Motivic zeta function explained

X is the formal power series:^[1] Z(X,t) = \sum ^{inf ty} _{n =0} [X ^{(n)} ]t ^{n} Here X ^{(n)} is the n -th symmetric power of X , i.e., the quotient of X ^{n} by the action of the symmetric group S _{n} , and [X ^{(n)} ] is the class of 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) , one obtains the local zeta function of X .

If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to

Z(X,t) , one obtains 1 /( 1- t) ^{\chi(X)} .

Motivic measures

A motivic measure is a map

\mu from the set of finite type schemes over a field k to a commutative ring A , satisfying the three properties\mu(X) depends only on the isomorphism class of X ,\mu(X) = \mu(Z) + \mu(X\setminus Z) if Z is a closed subscheme of X ,\mu(X _{1 x} X _{2)=\mu(X} _{1)\mu(X} _{2)} .For example if k is a finite field and A = { Z} is the ring of integers, then \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

\mu is the formal power series in A[[t]] given byZ _{\mu(X,t)=\sum} ^{inf ty\mu(X} ^{n =0} ^{(n)} )t ^{n} .

There is a universal motivic measure. It takes values in the K-ring of varieties,

A = K(V) , which is the ring generated by the symbols [X] , for all varieties X , subject to the relations[X'] = [X] if X' and X are isomorphic,[X] = [Z] + [X\setminus Z] if Z is a closed subvariety of X ,[X _{1 x} X _{2]=[X} _{1] ⋅ [X} _{2]} . The universal motivic measure gives rise to the motivic zeta function.

Examples

Let

L=[{ A} ^{1]} denote the class of the affine line.Z({ A},t)= (1)/(1-{ L) t} Z({ A} ^{n,t) = (1)/(1- { L)} ^{n} t} Z({ P} ^{n (1)/(1- { L)} _{i =0} ^{i} t}

If

X is a smooth projective irreducible curve of genus g admitting a line bundle of degree 1, and the motivic measure takes values in a field in which { L} is invertible, then Z(X,t) = (P(t))/(( 1- t)( 1- { L) t)}, where P(t) is a polynomial of degree 2 g . Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.

If

S is a smooth surface over an algebraically closed field of characteristic 0 , then the generating function for the motives of the Hilbert schemes of S can be expressed in terms of the motivic zeta function by Göttsche's Formula\sum ^{inf ty[S} _{n =0} ^{[n]} ]t ^{inf ty} _{m =1} Z(S,{ L} ^{m-1} t ^{m)}

Here

S ^{[n]} is the Hilbert scheme of length n subschemes of S . For the affine plane this formula gives\sum ^{inf ty[({ A}} _{n =0} ^{2)} ^{[n]} ]t ^{inf ty} _{m =1} (1)/(1- { L) ^{m+1} t ^{m}}

This is essentially the partition function.

Notes and References

1. Book: Marcolli . Matilde . Feynman Motives . 2010 . World Scientific . 9789814304481 . 115 . 26 April 2023.