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'', ''p''², ''p''³, and so on.

Definition

For a prime number ''p'' let ''K'' be a p-adic field, i.e. : \mathbb_p\infty , ''R'' the valuation ring and ''P'' the maximal ideal. For $z \in K$ we denote by $\operatorname(z)$ the valuation of ''z'', $\mid z \mid = q^$, and $ac(z)=z \pi^$ for a uniformizing parameter π of ''R''.

Furthermore let $\phi : K^n \to \mathbb$ be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let $\chi$ be a character of $R^\times$.

In this situation one associates to a non-constant polynomial f(x_1, \ldots, x_n) \in K _1,\ldots,x_n/math> the Igusa zeta function

: $Z_\phi(s,\chi) = \int_ \phi(x_1,\ldots,x_n) \chi(ac(f(x_1,\ldots,x_n))) , f(x_1,\ldots,x_n), ^s \, dx$

where $s \in \mathbb, \operatorname(s)>0,$ and ''dx'' is Haar measure so normalized that $R^n$ has measure 1.

Igusa's theorem

showed that $Z_\phi (s,\chi)$ is a rational function in $t=q^$. 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 $\phi$ to be the characteristic function of $R^n$ and $\chi$ to be the trivial character. Let $N_i$ denote the number of solutions of the congruence

:$f(x_1,\ldots,x_n) \equiv 0 \mod P^i$.

Then the Igusa zeta function

: $Z(t)= \int_ , f(x_1,\ldots,x_n), ^s \, dx$

is closely related to the Poincaré series

: $P(t)= \sum_^ q^N_i t^i$

by

: $P(t)= \frac.$

References

*{{Citation , last=Igusa , first=Jun-Ichi , year=1974 , title=Complex powers and asymptotic expansions. I. Functions of certain types , journal=Journal für die reine und angewandte Mathematik , volume=1974 , issue=268–269 , pages=110–130 , doi=10.1515/crll.1974.268-269.110 , zbl=0287.43007
*Information for this article was taken fro
J. Denef, Report on Igusa's Local Zeta Function, Séminaire Bourbaki 43 (1990-1991), exp. 741; Astérisque 201-202-203 (1991), 359-386

Zeta and L-functions
Diophantine geometry