# Statistics for biquadratic covers of the projective line over finite fields

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.
Keywords :
Type de document :
Article dans une revue
Journal of Number Theory, Elsevier, 2017, 173 (supplement C), pp.448-477. 〈10.1016/j.jnt.2016.09.007〉

https://hal.archives-ouvertes.fr/hal-01619460
Contributeur : Marie-Annick Guillemer <>
Soumis le : jeudi 19 octobre 2017 - 14:18:33
Dernière modification le : dimanche 25 novembre 2018 - 14:26:02

### Citation

Elisa Lorenzo Garcia, Giulio Meleleo, Piermarco Milione, Alina Bucur. Statistics for biquadratic covers of the projective line over finite fields. Journal of Number Theory, Elsevier, 2017, 173 (supplement C), pp.448-477. 〈10.1016/j.jnt.2016.09.007〉. 〈hal-01619460〉

### Métriques

Consultations de la notice