In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulop, p2, p3, and so on.
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.
Jun-IchiIgusa ( 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.)
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
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