Examples of CM curves of genus two defined over the reflex field

Bouyer, Florian; Streng, Marco

doi:10.1112/s1461157015000121

Cited by 20 publications

(38 citation statements)

References 29 publications

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“…Thus, the invariants of a sextic define a point in a weighted projective space [J 2 : J 4 : J 6 : J 10 ] ∈ WP 3 (2,4,6,10) . It was shown in [12] that points in the projective variety Proj C[J 2 , J 4 , J 6 , J 10 ] which are not on J 2 = 0 form the variety U 6 of moduli of sextics.…”

Section: Root Differences Let F (X Z) Be a Binary Sextic As Above Amentioning

confidence: 99%

Rational points in the moduli space of genus two

Beshaj¹,

Hidalgo²,

Kruk³

et al. 2018

Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

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We build a database of genus 2 curves defined over Q which contains all curves with minimal absolute height h ≤ 5, all curves with moduli height h ≤ 20, and all curves with extra automorphisms in standard form y 2 = f (x 2 ) defined over Q with height h ≤ 101. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space M 2 for which the field of moduli is a field of definition is discussed and some open problems are presented.

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Section: Root Differences Let F (X Z) Be a Binary Sextic As Above Amentioning

confidence: 99%

Rational points in the moduli space of genus two

Beshaj¹,

Hidalgo²,

Kruk³

et al. 2018

Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

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“…Remark 3.9. Bouyer and Streng [7,Algorithm 4.8] show how one can avoid factoring in the discriminant minimization of binary forms. Such a trick enabled them to eliminate the need for a loop like that in Step (i) of Algorithm 3.8 when considering curves of genus 2.…”

Section: 3mentioning

confidence: 99%

Plane quartics over $\mathbb {Q}$ with complex multiplication

Kılıçer¹,

Labrande²,

Lercier³

et al. 2018

Acta Arith.

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“…Here we give an example of such an evaluation for the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field. [8,10,17], the Jacobian J C of the genus 2 hyperelliptic curve C over Q ( √ 17) given by the equation…”

Section: A Chowla-selberg Formula For the Jacobian Of A Genus 2 Hypermentioning

confidence: 99%

On the Colmez conjecture for non-abelian CM fields

Sánchez

Masri

2018

Res Math Sci

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The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field E to logarithmic derivatives of Artin L-functions at s = 0. In this paper, we prove that if F is any fixed totally real number field of degree [F : Q] ≥ 3, then there are infinitely many effective, "positive density" sets of CM extensions E/F such that E/Q is non-abelian and the Colmez conjecture is true for E. Moreover, these CM extensions are explicitly constructed to be ramified at arbitrary prescribed sets of prime ideals of F. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be viewed as an explicit non-abelian Chowla-Selberg formula. Our results rely crucially on an averaged version of the Colmez conjecture which was recently proved independently by

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Examples of CM curves of genus two defined over the reflex field

Cited by 20 publications

References 29 publications

Rational points in the moduli space of genus two

Rational points in the moduli space of genus two

Plane quartics over $\mathbb {Q}$ with complex multiplication

On the Colmez conjecture for non-abelian CM fields

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