Local zeta function

In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as

${\displaystyle Z(V,s)=\exp \left(\sum _{m=1}^{\infty }{\frac {N_{m}}{m}}(q^{-s})^{m}\right)}$

where V is a non-singular n-dimensional projective algebraic variety over the field F_q with q elements and N_m is the number of points of V defined over the finite field extension F_q^m of F_q.

Making the variable transformation u = q^−s, gives

${\displaystyle {\mathit {Z}}(V,u)=\exp \left(\sum _{m=1}^{\infty }N_{m}{\frac {u^{m}}{m}}\right)}$

as the formal power series in the variable ${\displaystyle u}$.

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

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

${\displaystyle (2)\ \ {\frac {d}{du}}\log {\mathit {Z}}(V,u)=\sum _{m=1}^{\infty }N_{m}u^{m-1}\ .}$

In other words, the local zeta function Z(V, u) with coefficients in the finite field F_q is defined as a function whose logarithmic derivative generates the number N_m of solutions of the equation defining V in the degree m extension F_q^m.