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Igusa zeta function

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Type of generating function in mathematics

In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p, p, and so on.

Definition

For a prime number p let K be a p-adic field, i.e. [ K : Q p ] < ∞ {\displaystyle <\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} 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 ) | s d x {\displaystyle Z(t)=\int _{R^{n}}|f(x_{1},\ldots ,x_{n})|^{s}\,dx}

is closely related to the Poincaré series

P ( t ) = ∑ i = 0 ∞ q − i n N i t i {\displaystyle P(t)=\sum _{i=0}^{\infty }q^{-in}N_{i}t^{i}}

by

P ( t ) = 1 − t Z ( t ) 1 − t . {\displaystyle P(t)={\frac {1-tZ(t)}{1-t}}.}

References

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