• Source: Igusa zeta function

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.




Definition


For a prime number p let K be a p-adic field, i.e.



[
K
:


Q


p


]
<
∞


{\displaystyle [K:\mathbb {Q} _{p}]<\infty }

, R the valuation ring and P the maximal ideal. For



z
∈
K


{\displaystyle z\in K}

we denote by



ord
⁡
(
z
)


{\displaystyle \operatorname {ord} (z)}

the valuation of z,



∣
z
∣=

q

−
ord
⁡
(
z
)




{\displaystyle \mid z\mid =q^{-\operatorname {ord} (z)}}

, and



a
c
(
z
)
=
z

π

−
ord
⁡
(
z
)




{\displaystyle ac(z)=z\pi ^{-\operatorname {ord} (z)}}

for a uniformizing parameter π of R.
Furthermore let



ϕ
:

K

n


→

C



{\displaystyle \phi :K^{n}\to \mathbb {C} }

be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let



χ


{\displaystyle \chi }

be a character of




R

×




{\displaystyle R^{\times }}

.
In this situation one associates to a non-constant polynomial



f
(

x

1


,
…
,

x

n


)
∈
K
[

x

1


,
…
,

x

n


]


{\displaystyle f(x_{1},\ldots ,x_{n})\in K[x_{1},\ldots ,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


{\displaystyle Z_{\phi }(s,\chi )=\int _{K^{n}}\phi (x_{1},\ldots ,x_{n})\chi (ac(f(x_{1},\ldots ,x_{n})))|f(x_{1},\ldots ,x_{n})|^{s}\,dx}


where



s
∈

C

,
Re
⁡
(
s
)
>
0
,


{\displaystyle s\in \mathbb {C} ,\operatorname {Re} (s)>0,}

and dx is Haar measure so normalized that




R

n




{\displaystyle R^{n}}

has measure 1.




Igusa's theorem


Jun-Ichi Igusa (1974) showed that




Z

ϕ


(
s
,
χ
)


{\displaystyle Z_{\phi }(s,\chi )}

is a rational function in



t
=

q

−
s




{\displaystyle 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.)




Congruences modulo powers of P


Henceforth we take



ϕ


{\displaystyle \phi }

to be the characteristic function of




R

n




{\displaystyle R^{n}}

and



χ


{\displaystyle \chi }

to be the trivial character. Let




N

i




{\displaystyle N_{i}}

denote the number of solutions of the congruence




f
(

x

1


,
…
,

x

n


)
≡
0

mod



P

i




{\displaystyle f(x_{1},\ldots ,x_{n})\equiv 0\mod P^{i}}

.
Then the Igusa zeta function




Z
(
t
)
=

∫


R

n





|

f
(

x

1


,
…
,

x

n


)


|


s



d
x


{\displaystyle Z(t)=\int _{R^{n}}|f(x_{1},\ldots ,x_{n})|^{s}\,dx}


is closely related to the Poincaré series




P
(
t
)
=

∑

i
=
0


∞



q

−
i
n



N

i



t

i




{\displaystyle P(t)=\sum _{i=0}^{\infty }q^{-in}N_{i}t^{i}}


by




P
(
t
)
=



1
−
t
Z
(
t
)


1
−
t



.


{\displaystyle P(t)={\frac {1-tZ(t)}{1-t}}.}





References


Igusa, Jun-Ichi (1974), "Complex powers and asymptotic expansions. I. Functions of certain types", Journal für die reine und angewandte Mathematik, 1974 (268–269): 110–130, doi:10.1515/crll.1974.268-269.110, Zbl 0287.43007
Information for this article was taken from J. Denef, Report on Igusa's Local Zeta Function, Séminaire Bourbaki 43 (1990-1991), exp. 741; Astérisque 201-202-203 (1991), 359-386

Kata Kunci Pencarian:

• List of zeta functions
• Igusa zeta function
• Jun-Ichi Igusa
• Glossary of arithmetic and diophantine geometry
• François Loeser
• Period (algebraic geometry)
• Prakash Belkale
• Jan Denef
• Margaret M. Robinson
• Séminaire Nicolas Bourbaki (1950–1959)