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    • Source: Motivic zeta function

      • 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

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