Igusa zeta function explained

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: 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 π of R.

Furthermore let

\phi : K ^{n} \to C be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let \chi be a character of R ^{x} .f(x _{1,} \ldots, x _{n)} \in K[x _{1,\ldots,x} _{n]} the Igusa zeta functionZ _{\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} ^{s} ^{n)|} dx

where

s \in 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 congruencef(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} ^{s} ^{n)|} dx

is closely related to the Poincaré series

P(t) = \sum ^{inf ty} _{i =0} q ^{-in} N _{i} t ^{i}

by

P(t) = (1- t Z(t))/(1- t) .

References

• • 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