Motivic zeta function

In algebraic geometry, the motivic zeta function of a smooth algebraic variety ${\displaystyle X}${\displaystyle X} is the formal power series:^[1]

${\displaystyle Z(X,t)=\sum _{n=0}^{\infty }[X^{(n)}]t^{n}}${\displaystyle Z(X,t)=\sum _{n=0}^{\infty }[X^{(n)}]t^{n}}

Here ${\displaystyle X^{(n)}}${\displaystyle X^{(n)}} is the ${\displaystyle n}${\displaystyle n}-th symmetric power of ${\displaystyle X}${\displaystyle X}, i.e., the quotient of ${\displaystyle X^{n}}${\displaystyle X^{n}} by the action of the symmetric group ${\displaystyle S_{n}}${\displaystyle S_{n}}, and ${\displaystyle [X^{(n)}]}${\displaystyle [X^{(n)}]} is the class of ${\displaystyle X^{(n)}}${\displaystyle X^{(n)}} in the ring of motives (see below).

If the ground field is finite, and one applies the counting measure to ${\displaystyle Z(X,t)}${\displaystyle Z(X,t)}, one obtains the local zeta function of ${\displaystyle X}${\displaystyle X}.

If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to ${\displaystyle Z(X,t)}${\displaystyle Z(X,t)}, one obtains ${\displaystyle 1/(1-t)^{\chi (X)}}${\displaystyle 1/(1-t)^{\chi (X)}}.

A motivic measure is a map ${\displaystyle \mu }${\displaystyle \mu } from the set of finite type schemes over a field ${\displaystyle k}${\displaystyle k} to a commutative ring ${\displaystyle A}${\displaystyle A}, satisfying the three properties

${\displaystyle \mu (X),円}${\displaystyle \mu (X),円} depends only on the isomorphism class of ${\displaystyle X}${\displaystyle X},

${\displaystyle \mu (X)=\mu (Z)+\mu (X\setminus Z)}${\displaystyle \mu (X)=\mu (Z)+\mu (X\setminus Z)} if ${\displaystyle Z}${\displaystyle Z} is a closed subscheme of ${\displaystyle X}${\displaystyle X},

${\displaystyle \mu (X_{1}\times X_{2})=\mu (X_{1})\mu (X_{2})}${\displaystyle \mu (X_{1}\times X_{2})=\mu (X_{1})\mu (X_{2})}.

For example if ${\displaystyle k}${\displaystyle k} is a finite field and ${\displaystyle A={\mathbb {Z} }}${\displaystyle A={\mathbb {Z} }} is the ring of integers, then ${\displaystyle \mu (X)=\#(X(k))}${\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 ${\displaystyle \mu }${\displaystyle \mu } is the formal power series in ${\displaystyle A[[t]]}${\displaystyle A[[t]]} given by

${\displaystyle Z_{\mu }(X,t)=\sum _{n=0}^{\infty }\mu (X^{(n)})t^{n}}${\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, ${\displaystyle A=K(V)}${\displaystyle A=K(V)}, which is the ring generated by the symbols ${\displaystyle [X]}${\displaystyle [X]}, for all varieties ${\displaystyle X}${\displaystyle X}, subject to the relations

${\displaystyle [X']=[X],円}${\displaystyle [X']=[X],円} if ${\displaystyle X'}${\displaystyle X'} and ${\displaystyle X}${\displaystyle X} are isomorphic,

${\displaystyle [X]=[Z]+[X\setminus Z]}${\displaystyle [X]=[Z]+[X\setminus Z]} if ${\displaystyle Z}${\displaystyle Z} is a closed subvariety of ${\displaystyle X}${\displaystyle X},

${\displaystyle [X_{1}\times X_{2}]=[X_{1}]\cdot [X_{2}]}${\displaystyle [X_{1}\times X_{2}]=[X_{1}]\cdot [X_{2}]}.

The universal motivic measure gives rise to the motivic zeta function.

Let ${\displaystyle \mathbb {L} =[{\mathbb {A} }^{1}]}${\displaystyle \mathbb {L} =[{\mathbb {A} }^{1}]} denote the class of the affine line.

${\displaystyle Z({\mathbb {A} },t)={\frac {1}{1-{\mathbb {L} }t}}}${\displaystyle Z({\mathbb {A} },t)={\frac {1}{1-{\mathbb {L} }t}}}

${\displaystyle Z({\mathbb {A} }^{n},t)={\frac {1}{1-{\mathbb {L} }^{n}t}}}${\displaystyle Z({\mathbb {A} }^{n},t)={\frac {1}{1-{\mathbb {L} }^{n}t}}}

${\displaystyle Z({\mathbb {P} }^{n},t)=\prod _{i=0}^{n}{\frac {1}{1-{\mathbb {L} }^{i}t}}}${\displaystyle Z({\mathbb {P} }^{n},t)=\prod _{i=0}^{n}{\frac {1}{1-{\mathbb {L} }^{i}t}}}

If ${\displaystyle X}${\displaystyle X} is a smooth projective irreducible curve of genus ${\displaystyle g}${\displaystyle g} admitting a line bundle of degree 1, and the motivic measure takes values in a field in which ${\displaystyle {\mathbb {L} }}${\displaystyle {\mathbb {L} }} is invertible, then

${\displaystyle Z(X,t)={\frac {P(t)}{(1-t)(1-{\mathbb {L} }t)}},,円}${\displaystyle Z(X,t)={\frac {P(t)}{(1-t)(1-{\mathbb {L} }t)}},,円}

where ${\displaystyle P(t)}${\displaystyle P(t)} is a polynomial of degree ${\displaystyle 2g}${\displaystyle 2g}. Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.

If ${\displaystyle S}${\displaystyle S} is a smooth surface over an algebraically closed field of characteristic ${\displaystyle 0}${\displaystyle 0}, then the generating function for the motives of the Hilbert schemes of ${\displaystyle S}${\displaystyle S} can be expressed in terms of the motivic zeta function by Göttsche's Formula

${\displaystyle \sum _{n=0}^{\infty }[S^{[n]}]t^{n}=\prod _{m=1}^{\infty }Z(S,{\mathbb {L} }^{m-1}t^{m})}${\displaystyle \sum _{n=0}^{\infty }[S^{[n]}]t^{n}=\prod _{m=1}^{\infty }Z(S,{\mathbb {L} }^{m-1}t^{m})}

Here ${\displaystyle S^{[n]}}${\displaystyle S^{[n]}} is the Hilbert scheme of length ${\displaystyle n}${\displaystyle n} subschemes of ${\displaystyle S}${\displaystyle S}. For the affine plane this formula gives

${\displaystyle \sum _{n=0}^{\infty }[({\mathbb {A} }^{2})^{[n]}]t^{n}=\prod _{m=1}^{\infty }{\frac {1}{1-{\mathbb {L} }^{m+1}t^{m}}}}${\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.