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. $[K:\mathbb {Q} _{p}]<\infty$, R the valuation ring and P the maximal ideal. For $z\in K$ we denote by $\operatorname {ord} (z)$ the valuation of z, $\mid z\mid =q^{-\operatorname {ord} (z)}$, and $ac(z)=z\pi ^{-\operatorname {ord} (z)}$ for a uniformizing parameter π of R.

Furthermore let $\phi :K^{n}\to \mathbb {C}$ 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 $f(x_{1},\ldots ,x_{n})\in K[x_{1},\ldots ,x_{n}]$ the Igusa zeta function

$Z_{\phi }(s,\chi )=\int _{K^{n}}\phi (x_{1},\ldots ,x_{n})\chi (ac(f(x_{1},\ldots ,x_{n})))|f(x_{1},\ldots ,x_{n})|^{s}\,dx$

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

Igusa's theorem

Jun-Ichi Igusa (1974) showed that $Z_{\phi }(s,\chi )$ 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 $\phi$ 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

$Z(t)=\int _{R^{n}}|f(x_{1},\ldots ,x_{n})|^{s}\,dx$

is closely related to the Poincaré series

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

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, Zbl 0287.43007
• 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