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

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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

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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 P

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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

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  • 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