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.
, R the valuation ring and P the maximal ideal.
{\displaystyle ac(z)=z\pi ^{-\operatorname {ord} (z)}}
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
In this situation one associates to a non-constant polynomial
the Igusa zeta function where
and dx is Haar measure so normalized that
Jun-Ichi Igusa (1974) showed that
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.)
to be the characteristic function of
to be the trivial character.
denote the number of solutions of the congruence Then the Igusa zeta function is closely related to the Poincaré series by