Local zeta function: Difference between revisions

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In [[number theorymathematics]], the '''local zeta function''' <{{math>|''Z''(''V'',&nbsp;''s'')</math>}} (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''_''k''}} is the number of points of {{mvar|''V''}} defined over the finite field extension {{math|'''F'''_''q''^''k''}} of {{math|'''F'''_''q''}}.<ref>Section V.2 of {{Citation

where <math>N_m</math> is the number of points of <math>V</math> defined over the degree <math>m</math> extension{{explain|date=September 2017}} <math>\mathbf{F}_{q^m}</math> of <math>\mathbf{F}_q</math>, and <math>V</math> is a [[non-singular]] <math>n</math>-dimensional [[projective algebraic variety]] over the field <math>\mathbf{F}_q</math> with <math>q</math> elements. By the variable transformation <math>u=q^{-s}</math>, then it is defined by

| 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]] ofin the variable <math>ut</math>.

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

as the [[formal power series]] of the variable <math>u</math>.

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

:<math>

(1)\ \ \mathit{Z} (V,0) = 1 \,

</math>

:<math>

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

In other wordwords, the local zeta function <{{math>|''Z''(''V'',u&nbsp;''t'')</math>}} with coefficients in the [[finite field]] <math>\mathbf{{math|'''F}_q'''<sub>''q''</mathsub>}} is defined as a function whose [[logarithmic derivative]] generates the numbersnumber {{math|''N''_m''k''''}} of the solutions of the equation, defining ''{{mvar|V'',}} in the ''m''degree degree{{mvar|k}} extension <math>\mathbf{F}_{math|'''F'''_{''q^m}''<sup>''k''</mathsup>}.}}

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

:<math>Z(-t)</math>

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==Formulation==

Given a finite field ''F'', there is, up to [[isomorphism]], justonly one field ''F_k'' with

:<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>G(t) = N_1t +N_2t^2/2 + N_3t^3/3 +\cdots \,</math>.

The correct definition for ''Z''(''t'') is to makeset log ''Z'' equal to ''G'', and so

:<math>Z= \exp (G(t)) \, </math>

we will haveand ''Z''(0) = 1, since ''G''(0) = 0, and ''Z''(''t'') is ''a priori'' a [[formal power series]].

Note that theThe [[logarithmic derivative]]

:<math>Z'(t)/Z(t) \,</math>

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==Examples==

For example, assume all the ''N_k'' are 1; this happens for example if we start with an equation like ''X'' = 0, so that geometrically we are taking ''V'' to be a point. Then

:<math>G(t) = -\log(1 - t)</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 shall have

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

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:<math>G(t) = -\log(1 - t) -\log(1 - qt)</math>

for |''t''| small enough., and therefore

In this case we have

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

The first study of these functions was in the 1923 dissertation of [[Emil Artin]]. He obtained results for the case of a [[hyperelliptic curve]], and conjectured the further main points of the theory as applied to curves. The theory was then developed by [[F. K. Schmidt]] and [[Helmut Hasse]].<ref>[[Daniel Bump]], ''Algebraic Geometry'' (1998), p. 195.</ref> The earliest known non-trivialnontrivial cases of local zeta- functions were implicit in [[Carl Friedrich Gauss]]'s ''[[Disquisitiones Arithmeticae]]'', article 358;. thereThere, certain particular examples of [[elliptic curve]]s over finite fields having [[complex multiplication]] have their points counted by means of [[cyclotomy]].<ref>[[Barry Mazur]], ''Eigenvalues of Frobenius'', p. 244 in ''Algebraic Geometry, Arcata 1974: Proceedings American Mathematical Society'' (1974).</ref>

For the definition and some examples, see also.<ref>[[Robin Hartshorne]], ''Algebraic Geometry'', p. 449 Springer 1977 APPENDIX C "The Weil Conjectures"</ref>

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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'''. Those 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|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>

with ''P''(''t'') a polynomial, of degree 2''g'', where ''g'' is the [[genus (mathematics)|genus]] of ''C''. Rewriting

:<math>P(t)=\prod^{2g}_{i=1}(1-\omega_i t)\ ,</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 the [[scheme (mathematics)|scheme]] theory for the sakepurpose of resolving itthese.
A andgeneration finally,later [[Pierre Deligne]] hadcompleted proved a generationthe laterproof.
(See [[étale cohomology]] for the basic formulae of the general theory.)

==General formulas for the zeta function==

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It is a consequence of the [[Lefschetz trace formula]] for the [[Frobenius morphism]] that

:<math>Z(X,t)=\prod_{i=0}^{2\dim X}\det\big(1-t \mbox{Frob}_q |H^i_c(\overline{X},{\Bbbmathbb Q}_\ell)\big)^{(-1)^{i+1}}.</math>

Here <math>X</math> is a separated scheme of finite type over the finite field ''F'' with <math>q</math> elements, and Frob_q is the geometric Frobenius acting on <math>\ell</math>-adic étale cohomology with compact supports of <math>\overline{X}</math>, the lift of <math>X</math> to the algebraic closure of the field ''F''. This shows that the zeta function is a rational function of <math>t</math>.

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==References==

{{reflist}}

{{Bernhard Riemann}}

[[Category:Algebraic varieties]]