Local zeta function: Difference between revisions

Revision as of 00:24, 26 December 2020

Revision as of 23:47, 18 March 2021

Line 19:

In other words, the local zeta function {{math|''Z''(''V'', ''u'')}} with coefficients in the [[finite field]] {{math|'''F'''_''q''}} is defined as a function whose [[logarithmic derivative]] generates the number {{math|''N''_''m''}} of solutions of the equation defining {{mvar|V}} in the degree {{mvar|m}} extension {{math|'''F'''_''q''^''m''.}}

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

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

Line 76:

:<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>

Line 84:

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'''. 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-WeilHasse–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>.