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.
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Journal of Number Theory, Elsevier, 2017, 173 (supplement C), pp.448-477. 〈10.1016/j.jnt.2016.09.007〉
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Contributeur : Marie-Annick Guillemer <>
Soumis le : jeudi 19 octobre 2017 - 14:18:33
Dernière modification le : lundi 28 mai 2018 - 11:20:14

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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〉

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