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
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
is closely related to the Poincaré series
by
