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| In [[mathematics]], the '''Hasse–Weil zeta function''' attached to an [[algebraic variety]] ''V'' defined over an [[algebraic number field]] ''K'' is one of the two most important types of [[L-function]]. Such ''L''-functions are called 'global', in that they are defined as [[Euler product]]s in terms of [[local zeta-function|local zeta functions]]. They form one of the two major classes of global ''L''-functions, the other being the ''L''-functions associated to [[automorphic representations]]. 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 the [[Taniyama–Shimura conjecture]], itself a very deep and recent result ({{As of|2009|lc=on}}) in [[number theory]].
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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.
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| 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''<sub>''p''</sub> over the [[finite field]] '''F'''<sub>''p''</sub> with ''p'' elements, just by reducing equations for ''V''. Again for almost all ''p'' it will be non-singular. We define
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| :<math>Z_{V,Q}(s)</math>
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| to be the [[Dirichlet series]] of the [[complex variable]] ''s'', which is the [[infinite product]] of the [[local zeta-function|local zeta functions]]
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| :<math>\zeta_{V,p}\left(p^{-s}\right).</math> | |
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| Then ''Z''(''s''), according to our definition, is [[well-defined]] only up to multiplication by [[rational function]]s in a finite number of <math>p^{-s}</math>.
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| 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.
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| 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|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
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| :<math>\rho(\operatorname{Frob}(p)),</math> | |
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| 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.
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| ==Example: elliptic curve over Q==
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| Let ''E'' be an [[Elliptic curve#Elliptic curves over a general field|elliptic curve over '''Q''']] of [[Conductor of an abelian variety|conductor]] ''N''. Then, ''E'' has good reduction at all primes ''p'' not dividing ''N'', it has [[Semistable elliptic curve|multiplicative reduction]] at the primes ''p'' that ''exactly'' divide ''N'' (i.e. such that ''p'' divides ''N'', but ''p''<sup>2</sup> does not; this is written ''p'' || ''N''), and it has [[additive reduction]] elsewhere (i.e. at the primes where ''p''<sup>2</sup> divides ''N''). The Hasse–Weil zeta function of ''E'' then takes the form
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| :<math>Z_{E,Q}(s)= \frac{\zeta(s)\zeta(s-1)}{L(s,E)}. \,</math>
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| Here, ζ(''s'') is the usual [[Riemann zeta function]] and ''L''(''s'', ''E'') is called the ''L''-function of ''E''/'''Q''', which takes the form<ref>Section C.16 of {{Citation
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| | last=Silverman
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| | first=Joseph H.
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| | author-link=Joseph H. Silverman
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| | title=The arithmetic of elliptic curves
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| | publisher=[[Springer-Verlag]]
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| | location=New York
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| | series=[[Graduate Texts in Mathematics]]
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| | isbn=978-0-387-96203-0
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| | id={{MathSciNet | id = 1329092}}, ISBN 978-3-540-96203-8
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| | year=1992
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| | volume=106
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| }}</ref>
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| :<math>L(s,E)=\prod_pL_p(s,E)^{-1}\,</math>
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| where, for a given prime ''p'',
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| :<math>L_p(s,E)=\begin{cases}
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| (1-a_pp^{-s}+p^{1-2s}), & \text{if }p\nmid N \\
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| (1-a_pp^{-s}), & \text{if }p\|N \\
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| 1, & \text{if }p^2|N
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| \end{cases}</math>
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| where, in the case of good reduction ''a''<sub>''p''</sub> is ''p'' + 1 − (number of points of ''E'' mod ''p''), and in the case of multiplicative reduction ''a''<sub>''p''</sub> is ±1 depending on whether ''E'' has split or non-split multiplicative reduction at ''p''.
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| ==Hasse–Weil conjecture==
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| The Hasse–Weil conjecture states that the Hasse–Weil zeta function should extend to a meromorphic function for all complex ''s'', and should satisfy a functional equation similar to that of the [[Riemann zeta function]]. For elliptic curves over the rational numbers, the Hasse–Weil conjecture follows from the [[modularity theorem]].
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| ==See also==
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| *[[Arithmetic zeta function]]
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| ==References==
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| <references/>
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| ==Bibliography==
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| *[[Jean-Pierre Serre|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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| {{L-functions-footer}}
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| {{DEFAULTSORT:Hasse-Weil zeta function}}
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| [[Category:Zeta and L-functions]]
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| [[Category:Algebraic geometry]]
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Hello friend. Allow me introduce myself. I am Ron but I don't like when people use my full title. Interviewing is what she does. Camping is something that I've done for many years. Arizona is her birth location and she will by no means transfer.
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