| Hasse-weil L-function |
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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 and André 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'' a Non-singular Projective Variety , we can for Almost All Prime Number s ''p'' consider the reduction of ''V'' modulo ''p'', an algebraic variety ''V''''p'' over the Finite 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 the Complex Variable ''s'', which is the Infinite Product of the local zeta-functions : Then , according to our definition, is Well-defined only up to multiplication by Rational 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 became possible with the development of étale Cohomology ; this neatly explains what to do about the missing, 'bad reduction' factors. According to general principles visible in Ramification Theory , 'bad' primes carry good information (theory of the ''conductor''). This manifests itself in the étale theory in the Ogg-Néron-Shafarevich Criterion for Good Reduction ; namely that there is good reduction, in a definite sense, at all primes ''p'' for which the Galois 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 the Characteristic Polynomial of : being a Frobenius Element for ''p''. What happens at the ramified ''p'' is that ρ is non-trivial on the Inertia 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 the Trivial 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 by Serre and Deligne in the later 1960s ; the functional equation itself has not been proved in general. REFERENCE
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