We study the distribution of the traces of the Frobenius endomorphisms of genus g curves which are quartic non-cyclic covers of View the MathML sourcePFq1, 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+1q+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 View the MathML sourcePFq1 with Galois group isomorphic to r copies of Z/2ZZ/2Z. For r=1r=1 we recover the already known results for the family of hyperelliptic curves.
- Function fields
- Biquadratic curves
- Biquadratic covers
- Number of points over finite fields
- Arithmetic statistics
Lorenzo, E., Meleleo, G., Milione, P., & Bucur, A. (2017). Statistics for biquadratic covers of the projective line over finite fields. Journal of Number Theory, 173, 448-477. https://doi.org/10.1016/j.jnt.2016.09.007