Local zeta function: Difference between revisions

Revision as of 15:03, 28 May 2014

Revision as of 07:13, 29 May 2014

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In other word, the local zeta function ''Z(V,u)'' with coefficients in the [[finite field]] '''F''' is defined as a function whose [[logarithmic derivative]] generates the numbers ''N_m'' of the solutions of equation, defining ''V'', in the ''m'' degree extension '''F'''_''m''.

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

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

is a function whose [[logarithmic derivative]] is a [[generating function]]

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==Formulation==

Given ''F'', there is, up to [[isomorphism]], just one field ''F_k'' 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>

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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 make log ''Z'' equal to ''G'', and so

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

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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>

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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'' a point. Then

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

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is the expansion of a logarithm (for |''t''| < 1). In this case we have

:<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

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

and

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

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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>

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

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==Riemann hypothesis for curves over finite fields==

For projective curves ''C'' over ''F'' that are [[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''. Rewriteing

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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^-s''.

In the case where ''X'' is the variety ''V'' discussed above, the closed points

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*[[elliptic curve]]

==referencesReferences==

{{reflist}}