Abstract

Classical results due to Katz and Sarnak show that if the genus is fixed and q tends to infinity, then the number of points on a family of curves over a finite field of q elements is distributed as the trace of a random matrix in the monodromy group associated to the family.

Every smooth projective curve C corresponds to a finite Galois extension of the field of polynomials with coefficients in the finite field. Therefore, some natural families to consider are the curves that correspond to a extensions with a fixed Galois group. This thesis involves determining the distribution of the families with fixed abelian Galois group, G, when q is fixed and the genus tends to infinity.

Several authors determined that the distribution for the family of prime-cyclic curves as well as for the family of n-quadratic curves is that of a sum of q+1 random variables. This thesis shows that if we fix any abelian group, the distribution will be that of q+1 random variables.

The above results deal only with the distribution for the coarse irreducible moduli space of the families. It has been shown that if you look at the whole (coarse) moduli space, the distribution is the same in the case of prime-cyclic curves. We are able to show that the distribution is the same for the coarse moduli space of curves with G=(Z/QZ)^n, Q a prime. Some work is done towards proving this true for all abelian groups.