Fast evaluation of modular functions using Newton iterations and the AGM

DOI10.1090/S0025-5718-2011-01880-6zbMath1221.65075OpenAlexW2034285964MaRDI QIDQ3015060

Régis Dupont

Publication date: 8 July 2011

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

Full work available at URL: https://doi.org/10.1090/s0025-5718-2011-01880-6

zbMATH Keywords

modular functionsNewton iterationsnumerical evaluationAGM

Mathematics Subject Classification ID

Number-theoretic algorithms; complexity (11Y16) Modular and automorphic functions (11F03) Computation of special functions and constants, construction of tables (65D20) Holomorphic modular forms of integral weight (11F11) Elliptic functions and integrals (33E05) Analytic computations (11Y35)

Related Items

Efficient computation of multidimensional theta functions ⋮ Certified Newton schemes for the evaluation of low-genus theta functions ⋮ On the evaluation of some sparse polynomials ⋮ Computing Jacobi’s theta in quasi-linear time ⋮ Computing theta functions in quasi-linear time in genus two and above ⋮ The complex AGM, periods of elliptic curves over \(\mathbb C\) and complex elliptic logarithms ⋮ On the Computation of Class Polynomials with “Thetanullwerte” and Its Applications to the Unit Group Computation ⋮ The complexity of class polynomial computation via floating point approximations ⋮ Computing modular polynomials in quasi-linear time ⋮ Short addition sequences for theta functions ⋮ Modular polynomials on Hilbert surfaces ⋮ Computing Igusa class polynomials ⋮ Computing Class Polynomials for Abelian Surfaces

Uses Software

• gmp
• MPFR
• MPC

Cites Work

• The arithmetic-geometric mean of Gauss
• 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
• The complexity of class polynomial computation via floating point approximations
• MPFR
• How to Compute the Coefficients of the Elliptic Modular Functionj(z)
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