Local zeta function: Difference between revisions

Revision as of 01:29, 16 September 2020

Revision as of 02:01, 9 December 2020

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:<math>Z(V, s) = \exp\left(\sum_{m = 1}^\infty \frac{N_m}{m} (q^{-s})^m\right)</math>

where {{math|''N''_''m''}} is the number of points of {{mvar|''V''}} defined over the finite field extension {{math|'''F'''_''q''^''m''}} of {{math|'''F'''_''q'',}} and {{mvar|V}} is a [[non-singular]] {{mvar|n}}-dimensional [[projective algebraic variety]] over the field {{math|'''F'''_''q''}} with {{mvar|q}} elements. ByMaking the variable transformation {{math|''u''&nbsp;{{=}}&nbsp;''q''^−''s'',}} then it is defined bygives

:<math>

\mathit{Z} (V,u) = \exp

\left( \sum_{m=1}^{\infty} N_m \frac{u^m}{m} \right)

</math>

as the [[formal power series]] ofin the variable <math>u</math>.

Equivalently, the local zeta function sometimes is sometimes defined as follows:

:<math>

(1)\ \ \mathit{Z} (V,0) = 1 \,

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(2)\ \ \frac{d}{du} \log \mathit{Z} (V,u) = \sum_{m=1}^{\infty} N_m u^{m-1}\ .</math>

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

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