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 &pi; 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
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


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