  Let $\cC$ be a smooth absolutely irreducible curve of genus $g \ge 1$ defined
over $\F_q$, the finite field of $q$ elements. Let $# \cC(\F_{q^n})$ be the
number of $\F_{q^n}$-rational points on $\cC$. Under a certain multiplicative
independence condition on the roots of the zeta-function of $\cC$, we derive an
asymptotic formula for the number of $n =1, ..., N$ such that $(# \cC(\F_{q^n})
- q^n -1)/2gq^{n/2}$ belongs to a given interval $\cI \subseteq [-1,1]$. This
can be considered as an analogue of the Sato-Tate distribution which covers the
case when the curve $\E$ is defined over $\Q$ and considered modulo consecutive
primes $p$, although in our scenario the distribution function is different.
The above multiplicative independence condition has, recently, been considered
by E. Kowalski in statistical settings. It is trivially satisfied for ordinary
elliptic curves and we also establish it for a natural family of curves of
genus $g=2$.
