Is there a notion of a zeta function of a morphism? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:32:36Z http://mathoverflow.net/feeds/question/72488 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72488/is-there-a-notion-of-a-zeta-function-of-a-morphism Is there a notion of a zeta function of a morphism? James D. Taylor 2011-08-09T17:07:49Z 2011-08-09T22:04:28Z <p>The Hasse-Weil zeta function is defined only for arithmetic schemes. By an arithmetic scheme I will mean a scheme $X$ together with a morphism of finite type $X\rightarrow S$, where $S$ is an affine Dedekind scheme (a $0$ or $1$ dimensional nonsingular affine scheme). Actually, in the case where $S$ is $0$-dimensional, I believe it is only defined for $S$ being $Spec$ of a finite field.</p> <p>The way the Hasse-Weil zeta function is defined is like so: first you define it for varieties over finite fields, and then if $S$ is one dimensional, you define the zeta function as the product of the zeta function of every fiber.</p> <p>It seems rather arbitrary for it to be defined only in these cases. Is there a definition of a zeta function of a morphism of finite type (or maybe flat?) in general, even when $S$ is $\geq 2$ dimensional? I would be surprised if there isn't, but I've never heard of such an entity.</p>