Editing Local zeta function (section)

==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'' to be a point. Then

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

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 the one [[point at infinity]]. Therefore, we have

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

and

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

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

:<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 nontrivial 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>