# Explicit computation of the Abel-Jacobi map and its inverse. (Calcul explicite de l'application d'Abel-Jacobi et son inverse)

@inproceedings{Labrande2016ExplicitCO,
title={Explicit computation of the Abel-Jacobi map and its inverse. (Calcul explicite de l'application d'Abel-Jacobi et son inverse)},
author={Hugo Labrande},
year={2016}
}
The Abel-Jacobi map links the short Weierstrass form of a complex elliptic curve to the complex torus associated to it. One can compute it with a number of operations which is quasi-linear in the target precision, i.e. in time O(M(P) log P). Its inverse is given by Weierstrass's p-function, which can be written as a function of theta, an important function in number theory. The natural algorithm for evaluating theta requires O(M(P) sqrt(P)) operations, but some values (the theta-constants) can… Expand
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#### References

SHOWING 1-10 OF 119 REFERENCES
Computing Jacobi's in quasi-linear time
Jacobi’s θ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of θ(z, τ), for z, τ verifying certain conditions, with precision P inExpand
Computing l-Isogenies Using the p-Torsion
This paper proposes a general algorithm which does not use formal groups and takes advantage of the elementary Galois properties of the p-torsion, made exclusively of very classical routines in polynomial and elliptic curve arithmetic. Expand
Efficient Scalar Multiplication by Isogeny Decompositions
• Mathematics, Computer Science
• IACR Cryptol. ePrint Arch.
• 2005
This work derives a new tripling algorithm to find complexity improvements to triplication on a curve in certain projective coordinate systems, and combines this new operation to non-adjacent forms for l-adic expansions in order to obtain an improved strategy for scalar multiplication on elliptic curves. Expand
Fast evaluation of modular functions using Newton iterations and the AGM
• R. Dupont
• Computer Science, Mathematics
• Math. Comput.
• 2011
We present an asymptotically fast algorithm for the numerical evaluation of modular functions such as the elliptic modular function j. Our algorithm makes use of the natural connection between theExpand
The complexity of class polynomial computation via floating point approximations
• A. Enge
• Computer Science, Mathematics
• Math. Comput.
• 2009
The complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots, is analysed, using a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. Expand
Computing (ℓ, ℓ)-isogenies in polynomial time on Jacobians of genus 2 curves
• Computer Science, Mathematics
• Math. Comput.
• 2011
In this paper, we compute l-isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l, when l is an odd prime which is coprime to the characteristic.Expand
A generalization of Jacobi's derivative formula to dimension two.
Various 19th-century authors provided generalizations of (0. 2) to g-dimensional theta functions, expressing the Jacobian of g distinct odd theta functions at zero explicitly s a rational function inExpand
Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves,
We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3Expand
Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm
• Mathematics, Computer Science