Local zeta function

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

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

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

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

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

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

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

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

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

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