Igusa zeta function

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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:p]<, R the valuation ring and P the maximal ideal. For zK we denote by ord(z) the valuation of z, z=qord(z), and ac(z)=zπord(z) for a uniformizing parameter π of R. Furthermore let ϕ:Kn be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let χ be a character of R×. In this situation one associates to a non-constant polynomial f(x1,,xn)K[x1,,xn] the Igusa zeta function

Zϕ(s,χ)=Knϕ(x1,,xn)χ(ac(f(x1,,xn)))|f(x1,,xn)|sdx

where s,Re(s)>0, and dx is Haar measure so normalized that Rn has measure 1.

Igusa's theorem

Jun-Ichi Igusa (1974) showed that Zϕ(s,χ) is a rational function in t=qs. 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 ϕ to be the characteristic function of Rn and χ to be the trivial character. Let Ni denote the number of solutions of the congruence

f(x1,,xn)0modPi.

Then the Igusa zeta function

Z(t)=Rn|f(x1,,xn)|sdx

is closely related to the Poincaré series

P(t)=i=0qinNiti

by

P(t)=1tZ(t)1t.

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