Hasse–Weil zeta function

<h2 id="definition">Definition</h2>

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 <a href="/facts/Helmut_Hasse/WFtCX2jI">Helmut Hasse</a> and <a href="/facts/Andr%25C3%25A9_Weil/Q4Dd0Gr4">André Weil</a>, motivated by the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>, which results from the case when V is a single point.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
Taking the case of K the <a href="/facts/Rational_number/VJQmyY05">rational number</a> field 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
, and V a <a href="/facts/Algebraic_curve/LnWsPmDP">non-singular</a> <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a>, we can for <a href="/facts/Almost_all/KvHxUCCc">almost all</a> <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> p consider the reduction of V modulo p, an algebraic variety Vp over the <a href="/facts/Finite_field/UkMGJo3m">finite field</a> 
 
 
 
 
 
 F
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {F} _{p}}
 
 with p elements, just by reducing equations for V. <a href="/facts/Scheme_(mathematics)/c7uSqGqL">Scheme</a>-theoretically, this reduction is just the pullback of the <a href="/facts/N%25C3%25A9ron_model/6x7RKoCe">Néron model</a> of V along the canonical map Spec 
 
 
 
 
 
 F
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {F} _{p}}
 
 → Spec 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
. Again for almost all p it will be non-singular. We define a <a href="/facts/Dirichlet_series/uIAUaLcd">Dirichlet series</a> of the <a href="/facts/Complex_variable/hgCoQlH0">complex variable</a> s,

Z
          
            V
            
            ,
            
              Q
            
          
        
        (
        s
        )
        =
        
          ∏
          
            p
          
        
        
          Z
          
            V
            
            ,
            
            p
          
        
        (
        
          p
          
            −
            s
          
        
        )
        ,
      
    
    {\displaystyle Z_{V\!,\mathbb {Q} }(s)=\prod _{p}Z_{V\!,\,p}(p^{-s}),}

which is the <a href="/facts/Infinite_product/qaaNh5PM">infinite product</a> of the <a href="/facts/Local_zeta_function/jIu1cg89">local zeta functions</a>

Z
          
            V
            
            ,
            
            p
          
        
        (
        
          p
          
            −
            s
          
        
        )
        =
        exp
        ⁡
        
          (
          
            
              ∑
              
                k
                =
                1
              
              
                ∞
              
            
            
              
                
                  N
                  
                    k
                  
                
                k
              
            
            (
            
              p
              
                −
                s
              
            
            
              )
              
                k
              
            
          
          )
        
      
    
    {\displaystyle Z_{V\!,\,p}(p^{-s})=\exp \left(\sum _{k=1}^{\infty }{\frac {N_{k}}{k}}(p^{-s})^{k}\right)}

where Nk is the number of points of V defined over the finite field extension 
 
 
 
 
 
 F
 
 
 
 p
 
 k
 
 
 
 
 
 
 {\displaystyle \mathbb {F} _{p^{k}}}
 
 of 
 
 
 
 
 
 F
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {F} _{p}}
 
.
This 
 
 
 
 
 Z
 
 V
 
 ,
 
 Q
 
 
 
 (
 s
 )
 
 
 {\displaystyle Z_{V\!,\mathbb {Q} }(s)}
 
 is <a href="/facts/Well-defined/GAc0XbWz">well-defined</a> only up to multiplication by <a href="/facts/Rational_function/2nJGu0Ko">rational functions</a> in 
 
 
 
 
 p
 
 −
 s
 
 
 
 
 {\displaystyle p^{-s}}
 
 for finitely many primes p.
Since the indeterminacy is relatively harmless, and has <a href="/facts/Meromorphic_continuation/aBCuEgvt">meromorphic continuation</a> 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 <a href="/facts/Functional_equation_(L-function)/qKa0wBC5">functional equation</a> 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 <a href="/facts/%25C3%2589tale_cohomology/KXJ8a6Hw">étale cohomology</a>; this neatly explains what to do about the missing, 'bad reduction' factors. According to general principles visible in <a href="/facts/Ramification_(mathematics)/y4mDBqjo">ramification theory</a>, 'bad' primes carry good information (theory of the conductor). This manifests itself in the étale theory in the <a href="/facts/N%25C3%25A9ron%25E2%2580%2593Ogg%25E2%2580%2593Shafarevich_criterion/flRLUXf3">Néron–Ogg–Shafarevich criterion</a> for <a href="/facts/Good_reduction/MlmGW0Gm">good reduction</a>; namely that there is good reduction, in a definite sense, at all primes p for which the <a href="/facts/Galois_representation/fYDgqs8d">Galois representation</a> 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 <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> of

