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

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**Statistics for biquadratic covers of the projective line over finite fields.** / Lorenzo, Elisa; Meleleo, Giulio; Milione, Piermarco; Bucur, Alina.

Research output: Contribution to journal › Article › Scientific › peer-review

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*Journal of Number Theory*, vol. 173, pp. 448-477. https://doi.org/10.1016/j.jnt.2016.09.007

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*Journal of Number Theory*,

*173*, 448-477. https://doi.org/10.1016/j.jnt.2016.09.007

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TY - JOUR

T1 - Statistics for biquadratic covers of the projective line over finite fields

AU - Lorenzo, Elisa

AU - Meleleo, Giulio

AU - Milione, Piermarco

AU - Bucur, Alina

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Function fields

KW - Biquadratic curves

KW - Biquadratic covers

KW - Number of points over finite fields

KW - Arithmetic statistics

U2 - 10.1016/j.jnt.2016.09.007

DO - 10.1016/j.jnt.2016.09.007

M3 - Article

VL - 173

SP - 448

EP - 477

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -

ID: 11678241