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

Furthermore let ${\displaystyle \phi :K^{n}\mapsto \mathbb {C} }$ be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let ${\displaystyle \chi }$ be a character of ${\displaystyle R^{\times }}$.

In this situation one associates to a non-constant polynomial ${\displaystyle f(x_{1},\ldots ,x_{n})\in K[x_{1},\ldots ,x_{n}]}$ the Igusa zeta function

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

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

Igusa's theorem

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

Henceforth we take ${\displaystyle \phi }$ to be the characteristic function of ${\displaystyle R^{n}}$ and ${\displaystyle \chi }$ to be the trivial character. Let ${\displaystyle N_{i}}$ denote the number of solutions of the congruence

${\displaystyle f(x_{1},\ldots ,x_{n})\equiv 0\mod P^{i}}$.

Then the Igusa zeta function

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

is closely related to the Poincaré series

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

by

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

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