Moscow Mathematical Journal

Volume 22, Issue 4, October–December 2022 pp. 613–655.

Cyclic Isogenies for Abelian Varieties with Real Multiplication

Authors: Alina Dudeanu (1), Dimitar Jetchev (1), Damien Robert (2), and Marius Vuille (1)
Author institution:(1) Ecole Polytechnique Fédérale de Lausanne, Switzerland
(2) Université de Bordeaux, France

Summary:

We study quotients of principally polarized abelian varieties with real multiplication by finite Galois-stable subgroups and describe when these quotients are principally polarizable. We use this characterization to provide an algorithm to compute explicit cyclic isogenies from their kernels for ordinary and simple abelian varieties over finite fields. Our algorithm is polynomial in the logarithm of the order of the finite field as well as in the degree of the isogeny and is based on Mumford's theory of theta functions. Recently, the algorithm has been successfully applied to obtain new results on the discrete logarithm problem in genus 2 as well as to study the discrete logarithm problem in genus 3.

2020 Math. Subj. Class. 11G10, 14K02, 14H42, 14K25, 11G15, 14Q15.

Keywords: Abelian varieties, arithmetic geometry, isogenies, theta functions, cryptography.

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