Computing $(\ell ,\ell )$-isogenies in polynomial time on Jacobians of genus $2$ curves
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- by Romain Cosset and Damien Robert PDF
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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.References
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Additional Information
- Romain Cosset
- Affiliation: Campus Scientifique, Loria, 54506 Vandouevre-Les-Nancy, France
- Email: romain.cosset@crans.org
- Damien Robert
- Affiliation: Universite Bordeaux 1, Institut Mathematiques de Bordeaux, 351 Cours de la Liberation, Batiment A33, 33405 Talence, Cedex France
- Email: damien.robert@inria.fr
- Received by editor(s): March 24, 2011
- Received by editor(s) in revised form: October 4, 2013
- Published electronically: November 18, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 1953-1975
- MSC (2010): Primary 11Y40, 14K02; Secondary 94A60, 14G50, 11T71
- DOI: https://doi.org/10.1090/S0025-5718-2014-02899-8
- MathSciNet review: 3335899