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", "p"², "p"³, and so on.

Definition

For a prime number $p$ let $K$ be a p-adic field, i.e.

Furthermore let $phi\; :\; K^n\; mapsto\; mathbb\{C\}$ be a Schwartz-Bruhat function, i.e. a locally constant function with compact support and let $chi$ 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 $dx$ is Haar measure so normalized that $R^n$ has measure 1.

Igusa's theorem

Junichi Igusa 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. 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

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

References

*Information for this article was taken from [http://wis.kuleuven.be/algebra/denef.html J. Denef, Report on Igusa's Local Zeta Function, Séminaire Bourbaki 43 (1990-1991), exp. 741; Astérisque 201-202-203 (1991), 359-386]