Editing Hasse–Weil zeta function

{{Short description|Mathematical function associated to algebraic varieties}}
In [[mathematics]], the '''Hasse–Weil zeta function''' attached to an [[algebraic variety]] ''V'' defined over an [[algebraic number field]] ''K'' is a [[meromorphic function]] on the [[complex plane]] defined in terms of the number of points on the variety after reducing modulo each prime number ''p''. It is a global [[L-function|''L''-function]] defined as an [[Euler product]] of [[local zeta function]]s.

Hasse–Weil ''L''-functions form one of the two major classes of global ''L''-functions, alongside the ''L''-functions associated to [[automorphic representations]]. Conjecturally, these two types of global ''L''-functions are actually two descriptions of the same type of global ''L''-function; this would be a vast generalisation of the [[Taniyama-Weil conjecture]], itself an important result in [[number theory]].

For an [[elliptic curve]] over a number field ''K'', the Hasse–Weil zeta function is conjecturally related to the [[Group (mathematics)|group]] of [[rational point]]s of the elliptic curve over ''K'' by the [[Birch and Swinnerton-Dyer conjecture]].

==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.

==Hasse–Weil conjecture==

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<ref name="ThesisProblem">{{cite journal | last=Milne | first=James S. | title=The Riemann hypothesis over finite fields: from Weil to the present day | journal=Notices of the International Congress of Chinese Mathematicians | volume=4 | issue=2 | date=2016 | doi=10.4310/ICCM.2016.v4.n2.a4 | doi-access=free | pages=14–52| arxiv=1509.00797 }}</ref> from the [[modularity theorem]]: each elliptic curve {{mvar|E}} over <math>\Q</math> is modular.

==Birch and Swinnerton-Dyer conjecture==
The [[Birch and Swinnerton-Dyer conjecture]] states that the [[Rank of an abelian group|rank]] of the [[abelian group]] ''E''(''K'') of points of an elliptic curve ''E'' is the order of the zero of the Hasse–Weil ''L''-function ''L''(''E'',&nbsp;''s'') at ''s'' = 1, and that the first non-zero coefficient in the [[Taylor expansion]] of ''L''(''E'',&nbsp;''s'') at ''s'' = 1 is given by more refined arithmetic data attached to ''E'' over ''K''.<ref>{{Cite encyclopedia
| last=Wiles
| first=Andrew
| author-link=Andrew Wiles
| chapter=The Birch and Swinnerton-Dyer conjecture
| editor1-last=Carlson
| editor1-first=James
| editor2-last=Jaffe
| editor2-first=Arthur
| editor2-link=Arthur Jaffe
| editor3-last=Wiles
| editor3-first=Andrew
| editor3-link=Andrew Wiles
| title=The Millennium prize problems
| publisher=American Mathematical Society
| year=2006
| isbn=978-0-8218-3679-8
| chapter-url=http://www.claymath.org/sites/default/files/birchswin.pdf
| pages=31–44
| mr=2238272
| access-date=2022-04-13
| archive-date=2018-03-29
| archive-url=https://web.archive.org/web/20180329033023/http://www.claymath.org/sites/default/files/birchswin.pdf
| url-status=dead
}}</ref> The conjecture is one of the seven [[Millennium Prize Problems]] listed by the [[Clay Mathematics Institute]], which has offered a $1,000,000 prize for the first correct proof.<ref>[http://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture Birch and Swinnerton-Dyer Conjecture] at Clay Mathematics Institute</ref>

==Elliptic curves over Q==
An elliptic curve is a specific type of variety. Let ''E'' be an [[Elliptic curve#Elliptic curves over the rational numbers|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

:<math>Z_{V\!,\mathbb{Q}}(s)= \frac{\zeta(s)\zeta(s-1)}{L(E,s)}. \,</math>

Here, ζ(''s'') is the usual [[Riemann zeta function]] and ''L''(''E'', ''s'') is called the ''L''-function of ''E''/'''Q''', which takes the form<ref>Section C.16 of {{Citation
| last=Silverman
| first=Joseph H.
| author-link=Joseph H. Silverman
| title=The arithmetic of elliptic curves
| publisher=[[Springer-Verlag]]
| location=New York
| series=[[Graduate Texts in Mathematics]]
| isbn=978-0-387-96203-0
| mr=1329092
| year=1992
| volume=106
}}</ref>

:<math>L(E,s)=\prod_pL_p(E,s)^{-1} \,</math>

where, for a given prime ''p'',

:<math>L_p(E,s)=\begin{cases}
            (1-a_pp^{-s}+p^{1-2s}), & \text{if } p\nmid N \\
            (1-a_pp^{-s}), & \text{if }p\mid N \text{ and } p^2 \nmid N \\
            1, & \text{if }p^2\mid N
       \end{cases}</math>

where in the case of good reduction ''a''<sub>''p''</sub> is ''p''&nbsp;+&nbsp;1&nbsp;&minus;&nbsp;(number of points of ''E''&nbsp;mod&nbsp;''p''), and in the case of multiplicative reduction ''a''<sub>''p''</sub> is ±1 depending on whether ''E'' has split (plus sign) or non-split (minus sign) multiplicative reduction at&nbsp;''p''. A multiplicative reduction of curve ''E'' by the prime ''p'' is said to be split if -c<sub>6</sub> is a square in the finite field with p elements.<ref>{{cite web | url=https://math.stackexchange.com/questions/313170/testing-to-see-if-ell-is-of-split-or-nonsplit-multiplicative-reduction | title=Number theory - Testing to see if $\ell$ is of split or nonsplit multiplicative reduction }}</ref>

There is a useful relation not using the conductor:

# If ''p'' doesn't divide <math>\Delta</math> (where <math>\Delta</math> is the [[Elliptic_curve#Elliptic_curves_over_the_real_numbers|discriminant]] of the elliptic curve) then ''E'' has good reduction at ''p''.
# If ''p'' divides <math>\Delta</math> but not <math>c_4</math> then ''E'' has multiplicative bad reduction at ''p''.
# If ''p'' divides both <math>\Delta</math> and <math>c_4</math> then ''E'' has additive bad reduction at ''p''.

==See also==

*[[Arithmetic zeta function]]

==References==
<references/>

==Bibliography==
*[[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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[[Category:Zeta and L-functions]]
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