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


:''Z''(''-t'')


is a function whose [[logarithmic derivative]] is a [[generating function]]
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for the number of solutions of a set of equations defined over a [[finite field]] ''F'', in extension fields ''F<sub>k</sub>'' of ''F''.  
 
==Formulation==
 
Given ''F'', there is, up to [[isomorphism]], just one field ''F<sub>k</sub>'' with  
 
:<math>[ F_k : F ] = k \,</math>,
 
for ''k'' = 1, 2, ... . 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>
 
of solutions in ''F<sub>k</sub>'' and create the generating function
 
:<math>G(t) = N_1t +N_2t^2/2 + N_3t^3/3 +\cdots \,</math>.
 
The correct definition for ''Z''(''t'') is to make log ''Z'' equal to ''G'', and so
 
:<math>Z= \exp (G(t)) \, </math>
 
we will have ''Z''(0) = 1 since ''G''(0) = 0, and ''Z''(''t'') is ''a priori'' a [[formal power series]].
 
Note that the [[logarithmic derivative]]
 
:<math>Z'(t)/Z(t) \,</math>
 
equals the generating function
 
:<math>G'(t) = N_1 +N_2t^1 + N_3t^2 +\cdots \,</math>.
 
==Examples==
 
For example, assume all the ''N<sub>k</sub>'' are 1; this happens for example if we start with an equation like ''X'' = 0, so that geometrically we are taking ''V'' a point. Then
 
:''G''(''t'') = &minus;log(1 &minus; ''t'')
 
is the expansion of a logarithm (for |''t''| < 1). In this case we have
 
:Z(''t'') = 1/(1 &minus; ''t'').  
 
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
 
:''N<sub>k</sub>'' = ''q<sup>k</sup>'' + 1
 
and  
 
:G(''t'') = &minus;log(1 &minus; ''t'') &minus; log(1 &minus; ''qt''),
 
for |''t''| small enough.
 
In this case we have
 
:''Z''(''t'') = 1/{(1 &minus; ''t'')(1 &minus; ''qt'')}.
 
The first study of these functions was in the 1923 dissertation of [[Emil Artin]]. He obtained results for the case of [[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-trivial cases of local zeta-functions were implicit in [[Carl Friedrich Gauss]]'s ''[[Disquisitiones Arithmeticae]]'', article 358; there 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>
 
==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  finite field.
 
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<sup>-s</sup>'', where ''s'' is the complex variable traditionally used in [[Dirichlet series]]. (For details see [[Hasse-Weil zeta function|Hasse-Weil zeta-function]].)
 
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>.
 
==Riemann hypothesis for curves over finite fields==
 
For projective curves ''C'' over ''F'' that are [[non-singular]], it can be shown that
 
:''Z''(''t'') = ''P''(''t'')/{(1 &minus; ''t'')(1 &minus; ''qt'')},  
 
with ''P''(''t'') a polynomial, of degree 2''g'' where ''g'' is the [[genus (mathematics)|genus]] of ''C''. The '''Riemann hypothesis for curves over finite fields''' states that the roots of ''P'' have [[absolute value]]
 
:''q''<sup>&minus;1/2</sup>,  
 
where ''q'' = |''F''|.
 
For example, for the elliptic curve case there are two roots, and it is easy to show their product is ''q''<sup>&minus;1</sup>. [[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 writing up the [[algebraic geometry]] involved. This led him to the general [[Weil conjectures]], finally proved a generation later. See [[étale cohomology]] for the basic formulae of the general theory.
 
==General formulas for the zeta function==
 
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},{\Bbb 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<sub>q</sub> 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>.
 
An infinite product formula for  <math>Z(X, t)</math> is
 
:<math>Z(X, t)=\prod\ (1-t^{\deg(x)})^{-1}.</math>
 
Here, the product ranges over all closed points ''x'' of ''X'' and deg(''x'') is the degree of ''x''.
The local zeta function ''Z(X, t)'' is viewed as a function of the complex variable ''s'' via the change of
variables ''q<sup>-s</sup>''.
 
In the case where ''X'' is the variety ''V'' discussed above, the closed points
are the equivalence classes ''x=[P]'' of points ''P'' on <math>\overline{V}</math>, where two points are equivalent if they are conjugates over ''F''.  The degree of ''x'' is the degree of the field extension of ''F''
generated by the coordinates of ''P''. The logarithmic derivative of the infinite product ''Z(X, t)'' is easily seen to be the generating function discussed above, namely
 
:<math>N_1 +N_2t^1 + N_3t^2 +\cdots \,</math>.
 
==See also==
*[[zeta function]]
 
==Notes==
{{reflist}}
 
[[Category:Algebraic varieties]]
[[Category:Finite fields]]
[[Category:Diophantine geometry]]
[[Category:Zeta and L-functions]]
[[Category:Fixed points (mathematics)]]

Latest revision as of 04:34, 4 January 2015


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