Let C be a smooth absolutely irreducible curve of genus g ≥ 1 defined over Fq, the finite field of q elements. Let ≠C(Fqn) be the number of Fqn -rational points on C. Under a certain multiplicative independence condition on the roots of the zetafunction of C, we derive an asymptotic formula for the number of n = 1, . . . ,N such that (≠C(Fqn)-qn-1)/2gqn/2 belongs to a given interval I ⊆ [-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.
|Number of pages||11|
|Journal||Mathematical Research Letters|
|Publication status||Published - Jul 2010|