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In algebraic geometry, the motivic zeta function of a smooth algebraic variety is the formal power series:[1]
Here is the -th symmetric power of , i.e., the quotient of by the action of the symmetric group , and is the class of in the ring of motives (see below).
If the ground field is finite, and one applies the counting measure to , one obtains the local zeta function of .
If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to , one obtains .
^Marcolli, Matilde (2010). Feynman Motives. World Scientific. p. 115. ISBN 9789814304481. Retrieved 26 April 2023.
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In algebraic geometry, the motiviczetafunction of a smooth algebraic variety X {\displaystyle X} is the formal power series: Z ( X , t ) = ∑ n = 0 ∞...
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could be applied to the counting in local zeta-functions. For later history see motive (algebraic geometry), motivic cohomology. Weil conjectures The Weil...
Voevodsky, Vladimir (2000), "Triangulated categories of motives over a field", Cycles, Transfers, and Motivic Homology Theories, Princeton University Press, pp...
introduced by Goncharov (1995) suggesting that the cohomology of certain motivic complexes coincides with pieces of K-groups. It extends a conjecture due...
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group of the ring of integers of a number field to the field's Dedekind zetafunction. Bombieri–Lang conjectures on densities of rational points of algebraic...
equation for the zeta-function are needed to define the function for all s). The Dedekind zeta-function generalizes the Riemann zeta-function in that ζ Q {\displaystyle...
Dwork proved one of the four Weil conjectures (rationality of the local zetafunction) in 1960. Grothendieck developed étale cohomology theory to prove two...
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Global Information
![{\displaystyle Z(X,t)=\sum _{n=0}^{\infty }[X^{(n)}]t^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/552d3e2a314451de664723246abfeb2bb4c9242c)
![{\displaystyle [X^{(n)}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2219d8e43715930bfd4fa5305ac2f9d9b732027a)
