Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication
MARTINDALE, Chloe
Universiteit Leiden = Leiden University
Lithe and fast algorithmic number theory [LFANT]
Universiteit Leiden = Leiden University
Lithe and fast algorithmic number theory [LFANT]
SMITH, Benjamin
Laboratoire d'informatique de l'École polytechnique [Palaiseau] [LIX]
Geometry, arithmetic, algorithms, codes and encryption [GRACE]
< Reduce
Laboratoire d'informatique de l'École polytechnique [Palaiseau] [LIX]
Geometry, arithmetic, algorithms, codes and encryption [GRACE]
Language
en
Communication dans un congrès
This item was published in
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
English Abstract
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, ...Read more >
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.Read less <
ANR Project
Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020
Origin
Hal imported