2015/03/11 by Lorenzo, Elisa, Meleleo, Giulio, Milione, Piermarco +1
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1503.03276
We study the distribution of the traces of the Frobenius endomorphism of genus g curves which are quartic non-cyclic covers of ℙ1_\mathbbFq, as the curve varies in an irreducible component of the moduli space. We show that for q fixed, the limiting distribution of the trace 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 ℙ1_\mathbbFq with Galois group isomorphic to r copies of ℤ/2ℤ. For r = 1, we recover the already known hyperelliptic case. We also include an appendix by Alina Bucur giving the heuristic of these distributions.