Statistics for biquadratic covers of the projective line over finite fields

We study the distribution of the traces of the Frobenius endomorphisms of genus g curves which are quartic non-cyclic covers of PFq1, as the curve varies in an irreducible component of the moduli space. We show that for q fixed, the limiting distribution of the traces of Frobenius equals the sum of q+1 independent random discrete variables. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution. Finally, we extend these computations to the general case of arbitrary covers of PFq1 with Galois group isomorphic to r copies of Z/2Z. For r=1 we recover the already known results for the family of hyperelliptic curves.