BALLENTINE, Sean; GUILLEVIC, Aurore; LORENZO GARCÍA, Elisa; MARTINDALE, Chloe; MASSIERER, Maike; SMITH, Benjamin; TOP, Jaap

doi:10.1007/978-3-319-63931-4_3

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BALLENTINE, Sean
Department of Mathematics [College Park]

GUILLEVIC, Aurore
University of Calgary
Pacific Institute for the Mathematical Sciences [PIMS]

LORENZO GARCÍA, Elisa
Institut de Recherche Mathématique de Rennes [IRMAR]

Langue

Communication dans un congrès

Ce document a été publié dans

Association for Women in Mathematics Series, Association for Women in Mathematics Series, Algebraic Geometry for Coding Theory and Cryptography, 2016-02-22, Los Angeles. 2017, vol. 9, p. 63-94

Springer

Résumé en anglais

Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof's algorithm. While we are currently missing the tools we need to generalize Elkies' methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this article, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves.< Réduire

URI

https://oskar-bordeaux.fr/handle/20.500.12278/193505

DOI

http://dx.doi.org/10.1007/978-3-319-63931-4_3

Project ANR

Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020

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Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251