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 {{mathmvar|''N''_''m''V}} is thea number[[Singular point of pointsan ofalgebraic variety|non-singular]] {{mvar|''V''n}}-dimensional defined[[projective algebraic variety]] over the finite field extension {{math|'''F'''_''q''^''m''}} ofwith {{mvar|q}} elements and {{math|'''F'N''_''qk'',}} andis {{mvar|V}}the isnumber aof [[non-singular]]points of {{mvar|n''V''}}-dimensional [[projective algebraic variety]]defined over the finite field extension {{math|'''F'''_''q''^''k''}} with {{mvar|q}} elements. Making the variable transformationof {{math|''u'F'&nbsp;{{=}}&nbsp;''q''<supsub>−''sq''</supsub>,}}.<ref>Section givesV.2 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>

Making the variable transformation {{math|''t''&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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:<math>Z(t) = \frac{1}{(1 - t)}\ .</math>

To take something more interesting, let ''V'' be the [[projective line]] over ''F''. If ''F'' has ''q'' elements, then this has ''q'' + 1 points, including as we must the one [[point at infinity]]. Therefore, we have

:<math>N_k = q^k + 1</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,

It is the functions ''Z'' that are designed to multiply, to get '''global zeta functions'''. These involve different finite fields (for example the whole family of fields '''Z'''/''p'''''Z''' as ''p'' runs over all [[prime number]]s). In that connection, the variable ''t'' undergoes substitution by ''p^−s'', where ''s'' is the complex variable traditionally used in [[Dirichlet series]]. (For details see [[Hasse–Weil zeta function]].)

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

With that understanding, the products of the ''Z'' in the two cases used as examples come out as <math>\zeta(s)</math> and <math>\zeta(s)\zeta(s-1)</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]].)

WithThe that understanding, theglobal products of the ''Z'' in the two cases used as examples in the previous section therefore come out as <math>\zeta(s)</math> and <math>\zeta(s)\zeta(s-1)</math> after letting <math>q=p</math>.

==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.)

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==References==

{{reflist}}

{{Bernhard Riemann}}

[[Category:Algebraic varieties]]