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. [math]\displaystyle{ [K: \mathbb{Q}_p]\lt \infty }[/math], R the valuation ring and P the maximal ideal. For [math]\displaystyle{ z \in K }[/math] we denote by [math]\displaystyle{ \operatorname{ord}(z) }[/math] the valuation of z, [math]\displaystyle{ \mid z \mid = q^{-\operatorname{ord}(z)} }[/math], and [math]\displaystyle{ ac(z)=z \pi^{-\operatorname{ord}(z)} }[/math] for a uniformizing parameter π of R.
Furthermore let [math]\displaystyle{ \phi : K^n \mapsto \mathbb{C} }[/math] be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let [math]\displaystyle{ \chi }[/math] be a character of [math]\displaystyle{ R^\times }[/math].
In this situation one associates to a non-constant polynomial [math]\displaystyle{ f(x_1, \ldots, x_n) \in K[x_1,\ldots,x_n] }[/math] the Igusa zeta function
- [math]\displaystyle{ 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 }[/math]
where [math]\displaystyle{ s \in \mathbb{C}, \operatorname{Re}(s)\gt 0, }[/math] and dx is Haar measure so normalized that [math]\displaystyle{ R^n }[/math] has measure 1.
Igusa's theorem
Jun-Ichi Igusa (1974) showed that [math]\displaystyle{ Z_\phi (s,\chi) }[/math] is a rational function in [math]\displaystyle{ t=q^{-s} }[/math]. 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 [math]\displaystyle{ P }[/math]
Henceforth we take [math]\displaystyle{ \phi }[/math] to be the characteristic function of [math]\displaystyle{ R^n }[/math] and [math]\displaystyle{ \chi }[/math] to be the trivial character. Let [math]\displaystyle{ N_i }[/math] denote the number of solutions of the congruence
- [math]\displaystyle{ f(x_1,\ldots,x_n) \equiv 0 \mod P^i }[/math].
Then the Igusa zeta function
- [math]\displaystyle{ Z(t)= \int_{R^n} |f(x_1,\ldots,x_n)|^s \, dx }[/math]
is closely related to the Poincaré series
- [math]\displaystyle{ P(t)= \sum_{i=0}^{\infty} q^{-in}N_i t^i }[/math]
by
- [math]\displaystyle{ P(t)= \frac{1-t Z(t)}{1-t}. }[/math]
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
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