ρ
        (
        Frob
        ⁡
        (
        p
        )
        )
        ,
      
    
    {\displaystyle \rho (\operatorname {Frob} (p)),}

Frob(p) being a <a href="/facts/Frobenius_element/b9hkmc97">Frobenius element</a> for p. What happens at the ramified p is that ρ is non-trivial on the <a href="/facts/Inertia_group/UHXcYyGh">inertia group</a> 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 <a href="/facts/Trivial_representation/2o9Fvjuj">trivial representation</a>. 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 <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre</a> and <a href="/facts/Deligne/wYC8fhfq">Deligne</a> in the later 1960s; the functional equation itself has not been proved in general.

<h2 id="hasseweil-conjecture">Hasse–Weil conjecture</h2>
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 <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>. For elliptic curves over the rational numbers, the Hasse–Weil conjecture follows<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> from the <a href="/facts/Modularity_theorem/aI3HRxwj">modularity theorem</a>: each elliptic curve E over 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 is modular.

<h2 id="birch-and-swinnerton-dyer-conjecture">Birch and Swinnerton-Dyer conjecture</h2>
The <a href="/facts/Birch_and_Swinnerton-Dyer_conjecture/Issh7Yfr">Birch and Swinnerton-Dyer conjecture</a> states that the <a href="/facts/Rank_of_an_abelian_group/zo0y8Mj4">rank</a> of the <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> E(K) of points of an elliptic curve E is the order of the zero of the Hasse–Weil L-function L(E, s) at s = 1, and that the first non-zero coefficient in the <a href="/facts/Taylor_expansion/M0aTuftH">Taylor expansion</a> of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> The conjecture is one of the seven <a href="/facts/Millennium_Prize_Problems/WWSB5ddp">Millennium Prize Problems</a> listed by the <a href="/facts/Clay_Mathematics_Institute/9sCIUpD7">Clay Mathematics Institute</a>, which has offered a $1,000,000 prize for the first correct proof.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="elliptic-curves-over-q">Elliptic curves over Q</h2>
An elliptic curve is a specific type of variety. Let E be an <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curve over Q</a> of <a href="/facts/Conductor_of_an_abelian_variety/uqvn5gT8">conductor</a> N. Then, E has good reduction at all primes p not dividing N, it has <a href="/facts/Semistable_elliptic_curve/d6Tf9rtC">multiplicative reduction</a> at the primes p that exactly divide N (i.e. such that p divides N, but p2 does not; this is written p || N), and it has additive reduction elsewhere (i.e. at the primes where p2 divides N). The Hasse–Weil zeta function of E then takes the form

Z
          
            V
            
            ,
            
              Q
            
          
        
        (
        s
        )
        =
        
          
            
              ζ
              (
              s
              )
              ζ
              (
              s
              −
              1
              )
            
            
              L
              (
              E
              ,
              s
              )
            
          
        
        .
        
      
    
    {\displaystyle Z_{V\!,\mathbb {Q} }(s)={\frac {\zeta (s)\zeta (s-1)}{L(E,s)}}.\,}

Here, ζ(s) is the usual <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> and L(E, s) is called the L-function of E/Q, which takes the form<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

L
        (
        E
        ,
        s
        )
        =
        
          ∏
          
            p
          
        
        
          L
          
            p
          
        
        (
        E
        ,
        s
        
          )
          
            −
            1
          
        
        
      
    
    {\displaystyle L(E,s)=\prod _{p}L_{p}(E,s)^{-1}\,}

where, for a given prime p,

L
          
            p
          
        
        (
        E
        ,
        s
        )
        =
        
          
            {
            
              
                
                  (
                  1
                  −
                  
                    a
                    
                      p
                    
                  
                  
                    p
                    
                      −
                      s
                    
                  
                  +
                  
                    p
                    
                      1
                      −
                      2
                      s
                    
                  
                  )
                  ,
                
                
                  
                    if 
                  
                  p
                  ∤
                  N
                
              
              
