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, p², p³, and so on.

Definition

For a prime number p let K be a p-adic field, i.e. $[K:{\mathbb{\mathbb{Q}}}_p]<\mathrm{\infty }$, R the valuation ring and P the maximal ideal. For $z\in K$ we denote by $\mathrm{ord}(z)$ the valuation of z, $\mid z\mid =q^{-\mathrm{ord}(z)}$, and $ac(z)=z{\pi }^{-\mathrm{ord}(z)}$ for a uniformizing parameter π of R.

Furthermore let $\varphi :K^n\mapsto \mathbb{ℂ}$ 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 $f(x_1,\dots ,x_n)\in K[x_1,\dots ,x_n]$ the Igusa zeta function

$Z_{\varphi }(s,\chi )=\underset{K^n}{\int }\varphi (x_1,\dots ,x_n)\chi (ac(f(x_1,\dots ,x_n)))|f(x_1,\dots ,x_n){|}^{s}dx$

where $s\in \mathbb{ℂ},\mathrm{Re}(s)>0,$ and dx is Haar measure so normalized that $R^n$ has measure 1.

Igusa's theorem

Jun-Ichi Igusa (1974) showed that $Z_{\varphi }(s,\chi )$ is a rational function in $t=q^{-s}$. 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 $\varphi$ 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

$f(x_1,\dots ,x_n)\equiv 0mod{P}^i$.

Then the Igusa zeta function

$Z(t)=\underset{R^n}{\int }|f(x_1,\dots ,x_n){|}^{s}dx$

is closely related to the Poincaré series

$P(t)=\sum _{i=0}^{\mathrm{\infty }}{q}^{-in}{N}_i{t}^i$

by

$P(t)=\frac{1-tZ(t)}{1-t}.$

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 
• 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