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Mathematics of Computation

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Computing $(\ell ,\ell )$-isogenies in polynomial time on Jacobians of genus $2$ curves

Authors: Romain Cosset and Damien Robert
Journal: Math. Comp. 84 (2015), 1953-1975
MSC (2010): Primary 11Y40, 14K02; Secondary 94A60, 14G50, 11T71
Published electronically: November 18, 2014
MathSciNet review: 3335899
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Abstract: In this paper, we compute $\ell$-isogenies between abelian varieties over a field of characteristic different from $2$ in polynomial time in $\ell$, when $\ell$ is an odd prime which is coprime to the characteristic. We use level $n$ symmetric theta structure where $n=2$ or $n=4$. In the second part of this paper we explain how to convert between Mumford coordinates of Jacobians of genus $2$ hyperelliptic curves to theta coordinates of level $2$ or $4$. Combined with the preceding algorithm, this gives a method to compute $(\ell ,\ell )$-isogenies in polynomial time on Jacobians of genus $2$ curves.

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Additional Information

Romain Cosset
Affiliation: Campus Scientifique, Loria, 54506 Vandouevre-Les-Nancy, France

Damien Robert
Affiliation: Universite Bordeaux 1, Institut Mathematiques de Bordeaux, 351 Cours de la Liberation, Batiment A33, 33405 Talence, Cedex France

Received by editor(s): March 24, 2011
Received by editor(s) in revised form: October 4, 2013
Published electronically: November 18, 2014
Article copyright: © Copyright 2014 American Mathematical Society