Type of generating function in mathematics
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.
, R the
valuation ring and P the maximal
ideal. For
we denote by
the
valuation of z,
, and
for a uniformizing parameter π of R.
Furthermore let
be a
Schwartz–Bruhat function, i.e. a locally constant function with
compact support and let
be a
character of
.
In this situation one associates to a non-constant
polynomial
the Igusa zeta function
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6964de690d70d6181c7e1afb5895cd1a65da8e2)
where
and dx is
Haar measure so normalized that
has measure 1.
Igusa's theorem
Jun-Ichi Igusa (
1974) showed that
is a rational function in
. 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
to be the
characteristic function of
and
to be the trivial character. Let
denote the number of solutions of the
congruence
.
Then the Igusa zeta function
![{\displaystyle Z(t)=\int _{R^{n}}|f(x_{1},\ldots ,x_{n})|^{s}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9141fdc079379c24ab0c56dac960b538de8dead)
is closely related to the Poincaré series
![{\displaystyle P(t)=\sum _{i=0}^{\infty }q^{-in}N_{i}t^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae07bd9f9ec9462622b00735935d7c747a4bed73)
by
![{\displaystyle P(t)={\frac {1-tZ(t)}{1-t}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df998d16f4f52352dda0ee15089957d8c5aee5af)
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