                
                  (
                  1
                  −
                  
                    a
                    
                      p
                    
                  
                  
                    p
                    
                      −
                      s
                    
                  
                  )
                  ,
                
                
                  
                    if 
                  
                  p
                  ∣
                  N
                  
                     and 
                  
                  
                    p
                    
                      2
                    
                  
                  ∤
                  N
                
              
              
                
                  1
                  ,
                
                
                  
                    if 
                  
                  
                    p
                    
                      2
                    
                  
                  ∣
                  N
                
              
            
            
          
        
      
    
    {\displaystyle L_{p}(E,s)={\begin{cases}(1-a_{p}p^{-s}+p^{1-2s}),&{\text{if }}p\nmid N\\(1-a_{p}p^{-s}),&{\text{if }}p\mid N{\text{ and }}p^{2}\nmid N\\1,&{\text{if }}p^{2}\mid N\end{cases}}}

where in the case of good reduction ap is p + 1 − (number of points of E mod p), and in the case of multiplicative reduction ap is ±1 depending on whether E has split (plus sign) or non-split (minus sign) multiplicative reduction at p. A multiplicative reduction of curve E by the prime p is said to be split if -c6 is a square in the finite field with p elements.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
There is a useful relation not using the conductor:

<ol><li>If p doesn't divide 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 (where 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 is the <a href="/facts/Elliptic_curve/60jvaHSA">discriminant</a> of the elliptic curve) then E has good reduction at p.</li>
<li>If p divides 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 but not 
 
 
 
 
 c
 
 4
 
 
 
 
 {\displaystyle c_{4}}
 
 then E has multiplicative bad reduction at p.</li>
<li>If p divides both 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 and 
 
 
 
 
 c
 
 4
 
 
 
 
 {\displaystyle c_{4}}
 
 then E has additive bad reduction at p.</li></ol>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Arithmetic_zeta_function/yP1wFGnK">Arithmetic zeta function</a></li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">J.-P. Serre</a>, Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures), 1969/1970, Sém. Delange–Pisot–Poitou, exposé 19</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"The Hasse-Weil Zeta Function of a Quotient Variety" (PDF). Archived from the original (PDF) on 2022-10-19. Retrieved 2024-04-29. <a href="https://web.archive.org/web/20221019203429/https://mast.queensu.ca/~kani/lectures/hwquot.pdf" target="_blank">https://web.archive.org/web/20221019203429/https://mast.queensu.ca/~kani/lectures/hwquot.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Milne, James S. (2016). "The Riemann hypothesis over finite fields: from Weil to the present day". Notices of the International Congress of Chinese Mathematicians. 4 (2): 14–52. arXiv:1509.00797. doi:10.4310/ICCM.2016.v4.n2.a4. <a href="https://doi.org/10.4310%2FICCM.2016.v4.n2.a4" target="_blank">https://doi.org/10.4310%2FICCM.2016.v4.n2.a4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Wiles, Andrew (2006). "The Birch and Swinnerton-Dyer conjecture" (PDF). In Carlson, James; Jaffe, Arthur; Wiles, Andrew (eds.). The Millennium prize problems. American Mathematical Society. pp. 31–44. ISBN 978-0-8218-3679-8. MR 2238272. Archived from the original (PDF) on 2018-03-29. Retrieved 2022-04-13. <a href="978-0-8218-3679-8" target="_blank">978-0-8218-3679-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Birch and Swinnerton-Dyer Conjecture at Clay Mathematics Institute <a href="http://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture" target="_blank">http://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Section C.16 of Silverman, Joseph H. (1992), The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, New York: Springer-Verlag, ISBN 978-0-387-96203-0, MR 1329092 <a href="978-0-387-96203-0" target="_blank">978-0-387-96203-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">"Number theory - Testing to see if $\ell$ is of split or nonsplit multiplicative reduction". <a href="https://math.stackexchange.com/questions/313170/testing-to-see-if-ell-is-of-split-or-nonsplit-multiplicative-reduction" target="_blank">https://math.stackexchange.com/questions/313170/testing-to-see-if-ell-is-of-split-or-nonsplit-multiplicative-reduction</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Hasse–Weil zeta function open-in-new

Hasse–Weil zeta function