Statistics for biquadratic covers of the projective line over finite fields

TL;DR

This paper analyzes the distribution of Frobenius traces for genus g curves that are quartic non-cyclic covers of the projective line over finite fields, revealing limiting behaviors and extending to general covers with specific Galois groups.

Contribution

It provides new statistical descriptions of Frobenius traces for a broad class of covers, extending known results from hyperelliptic to more general cases.

Findings

Limiting distribution of Frobenius trace as q fixed is sum of q+1 independent variables.

Normalized trace converges to a standard complex Gaussian as g and q grow.

Results extend to covers with Galois group isomorphic to r copies of Z/2Z.

Abstract

We study the distribution of the traces of the Frobenius endomorphism of genus $g$ curves which are quartic non-cyclic covers of $\mathbb{P}^{1}_{\mathbb{F}_{q}}$, 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 $\mathbb{P}^{1}_{\mathbb{F}_{q}}$ with Galois group isomorphic to $r$ copies of $\mathbb{Z}/2\mathbb{Z}$. For $r = 1$, we recover the already known hyperelliptic case. We also include an appendix by Alina Bucur giving the heuristic of these distributions.