Asymptotically fast group operations on Jacobians of general curves

DOI10.1090/S0025-5718-07-01989-8zbMath1131.14063arXivmath/0409209MaRDI QIDQ3592703

Kamal Khuri-Makdisi

Publication date: 13 September 2007

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

Full work available at URL: https://arxiv.org/abs/math/0409209

zbMATH Keywords

computing in Jacobian

Mathematics Subject Classification ID

Number-theoretic algorithms; complexity (11Y16) Jacobians, Prym varieties (14H40) Curves over finite and local fields (11G20) Computational aspects of algebraic curves (14Q05)

Related Items (20)

MODULI INTERPRETATION OF EISENSTEIN SERIES ⋮ Group Law Computations on Jacobians of Hyperelliptic Curves ⋮ Companion forms and explicit computation of \(\mathrm{PGL}_2\) number fields with very little ramification ⋮ Moduli-friendly Eisenstein series over the \(p\)-adics and the computation of modular Galois representations ⋮ Computing Riemann-Roch spaces via Puiseux expansions ⋮ Computing modular Galois representations ⋮ Computing functions on Jacobians and their quotients ⋮ Computational Number Theory, Past, Present, and Future ⋮ A fast randomized geometric algorithm for computing Riemann-Roch spaces ⋮ Computation of étale cohomology on curves in single exponential time ⋮ On the computation of coefficients of modular forms: The reduction modulo $p$ approach ⋮ Asymptotically fast group operations on Jacobians of general curves ⋮ A generic approach to searching for Jacobians ⋮ Hensel-lifting torsion points on Jacobians and Galois representations ⋮ Factoring polynomials over finite fields ⋮ ON THE MAPS FROM X(4p) TO X(4) ⋮ On Jacobian group arithmetic for typical divisors on curves ⋮ Computing in Picard groups of projective curves over finite fields ⋮ Modular forms constructed from moduli of elliptic curves, with applications to explicit models of modular curves ⋮ Two Recent p-adic Approaches Towards the (Effective) Mordell Conjecture

Uses Software

• PARI/GP

Cites Work

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• Matrix multiplication via arithmetic progressions
• Computing Riemann-Roch spaces in algebraic function fields and related topics.
• The calculation of radical ideals in positive characteristic
• Abeliants and their application to an elementary construction of Jacobians.
• Efficient algorithms for the Riemann-Roch problem and for addition in the Jacobian of a curve
• Efficient decomposition of associative algebras over finite fields
• A linear lower bound on the gonality of modular curves
• Asymptotically fast group operations on Jacobians of general curves
• Computing in the Jacobian of a Hyperelliptic Curve
• Linear algebra algorithms for divisors on an algebraic curve
• The Jacobian Variety of an Algebraic Curve

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