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 : Q p ] < ∞ {\displaystyle [K:\mathbb {Q} _{p}]<\infty } , R the valuation ring and P the maximal ideal. For z ∈ K {\displaystyle z\in K} we denote by ord ( z ) {\displaystyle \operatorname {ord} (z)} the valuation of z, ∣ z ∣= q − ord ( z ) {\displaystyle \mid z\mid =q^{-\operatorname {ord} (z)}} , and a c ( z ) = z π − ord ( z ) {\displaystyle ac(z)=z\pi ^{-\operatorname {ord} (z)}} for a uniformizing parameter π of R. Furthermore let ϕ : K n → C {\displaystyle \phi :K^{n}\to \mathbb {C} } be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let χ {\displaystyle \chi } be a character of R × {\displaystyle R^{\times }} . In this situation one associates to a non-constant polynomial f ( x 1 , … , x n ) ∈ K [ x 1 , … , x n ] {\displaystyle f(x_{1},\ldots ,x_{n})\in K[x_{1},\ldots ,x_{n}]} the Igusa zeta function
Z ϕ ( s , χ ) = ∫ K n ϕ ( x 1 , … , x n ) χ ( a c ( f ( x 1 , … , x n ) ) ) | f ( x 1 , … , x n ) | s d x {\displaystyle Z_{\phi }(s,\chi )=\int _{K^{n}}\phi (x_{1},\ldots ,x_{n})\chi (ac(f(x_{1},\ldots ,x_{n})))|f(x_{1},\ldots ,x_{n})|^{s}\,dx}
where s ∈ C , Re ( s ) > 0 , {\displaystyle s\in \mathbb {C} ,\operatorname {Re} (s)>0,} and dx is Haar measure so normalized that R n {\displaystyle R^{n}} has measure 1.
Igusa's theorem Jun-Ichi Igusa (1974) showed that Z ϕ ( s , χ ) {\displaystyle Z_{\phi }(s,\chi )} is a rational function in t = q − s {\displaystyle 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.)
Congruences modulo powers of P Henceforth we take ϕ {\displaystyle \phi } to be the characteristic function of R n {\displaystyle R^{n}} and χ {\displaystyle \chi } to be the trivial character. Let N i {\displaystyle N_{i}} denote the number of solutions of the congruence
f ( x 1 , … , x n ) ≡ 0 mod P i {\displaystyle f(x_{1},\ldots ,x_{n})\equiv 0\mod P^{i}} . Then the Igusa zeta function
Z ( t ) = ∫ R n | f ( x 1 , … , x n ) | t d x {\displaystyle Z(t)=\int _{R^{n}}|f(x_{1},\ldots ,x_{n})|^{t}\,dx}
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