On the distribution of the number of points on algebraic curves in extensions of finite fields

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 $n =1, \ldots, N$ 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 $g=2$.

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Published 1 January 2010