Distribution of points on Abelian covers over finite fields

Abstract

We determine in this paper the distribution of the number of points on the covers of P 1 (Fq) such that K(C) is a Galois extension and Gal(K(C)/K) is abelian when q is fixed and the genus, g, tends to infinity. This generalizes 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 q + 1 random variables.

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

“…Since F 1 and F 2 are square-free, we get that f 1 , f 2 , f 3 are square-free and pairwise coprime. [9] and [10]. The same argument works for N n (C).…”

“…Now we will take the result of Proposition 6•3 and sum over all primes. Corollary 1•7 of [11] shows that there exists a quadratic polynomial, A, such that…”

“…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.…”