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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Licencia de uso del documento

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]

Idioma

Communication dans un congrès

Este ítem está publicado en

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

Resumen en inglés

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.< Leer menos

URI

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

DOI

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

Proyecto ANR

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

Orígen

Importado de Hal

Centros de investigación

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251