In mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as
Z ( V , s ) = exp ( ∑ k = 1 ∞ N k k ( q − s ) k ) {\displaystyle Z(V,s)=\exp \left(\sum _{k=1}^{\infty }{\frac {N_{k}}{k}}(q^{-s})^{k}\right)}
where V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements and Nk is the number of points of V defined over the finite field extension Fqk of Fq. Making the variable transformation t = q−s, gives
Z ( V , t ) = exp ( ∑ k = 1 ∞ N k t k k ) {\displaystyle {\mathit {Z}}(V,t)=\exp \left(\sum _{k=1}^{\infty }N_{k}{\frac {t^{k}}{k}}\right)}
as the formal power series in the variable t {\displaystyle t} . Equivalently, the local zeta function is sometimes defined as follows:
( 1 ) Z ( V , 0 ) = 1 {\displaystyle (1)\ \ {\mathit {Z}}(V,0)=1\,}
( 2 ) d d t log Z ( V , t ) = ∑ k = 1 ∞ N k t k − 1 . {\displaystyle (2)\ \ {\frac {d}{dt}}\log {\mathit {Z}}(V,t)=\sum _{k=1}^{\infty }N_{k}t^{k-1}\ .}
In other words, the local zeta function Z(V, t) with coefficients in the finite field Fq is defined as a function whose logarithmic derivative generates the number Nk of solutions of the equation defining V in the degree k extension Fqk.
Formulation Given a finite field F, there is, up to isomorphism, only one field Fk with
[ F k : F ] = k {\displaystyle [F_{k}:F]=k\,} , for k = 1, 2, ... . When F is the unique field with q elements, Fk is the unique field with q k {\displaystyle q^{k}} elements. Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number
N k {\displaystyle N_{k}\,}
of solutions in Fk and create the generating function
G ( t ) = N 1 t + N 2 t 2 / 2 + N 3 t 3 / 3 + ⋯ {\displaystyle G(t)=N_{1}t+N_{2}t^{2}/2+N_{3}t^{3}/3+\cdots \,} . The correct definition for Z(t) is to set log Z equal to G, so
Z = exp ( G ( t ) ) {\displaystyle Z=\exp(G(t))\,}
and Z(0) = 1, since G(0) = 0, and Z(t) is a priori a formal power series. The logarithmic derivative
Z ′ ( t ) / Z ( t ) {\displaystyle Z'(t)/Z(t)\,}
equals the generating function
G ′ ( t ) = N 1 + N 2 t 1 + N 3 t 2 + ⋯ {\displaystyle G'(t)=N_{1}+N_{2}t^{1}+N_{3}t^{2}+\cdots \,} .
Examples For example, assume all the Nk 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
G ( t ) = − log ( 1 − t ) {\displaystyle G(t)=-\log(1-t)}
is the expansion of a logarithm (for |t| < 1). In this case we have
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