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Motivic zeta function
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<h2 id="motivic-measures">Motivic measures</h2> <p>A motivic measure is a map μ {\displaystyle \mu } from the set of finite type <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a> over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> k {\displaystyle k} to a commutative <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> A {\displaystyle A} , satisfying the three properties </p> 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})} . <p>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 <i>counting measure</i>. </p><p>If the ground field is the complex numbers, then Euler characteristic with compact supports defines a motivic measure with values in the integers. </p><p>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 </p> 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}} . <p>There is a <i>universal motivic measure</i>. 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 </p> [ 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}]} . <p>The universal motivic measure gives rise to the motivic zeta function. </p> <h2 id="examples">Examples</h2> <p>Let L = [ A 1 ] {\displaystyle \mathbb {L} =[{\mathbb {A} }^{1}]} denote the class of the <a href="/facts/Affine_space/tlvI3oX1">affine line</a>. </p> 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}}} <p>If X {\displaystyle X} is a smooth projective irreducible <a href="/facts/Algebraic_curve/LnWsPmDP">curve</a> of <a href="/facts/Genus_(curve)/o1Yhf8ty">genus</a> g {\displaystyle g} admitting a <a href="/facts/Line_bundle/CupAl1ft">line bundle</a> of degree 1, and the motivic measure takes values in a field in which L {\displaystyle {\mathbb {L} }} is invertible, then </p> Z ( X , t ) = P ( t ) ( 1 − t ) ( 1 − L t ) , {\displaystyle Z(X,t)={\frac {P(t)}{(1-t)(1-{\mathbb {L} }t)}}\,,} <p>where P ( t ) {\displaystyle P(t)} is a polynomial of degree 2 g {\displaystyle 2g} . Thus, in this case, the motivic zeta function is <a href="/facts/Rational_function/2nJGu0Ko">rational</a>. In higher dimension, the motivic zeta function is not always rational. </p><p>If S {\displaystyle S} is a smooth <a href="/facts/Algebraic_surface/2zHU0o9J">surface</a> over an algebraically closed field of characteristic 0 {\displaystyle 0} , then the generating function for the motives of the <a href="/facts/Hilbert_scheme/kJcbts4e">Hilbert schemes</a> of S {\displaystyle S} can be expressed in terms of the motivic zeta function by <i><a href="/facts/Lothar_G%25C3%25B6ttsche/T3sTQP5V">Göttsche's</a> Formula</i> </p> ∑ 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})} <p>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 </p> ∑ 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}}}} <p>This is essentially the <a href="/facts/Partition_function_(number_theory)/mBUDy5vU">partition function</a>. </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Marcolli, Matilde (2010). Feynman Motives. World Scientific. p. 115. ISBN 9789814304481. Retrieved 26 April 2023. <a href="9789814304481" target="_blank">9789814304481</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>