Local zeta function: Difference between revisions

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In [[number theorymathematics]], the '''local zeta function''' {{math|''Z''(''V'', ''s'')}} (sometimes called the '''congruent zeta function''' or the [[Hasse–Weil zeta function]]) is defined as

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

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}}</ref>

Making the variable transformation {{math|''ut''&nbsp;{{=}}&nbsp;''q''^−''s'',}} gives

:<math>

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

\left( \sum_{k=1}^{\infty} N_k \frac{ut^k}{k} \right)

</math>

as the [[formal power series]] in the variable <math>ut</math>.

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

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</math>

:<math>

(2)\ \ \frac{d}{dudt} \log \mathit{Z} (V,ut) = \sum_{k=1}^{\infty} N_k ut^{k-1}\ .</math>

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

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

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:<math>[ F_k : F ] = k \,</math>,

for ''k'' = 1, 2, ... . When ''F'' hasis the unique field with ''q'' elements, ''F_k'' hasis the unique field with <math>q^k</math> elements. 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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==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 a finite field.

The local ''Z'' zeta functions are multiplied to get global ''<math>\zeta</math>'' zeta functions,