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Computing separable isogenies in quasi-optimal time

Part of: Arithmetic algebraic geometry Abelian varieties and schemes

Published online by Cambridge University Press:  01 February 2015

David Lubicz  and
Damien Robert
David Lubicz
Affiliation:
Université de Rennes 1, Campus de Beaulier, 35042 Rennes, France email david.lubicz@univ-rennes1.fr
Damien Robert
Affiliation:
INRIA Bordeaux Sud-Ouest, 200 avenue de la Vieille Tour, 33405 Talence, France email damien.robert@inria.fr

Abstract

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Let $A$ be an abelian variety of dimension $g$ together with a principal polarization ${\it\phi}:A\rightarrow \hat{A}$ defined over a field $k$. Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of $A[\ell ]$ which is maximal isotropic for the Riemann form associated to ${\it\phi}$. We suppose that $K$ is defined over $k$ and let $B=A/K$ be the quotient abelian variety together with a polarization compatible with ${\it\phi}$. Then $B$, as a polarized abelian variety, and the isogeny $f:A\rightarrow B$ are also defined over $k$. In this paper, we describe an algorithm that takes as input a theta null point of $A$ and a polynomial system defining $K$ and outputs a theta null point of $B$ as well as formulas for the isogeny $f$. We obtain a complexity of $\tilde{O} (\ell ^{(rg)/2})$ operations in $k$ where $r=2$ (respectively, $r=4$) if $\ell$ is a sum of two (respectively, four) squares which constitutes an improvement over the algorithm described in Cosset and Robert (Math. Comput. (2013) accepted for publication). We note that the algorithm is quasi-optimal if $\ell$ is a sum of two squares since its complexity is quasi-linear in the degree of $f$.

MSC classification

Primary: 14K02: Isogeny 14K25: Theta functions 11G10: Abelian varieties of dimension $> 1$
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 198 - 216
Copyright
© The Author(s) 2015 

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