Construction of CM Picard curves

DOI10.1090/S0025-5718-04-01656-4zbMath1049.14014OpenAlexW2036117875MaRDI QIDQ4821060

Kenji Koike, Annegret Weng

Publication date: 7 October 2004

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0025-5718-04-01656-4

zbMATH Keywords

Picard curvesCM method

Mathematics Subject Classification ID

Special algebraic curves and curves of low genus (14H45) Complex multiplication and moduli of abelian varieties (11G15) Complex multiplication and abelian varieties (14K22) Applications to coding theory and cryptography of arithmetic geometry (14G50)

Related Items (14)

Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal complex multiplication ⋮ Picard Curves with Small Conductor ⋮ Explicit bounds of polynomial coefficients and counting points on Picard curves over finite fields ⋮ Point counting on Picard curves in large characteristic ⋮ A bound on the primes of bad reduction for CM curves of genus 3 ⋮ Constructing Picard curves with complex multiplication using the Chinese remainder theorem ⋮ Principally polarized squares of elliptic curves with field of moduli equal to \(\mathbb{Q}\) ⋮ Constructing genus-3 hyperelliptic Jacobians with CM ⋮ Reduction type of smooth plane quartics ⋮ An inverse Jacobian algorithm for Picard curves ⋮ Bad Reduction of Genus Three Curves with Complex Multiplication ⋮ Galois representations attached to Picard curves ⋮ Primes Dividing Invariants of CM Picard Curves ⋮ Index calculus in class groups of non-hyperelliptic curves of genus three

Uses Software

• PARI/GP
• ECPP

Cites Work

• Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades
• On unramified Galois extensions of real quadratic number fields
• Hyperelliptic cryptosystems
• On the representation of the Picard modular function by \(\theta\) constants. I-II
• Tata lectures on theta. I: Introduction and motivation: Theta functions in one variable. Basic results on theta functions in several variables. With the assistance of C. Musili, M. Nori, E. Previato, and M. Stillman
• Isogeny classes of abelian varieties over finite fields
• Arithmetic on superelliptic curves
• Elliptic Curves and Primality Proving
• Elliptic Curve Cryptosystems
• An improved algorithm for computing logarithms over<tex>GF(p)</tex>and its cryptographic significance (Corresp.)
• Examples of genus two CM curves defined over the rationals
• The arithmetic of certain cubic function fields
• Constructing hyperelliptic curves of genus 2 suitable for cryptography
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