Editing Hasse–Weil zeta function (section)

==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''.