In algebraic geometry, the motivic zeta function of a smooth algebraic variety X {\displaystyle X} is the formal power series:
Z ( X , t ) = ∑ n = 0 ∞ [ X ( n ) ] t n {\displaystyle Z(X,t)=\sum _{n=0}^{\infty }[X^{(n)}]t^{n}}Here X ( n ) {\displaystyle X^{(n)}} is the n {\displaystyle n} -th symmetric power of X {\displaystyle X} , i.e., the quotient of X n {\displaystyle X^{n}} by the action of the symmetric group S n {\displaystyle S_{n}} , and [ X ( n ) ] {\displaystyle [X^{(n)}]} is the class of X ( n ) {\displaystyle X^{(n)}} in the ring of motives (see below).
If the ground field is finite, and one applies the counting measure to Z ( X , t ) {\displaystyle Z(X,t)} , one obtains the local zeta function of X {\displaystyle X} .
If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to Z ( X , t ) {\displaystyle Z(X,t)} , one obtains 1 / ( 1 − t ) χ ( X ) {\displaystyle 1/(1-t)^{\chi (X)}} .
Motivic measures
A motivic measure is a map μ {\displaystyle \mu } from the set of finite type schemes over a field k {\displaystyle k} to a commutative ring A {\displaystyle A} , satisfying the three properties
μ ( X ) {\displaystyle \mu (X)\,} depends only on the isomorphism class of X {\displaystyle X} , μ ( X ) = μ ( Z ) + μ ( X ∖ Z ) {\displaystyle \mu (X)=\mu (Z)+\mu (X\setminus Z)} if Z {\displaystyle Z} is a closed subscheme of X {\displaystyle X} , μ ( X 1 × X 2 ) = μ ( X 1 ) μ ( X 2 ) {\displaystyle \mu (X_{1}\times X_{2})=\mu (X_{1})\mu (X_{2})} .For example if k {\displaystyle k} is a finite field and A = Z {\displaystyle A={\mathbb {Z} }} is the ring of integers, then μ ( X ) = # ( X ( k ) ) {\displaystyle \mu (X)=\#(X(k))} defines a motivic measure, the counting measure.
If the ground field is the complex numbers, then Euler characteristic with compact supports defines a motivic measure with values in the integers.
The zeta function with respect to a motivic measure μ {\displaystyle \mu } is the formal power series in A [ [ t ] ] {\displaystyle A[[t]]} given by
Z μ ( X , t ) = ∑ n = 0 ∞ μ ( X ( n ) ) t n {\displaystyle Z_{\mu }(X,t)=\sum _{n=0}^{\infty }\mu (X^{(n)})t^{n}} .There is a universal motivic measure. It takes values in the K-ring of varieties, A = K ( V ) {\displaystyle A=K(V)} , which is the ring generated by the symbols [ X ] {\displaystyle [X]} , for all varieties X {\displaystyle X} , subject to the relations
[ X ′ ] = [ X ] {\displaystyle [X']=[X]\,} if X ′ {\displaystyle X'} and X {\displaystyle X} are isomorphic, [ X ] = [ Z ] + [ X ∖ Z ] {\displaystyle [X]=[Z]+[X\setminus Z]} if Z {\displaystyle Z} is a closed subvariety of X {\displaystyle X} , [ X 1 × X 2 ] = [ X 1 ] ⋅ [ X 2 ] {\displaystyle [X_{1}\times X_{2}]=[X_{1}]\cdot [X_{2}]} .The universal motivic measure gives rise to the motivic zeta function.
Examples
Let L = [ A 1 ] {\displaystyle \mathbb {L} =[{\mathbb {A} }^{1}]} denote the class of the affine line.
Z ( A , t ) = 1 1 − L t {\displaystyle Z({\mathbb {A} },t)={\frac {1}{1-{\mathbb {L} }t}}} Z ( A n , t ) = 1 1 − L n t {\displaystyle Z({\mathbb {A} }^{n},t)={\frac {1}{1-{\mathbb {L} }^{n}t}}} Z ( P n , t ) = ∏ i = 0 n 1 1 − L i t {\displaystyle Z({\mathbb {P} }^{n},t)=\prod _{i=0}^{n}{\frac {1}{1-{\mathbb {L} }^{i}t}}}If X {\displaystyle X} is a smooth projective irreducible curve of genus g {\displaystyle g} admitting a line bundle of degree 1, and the motivic measure takes values in a field in which L {\displaystyle {\mathbb {L} }} is invertible, then
Z ( X , t ) = P ( t ) ( 1 − t ) ( 1 − L t ) , {\displaystyle Z(X,t)={\frac {P(t)}{(1-t)(1-{\mathbb {L} }t)}}\,,}where P ( t ) {\displaystyle P(t)} is a polynomial of degree 2 g {\displaystyle 2g} . Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.
If S {\displaystyle S} is a smooth surface over an algebraically closed field of characteristic 0 {\displaystyle 0} , then the generating function for the motives of the Hilbert schemes of S {\displaystyle S} can be expressed in terms of the motivic zeta function by Göttsche's Formula
∑ n = 0 ∞ [ S [ n ] ] t n = ∏ m = 1 ∞ Z ( S , L m − 1 t m ) {\displaystyle \sum _{n=0}^{\infty }[S^{[n]}]t^{n}=\prod _{m=1}^{\infty }Z(S,{\mathbb {L} }^{m-1}t^{m})}Here S [ n ] {\displaystyle S^{[n]}} is the Hilbert scheme of length n {\displaystyle n} subschemes of S {\displaystyle S} . For the affine plane this formula gives
∑ n = 0 ∞ [ ( A 2 ) [ n ] ] t n = ∏ m = 1 ∞ 1 1 − L m + 1 t m {\displaystyle \sum _{n=0}^{\infty }[({\mathbb {A} }^{2})^{[n]}]t^{n}=\prod _{m=1}^{\infty }{\frac {1}{1-{\mathbb {L} }^{m+1}t^{m}}}}This is essentially the partition function.
References
Marcolli, Matilde (2010). Feynman Motives. World Scientific. p. 115. ISBN 9789814304481. Retrieved 26 April 2023. 9789814304481 ↩