Number of Points on the Full Moduli Space of Curves over Finite Fields

Abstract

The distribution of the number of points on abelian covers of P 1 (Fq) ranging over an irreducible moduli space has been answered in a recent work by the author [10], [11]. The authors of [1] determined the distribution over the whole moduli space for curves with Gal(K(C)/K) a prime cyclic. In this paper, we prove a result towards determining the distribution over the whole moduli space of curves with Gal(K(C)/K) any abelian group. We successfully determine the distribution in the case Gal(K(C)/K) is a power of a prime cyclic.

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

“…Now we will take the result of Proposition 6.2 and sum over all primes. Corollary 1.7 of [11] shows that there exists a quadratic polynomials, A, such that…”

“…This was first discussed by Kurlberg and Rudnick in [8] in which they considered the distribution of Tr(Θ 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 vairables.…”