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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. , R the valuation ring and P the maximal ideal. For we denote by the valuation of z, , and for a uniformizing parameter π of R.

Furthermore let be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let be a character of .

In this situation one associates to a non-constant polynomial the Igusa zeta function

where and dx is Haar measure so normalized that has measure 1.

Igusa's theorem

Jun-Ichi Igusa (1974) showed that is a rational function in . 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

Henceforth we take to be the characteristic function of and to be the trivial character. Let denote the number of solutions of the congruence

.

Then the Igusa zeta function

is closely related to the Poincaré series

by

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