Explicit computation of the Abel-Jacobi map and its inverse

Hugo Labrande 1, 2
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : 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 be computed in O(M(P) log P) operations by exploiting the links with the arithmetico-geometric mean (AGM). In this manuscript, we generalize this algorithm in order to compute theta in O(M(P) log P). We give a function F which has similar properties to the AGM. As with the algorithm for theta-constants, we can then use Newton's method to compute the value of theta. We implemented this algorithm, which is faster than the naive method for precisions larger than 300,000 decimal digits. We then study the generalization of this algorithm in higher genus, and in particular how to generalize the F function. In genus 2, we managed to prove that the same method leads to a O(M(P) log P) algorithm for theta; the same complexity applies to the Abel-Jacobi map. This algorithm is faster than the naive method for precisions smaller than in genus 1, of about 3,000 decimal digits. We also outline a way one could reach the same complexity in any genus. Finally, we study a new algorithm which computes an isogeny of elliptic curves with given kernel. This algorithm uses the Abel-Jacobi map because it is easy to evaluate the isogeny on the complex torus; this algorithm may be generalizable to higher genera
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Hugo Labrande. Explicit computation of the Abel-Jacobi map and its inverse. Computational Geometry [cs.CG]. Université de Lorraine, 2016. English. ⟨NNT : 2016LORR0142⟩. ⟨tel-01403849v2⟩

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