Editing Hasse–Weil zeta function (section)

==Definition==
{{No footnotes|section|date=April 2022}}
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 [[Riemann zeta function]], which results from the case when ''V'' is a single point.<ref>{{Cite web |title=The Hasse-Weil Zeta Function of a Quotient Variety |url=https://mast.queensu.ca/~kani/lectures/hwquot.pdf |access-date=2024-04-29 |archive-url=https://web.archive.org/web/20221019203429/https://mast.queensu.ca/~kani/lectures/hwquot.pdf |archive-date=2022-10-19 }}</ref>

Taking the case of ''K'' the [[rational number]] field <math>\mathbb{Q}</math>, and ''V'' a [[Algebraic curve#Singularities|non-singular]] [[projective variety]], we can for [[almost all]] [[prime number]]s ''p'' consider the reduction of ''V'' modulo ''p'', an algebraic variety ''V''<sub>''p''</sub> over the [[finite field]] <math>\mathbb{F}_{p}</math> with ''p'' elements, just by reducing equations for ''V''. [[Scheme (mathematics)|Scheme]]-theoretically, this reduction is just the pullback of the [[Néron model]] of ''V'' along the canonical map Spec <math>\mathbb{F}_{p}</math>  → Spec <math>\mathbb{Z}</math>. Again for almost all ''p'' it will be non-singular. We define a [[Dirichlet series]] of the [[complex variable]] ''s'',

:<math>Z_{V\!,\mathbb{Q}}(s) = \prod_{p} Z_{V\!,\,p}(p^{-s}), </math>

which is the [[infinite product]] of the [[local zeta function]]s

: <math>Z_{V\!,\,p}(p^{-s}) = \exp\left(\sum_{k = 1}^\infty \frac{N_k}{k} (p^{-s})^k\right)</math>

where ''N<sub>k</sub>'' is the number of points of ''V'' defined over the finite field extension <math>\mathbb{F}_{p^k}</math> of <math>\mathbb{F}_{p}</math>.

This <math>Z_{V\!,\mathbb{Q}}(s)</math> is [[well-defined]] only up to multiplication by [[rational function]]s in <math>p^{-s}</math> for finitely many primes ''p''.

Since the indeterminacy is relatively harmless, and has [[meromorphic continuation]] everywhere, there is a sense in which the properties of ''Z(s)'' do not essentially depend on it. In particular, while the exact form of the [[functional equation (L-function)|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 (mathematics)|ramification theory]], 'bad' primes carry good information (theory of the ''conductor''). This manifests itself in the étale theory in the [[Néron–Ogg–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

:<math>\rho(\operatorname{Frob}(p)),</math>

Frob(''p'') being a [[Frobenius element]] for ''p''. What happens at the ramified ''p'' is that ρ is non-trivial on the [[inertia group]] ''I''(''p'') 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 ''Z''(''s'') 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 [[Jean-Pierre Serre|Serre]] and [[Deligne]] in the later 1960s; the functional equation itself has not been proved in general.