In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on.
For a prime number p let K be a p-adic field, i.e.
[
K
:
Q
p
]
<
∞
, R the valuation ring and P the maximal ideal. For
z
∈
K
we denote by
ord
(
z
)
the valuation of z,
∣
z
∣=
q
−
ord
(
z
)
, and
a
c
(
z
)
=
z
π
−
ord
(
z
)
for a uniformizing parameter π of R.
Furthermore let
ϕ
:
K
n
↦
C
be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let
χ
be a character of
R
×
.
In this situation one associates to a non-constant polynomial
f
(
x
1
,
…
,
x
n
)
∈
K
[
x
1
,
…
,
x
n
]
the Igusa zeta function
Z
ϕ
(
s
,
χ
)
=
∫
K
n
ϕ
(
x
1
,
…
,
x
n
)
χ
(
a
c
(
f
(
x
1
,
…
,
x
n
)
)
)
|
f
(
x
1
,
…
,
x
n
)
|
s
d
x
where
s
∈
C
,
Re
(
s
)
>
0
,
and dx is Haar measure so normalized that
R
n
has measure 1.
Jun-Ichi Igusa (1974) showed that
Z
ϕ
(
s
,
χ
)
is a rational function in
t
=
q
−
s
. The proof uses Heisuke Hironaka's theorem about the resolution of singularities. Later, an entirely different proof was given by Jan Denef using p-adic cell decomposition. Little is known, however, about explicit formulas. (There are some results about Igusa zeta functions of Fermat varieties.)
Henceforth we take
ϕ
to be the characteristic function of
R
n
and
χ
to be the trivial character. Let
N
i
denote the number of solutions of the congruence
f
(
x
1
,
…
,
x
n
)
≡
0
mod
P
i
.
Then the Igusa zeta function
Z
(
t
)
=
∫
R
n
|
f
(
x
1
,
…
,
x
n
)
|
s
d
x
is closely related to the Poincaré series
P
(
t
)
=
∑
i
=
0
∞
q
−
i
n
N
i
t
i
by
P
(
t
)
=
1
−
t
Z
(
t
)
1
−
t
.
