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 $n =1, ..., N$ 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 $g=2$.Comment: 14 page

Most hyperelliptic curves over Q have no rational points

By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible, complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in weighted projective space P(1,1,g+1) via an equation of the form C : z^2 = f(x,y) = f_0 x^n + f_1 x^{n-1} y + ... + f_n y^n where n=2g+2, the coefficients f_i lie in Z, and f factors into distinct linear factors over Q-bar. Define the height H(C) of C by H(C):=max{|f_i|}, and order all hyperelliptic curves over Q of genus g by height. Then we prove that, as g tends to infinity: 1) a density approaching 100% of hyperelliptic curves of genus g have no rational points; 2) a density approaching 100% of those hyperelliptic curves of genus g that have points everywhere locally fail the Hasse principle; and 3) a density approaching 100% of hyperelliptic curves of genus g have empty Brauer set, i.e., have a Brauer-Manin obstruction to having a rational point. We also prove positive proportion results of this type for individual genera, including g = 1.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1208.100