Local zeta function: Difference between revisions

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In [[number theorymathematics]], the '''local zeta function''' {{math|''Z''(''V'', ''s'')}} (sometimes called the '''congruent zeta function''' or the [[Hasse–Weil zeta function]]) is defined as

:<math>Z(V, s) = \exp\left(\sum_{mk = 1}^\infty \frac{N_mN_k}{mk} (q^{-s})^mk\right)</math>

where {{mvar|V}} is a [[Singular point of an algebraic variety|non-singular]] {{mvar|n}}-dimensional [[projective algebraic variety]] over the field {{math|'''F'''_''q''}} with {{mvar|q}} elements and {{math|''N''_''mk''}} is the number of points of {{mvar|''V''}} defined over the finite field extension {{math|'''F'''_''q''^''mk''}} of {{math|'''F'''_''q''}}.<ref>Section V.2 of {{Citation

| last=Silverman

| first=Joseph H.

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}}</ref>

Making the variable transformation {{math|''ut''&nbsp;{{=}}&nbsp;''q''^−''s'',}} gives

:<math>

\mathit{Z} (V,ut) = \exp

\left( \sum_{mk=1}^{\infty} N_mN_k \frac{ut^mk}{mk} \right)

</math>

as the [[formal power series]] in the variable <math>ut</math>.

Equivalently, the local zeta function is sometimes defined as follows:

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</math>

:<math>

(2)\ \ \frac{d}{dudt} \log \mathit{Z} (V,ut) = \sum_{mk=1}^{\infty} N_mN_k ut^{mk-1}\ .</math>

In other words, the local zeta function {{math|''Z''(''V'',&nbsp;''ut'')}} with coefficients in the [[finite field]] {{math|'''F'''_''q''}} is defined as a function whose [[logarithmic derivative]] generates the number {{math|''N''_''mk''}} of solutions of the equation defining {{mvar|V}} in the degree {{mvar|mk}} extension {{math|'''F'''_''q''^''mk''.}}

<!--In [[number theory]], a '''local zeta function'''

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:<math>[ F_k : F ] = k \,</math>,

for ''k'' = 1, 2, ... . When ''F'' is the unique field with ''q'' elements, ''F_k'' is the unique field with <math>q^k</math> elements. Given a set of polynomial equations &mdash; or an [[algebraic variety]] ''V'' &mdash; defined over ''F'', we can count the number

:<math>N_k \,</math>

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==Motivations==

The relationship between the definitions of ''G'' and ''Z'' can be explained in a number of ways. (See for example the infinite product formula for ''Z'' below.) In practice it makes ''Z'' a [[rational function]] of ''t'', something that is interesting even in the case of ''V'' an [[elliptic curve]] over a finite field.

The local ''Z'' zeta functions are multiplied to get global ''<math>\zeta</math>'' zeta functions.,

<math>\zeta = \prod Z</math>

These generally involve different finite fields (for example the whole family of fields '''Z'''/''p'''''Z''' as ''p'' runs over all [[prime number]]s).

In these fields, the variable ''t'' is substituted by ''p^−s'', where ''s'' is the complex variable traditionally used in [[Dirichlet series]]. (For details see [[Hasse–Weil zeta function]].)

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==Riemann hypothesis for curves over finite fields==

For projective curves ''C'' over ''F'' that are [[Algebraic curve#Singularities|non-singular]], it can be shown that

:<math>Z(t) = \frac{P(t)}{(1 - t)(1 - qt)}\ ,</math>

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For example, for the elliptic curve case there are two roots, and it is easy to show the absolute values of the roots are ''q''^1/2. [[Hasse's theorem on elliptic curves|Hasse's theorem]] is that they have the same absolute value; and this has immediate consequences for the number of points.

[[André Weil]] proved this for the general case, around 1940 (''Comptes Rendus'' note, April 1940): he spent much time in the years after that [[Foundations of Algebraic Geometry|writing]] up the [[algebraic geometry]] involved. This led him to the general [[Weil conjectures]]. [[Alexander Grothendieck]] developed [[scheme (mathematics)|scheme]] theory for the purpose of resolving these.

A generation later [[Pierre Deligne]] completed the proof.

(See [[étale cohomology]] for the basic formulae of the general theory.)

If {{mvar|E}} is an elliptic curve over a finite field with {{mvar|q}} elements, then the number of points of {{mvar|E}} defined over the field with {{math|''q''^''m''}} elements is {{math|1 − ''α''^''m'' − ''β''^''m'' + ''q''^''m''}}, where {{mvar|α}} and {{mvar|β}} are complex conjugates equal to <math>q^\frac{m}{2}e^{\pm i\theta}</math>.

The zeta function is

:<math>\frac{(1-\alpha q^{-s})(1-\beta q^{-s})}{(1-q^{-s})(1-q^{1-s})}</math>.

==General formulas for the zeta function==