Igusa zeta function

For a prime number p let K be a p-adic field, i.e. , R the valuation ring and P the maximal ideal. For $z\; \backslash in\; K$ we denote by $\backslash operatorname\{ord\}(z)$ the valuation of z, $\backslash mid\; z\; \backslash mid\; =\; q^\{-\backslash operatorname\{ord\}(z)\}$, and $ac(z)=z\; \backslash pi^\{-\backslash operatorname\{ord\}(z)\}$ for a uniformizing parameter π of R.

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

In this situation one associates to a non-constant polynomial $f(x\_1,\; \backslash ldots,\; x\_n)\; \backslash in\; K[x\_1,\backslash ldots,x\_n]$ the Igusa zeta function

: $Z\_\backslash phi(s,\backslash chi)\; =\; \backslash int\_\{K^n\}\; \backslash phi(x\_1,\backslash ldots,x\_n)\; \backslash chi(ac(f(x\_1,\backslash ldots,x\_n)))\; |f(x\_1,\backslash ldots,x\_n)|^s\; \backslash ,\; dx$

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

{{harvs|txt|authorlink=Jun-Ichi Igusa|first=Jun-Ichi|last=Igusa|year=1974}} showed that $Z\_\backslash phi\; (s,\backslash chi)$ 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.)

Henceforth we take $\backslash phi$ to be the characteristic function of $R^n$ and $\backslash chi$ to be the trivial character. Let $N\_i$ denote the number of solutions of the congruence

:$f(x\_1,\backslash ldots,x\_n)\; \backslash equiv\; 0\; \backslash mod\; P^i$.

Then the Igusa zeta function

: $Z(t)=\; \backslash int\_\{R^n\}\; |f(x\_1,\backslash ldots,x\_n)|^s\; \backslash ,\; dx$

is closely related to the Poincaré series

: $P(t)=\; \backslash sum\_\{i=0\}^\{\backslash infty\}\; q^\{-in\}N\_i\; t^i$

by

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

• {{Citation |last=Igusa |first=Jun-Ichi |year=1974 |title=Complex powers and asymptotic expansions. I. Functions of certain types |journal=Journal für die reine und angewandte Mathematik |volume=1974 |issue=268–269 |pages=110–130 |doi=10.1515/crll.1974.268-269.110 | zbl=0287.43007 }}
• Information for this article was taken from [http://www.numdam.org/item/SB_1990-1991__33__359_0 J. Denef, Report on Igusa's Local Zeta Function, Séminaire Bourbaki 43 (1990-1991), exp. 741; Astérisque 201-202-203 (1991), 359-386]

Category:Zeta and L-functions

Category:Diophantine geometry