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m fix minus sign, cleanup sup/sub, replaced: <sup>- → <sup>− (2) using AWB |
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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>
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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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>
In the case where ''X'' is the variety ''V'' discussed above, the closed points
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