- Hasse-Weil zeta function
In
mathematics , the Hasse-Weil zeta-function attached to analgebraic variety "V" defined over anumber field "K" is one of the two most important types ofL-function . Such "L"-functions are called 'global', in that they are defined asEuler product s in terms of local zeta-functions. They form one of the two major classes of global "L"-functions, the other being the "L"-functions associated toautomorphic representation s. Conjecturally there is just one essential type of global "L"-function, with two descriptions (coming from an algebraic variety, coming from an automorphic representation); this would be a vast generalisation of theTaniyama-Shimura conjecture , itself a very deep and recent result (as of 2004 ) innumber theory .The description of the Hasse-Weil zeta-function "up to finitely many factors of its Euler product" is relatively simple. This follows the initial suggestions of
Helmut Hasse andAndré Weil , motivated by the case in which "V" is a single point, and the Riemann zeta-function results.Taking the case of "K" the
rational number field "Q", and "V" anon-singular projective variety , we can foralmost all prime number s "p" consider the reduction of "V" modulo "p", an algebraic variety "V""p" over thefinite field with "p" elements, just by reducing equations for "V". Again for almost all "p" it will be non-singular. We define:
to be the
Dirichlet series of thecomplex variable "s", which is theinfinite product of the local zeta-functions:
Then , according to our definition, is
well-defined only up to multiplication byrational function s in a finite number of .Since the indeterminacy is relatively anodyne, and has
meromorphic continuation everywhere, there is a sense in which the properties of do not essentially depend on it. In particular, while the exact form of the functional equation for "Z"("s"), reflecting in a vertical line in the complex plane, will definitely depend on the 'missing' factors, the existence of some such functional equation does not.A more refined definition (as an
Artin L-function ) became possible with the development ofétale cohomology ; this neatly explains what to do about the missing, 'bad reduction' factors. According to general principles visible inramification theory , 'bad' primes carry good information (theory of the "conductor"). This manifests itself in the étale theory in theOgg-Néron-Shafarevich criterion forgood reduction ; namely that there is good reduction, in a definite sense, at all primes "p" for which theGalois representation ρ on the étale cohomology groups of "V" is "unramified". For those, the definition of local zeta-function can be recovered in terms of thecharacteristic polynomial of:
being a
Frobenius element for "p". What happens at the ramified "p" is that ρ is non-trivial on theinertia group for "p". At those primes the definition must be 'corrected', taking the largest quotient of the representation ρ on which the inertia group acts by thetrivial representation . With this refinement, the definition of can be upgraded successfully from 'almost all' "p" to "all" "p" participating in the Euler product. The consequences for the functional equation were worked out bySerre andDeligne in the later 1960s; the functional equation itself has not been proved in general.References
*
J.-P. Serre , "Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures)", 1969/1970, Sém. Delange-Pisot-Poitou, exposé 19
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