Igusa zeta-function

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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.

1 Definition
2 Igusa's theorem
3 Congruences modulo powers of P
4 Reference

[edit] 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 $z \in K$ $\operatorname{ord}(z)$ denotes the valuation of $z$ , $\mid z \mid = q^{-\operatorname{ord}(z)}$ , and Failed to parse (Cannot write to or create math output directory): ac(z)=z \pi^{-\operatorname{ord}(z)}

for a uniformizing parameter  $π$  of  $R$ .

Furthermore let $\phi : K^n \mapsto \mathbb{C}$ be a Schwartz-Bruhat function, i.e. a locally constant function with compact support and let $χ$ be a character of $K *$ .

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 $d x$ is Haar measure so normalized that $R n$ has measure 1.

[edit] Igusa's theorem

Junichi Igusa showed that $Z φ (s,χ)$ is a rational function in $t = q - s$ . The proof uses Heisuke Hironaka's theorem about the resolution of singularities. Little is known, however, about explicit formulas. (There are some results about Igusa zeta functions of Fermat varieties.)

[edit] Congruences modulo powers of $P$

Henceforth we take $φ$ to be the characteristic function of $R n$ and $χ$ to be the trivial character. Let $N i$ denote the number of solutions of the congruence

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