
(a
i
1
:
a
i
2
)
with respect to the homogeneous coordinates
(y
1
:
y
2
)
on
E
r
⊂
Y
r
. Let
j
be the unique element of
T
\{
r
}
such that
E
j
passes through
F
i
in
Y
. Let
ρ
be the number of blowingups among
g
r
,...,g
t
which are centered at
F
i
.Then, the announced description is
α
i
=
ν
j
+
s
0
N
j
−
(
2
ρkπ
√ −
1
)/(
log
q)
. The second author proved this in [10, Sect. 2.7] in the case
k
=
0, and the general case istreated in a similar way. It follows that Re
(α
i
)<
0 if and only if
−
ν
r
/N
r
<
−
ν
j
/N
j
.One checks also easily that
s
0
is a candidate pole of
E
j
⇐⇒
ν
j
+
s
0
N
j
is a multiple of 2
π
√ −
1
/(
log
q)
⇐⇒
α
i
is a multiple of 2
π
√ −
1
/(
log
q).