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 P
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
Wikiwand in your browser!
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.