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

DOI10.4064/aa170227-16-3zbMath1409.14051arXiv1701.06489OpenAlexW3125903495MaRDI QIDQ4963277

Reynald Lercier, Jeroen Sijsling, Pınar Kılıçer, Hugo Labrande, Christophe Ritzenthaler, Marco Streng

Publication date: 1 November 2018

Published in: Acta Arithmetica (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1701.06489

zbMATH Keywords

complex multiplicationplane quarticsgenus 3explicit aspects

Mathematics Subject Classification ID

Arithmetic ground fields for curves (14H25) Special algebraic curves and curves of low genus (14H45) Theta functions and abelian varieties (14K25) Actions of groups on commutative rings; invariant theory (13A50) Complex multiplication and moduli of abelian varieties (11G15) Algebraic number theory computations (11Y40) Computational aspects of algebraic curves (14Q05) Complex multiplication and abelian varieties (14K22)

Related Items (14)

curve_reconstruction ⋮ Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal complex multiplication ⋮ Computing period matrices and the Abel-Jacobi map of superelliptic curves ⋮ The CM class number one problem for curves of genus 2 ⋮ Theta nullvalues of supersingular abelian varieties ⋮ A bound on the primes of bad reduction for CM curves of genus 3 ⋮ Principally polarized squares of elliptic curves with field of moduli equal to \(\mathbb{Q}\) ⋮ Reduction type of smooth plane quartics ⋮ An inverse Jacobian algorithm for Picard curves ⋮ Reconstructing plane quartics from their invariants ⋮ Plane quartics over $\mathbb {Q}$ with complex multiplication ⋮ Primes Dividing Invariants of CM Picard Curves ⋮ Modular invariants for genus 3 hyperelliptic curves ⋮ Gluing curves of genus 1 and 2 along their 2-torsion

Uses Software

• Magma
• PARI/GP
• SageMath
• GitHub
• genus3
• RECIP
• quartic_reconstruction
• endomorphisms

Cites Work

• Explicit computations of invariants of plane quartic curves
• Reduction theory of point clusters in projective space
• Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group
• Computing Igusa class polynomials
• Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields
• Computing theta functions in quasi-linear time in genus two and above
• Constructing genus-3 hyperelliptic Jacobians with CM
• Computing Jacobi’s theta in quasi-linear time
• Rigorous computation of the endomorphism ring of a Jacobian
• Solving quadratic equations using reduced unimodular quadratic forms
• Computing Riemann theta functions
• Plane quartics over $\mathbb {Q}$ with complex multiplication
• The Fields of Moduli for Polarized Abelian Varieties and for Curves
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