Distribution of points on cyclic curves over finite fields

Abstract

We determine in this paper the distribution of the number of points on the cyclic covers of P 1 (Fq) with affine models C :and r th -power free when q is fixed and the genus, g, tends to infinity. This generalize the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over Fq. In all cases, the distribution is given by a sum of random variables.

Distribution of points on Abelian covers over finite fields

“…Iterating this process then we get the result with the Chinese Remainder Theorem. As in the cyclic case in [9], it will be important to keep track of when and how an admissible set can have zero values. Fix a β such that f β (x) = 0.…”

“…First, note that |G| = r 1 · · · r n . Moreover, Proposition 3.4 of [9] does not rely on the fact that the r j |r j+1 and hence define a group. Secondly, observe that the size of the set is independent of the choices of ǫ ρj , 1,i as long as they are non-zero.…”

“…Lorenzo, Meleleo and Milione [8] then determined this for n-quadratic curves (G = (Z/2Z) n ). More recently the author [9] extended the work of Bucur, David, Feigon and Lalin to the case of arbitrary cyclic curves (G = Z/rZ, r not necessarily a prime).…”

Expected value of high powers of trace of frobenius of biquadratic curves over a finite field

“…Equation (1.3) shows that knowing Tr(Θ n C ) is equivalent to knowing N n (C), the number of F q n -points on C. A formula for N 1 (C) is developed in [9] and [10]. The same argument works for N n (C).…”

“…This was first discussed by Kurlberg and Rudnick in [8] in which they considered the distribution of Tr( C ) over the family of hyperelliptic curves. This was extended by various authors to several different families [2][3][4][9][10][11][12]. For all cases, the statistics can be given as a sum of q + 1 random variables.…”