Motivic zeta function

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, the motivic zeta function of a <a href="/facts/Smooth_algebraic_variety/8nvkrkE9">smooth algebraic variety</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a>:

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 <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> 
 
 
 
 
 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 <a href="/facts/Ground_field/cUNvxI33">ground field</a> is finite, and one applies the counting measure to 
 
 
 
 Z
 (
 X
 ,
 t
 )
 
 
 {\displaystyle Z(X,t)}
 
, one obtains the <a href="/facts/Local_zeta_function/jIu1cg89">local zeta function</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
.
If the ground field is the complex numbers, and one applies <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a> 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 zeta function open-in-new

Motivic zeta function