Computing modular correspondences for abelian varieties

Jean-Charles Faugère 1 David Lubicz 2 Damien Robert 3, 4, 5
1 SALSA - Solvers for Algebraic Systems and Applications
LIP6 - Laboratoire d'Informatique de Paris 6, Inria Paris-Rocquencourt
3 CACAO - Curves, Algebra, Computer Arithmetic, and so On
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
4 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : The aim of this paper is to give a higher dimensional equivalent of the classical modular polynomials $\Phi_\ell(X,Y)$. If $j$ is the $j$-invariant associated to an elliptic curve $E_k$ over a field $k$ then the roots of $\Phi_\ell(j,X)$ correspond to the $j$-invariants of the curves which are $\ell$-isogeneous to $E_k$. Denote by $X_0(N)$ the modular curve which parametrizes the set of elliptic curves together with a $N$-torsion subgroup. It is possible to interpret $\Phi_\ell(X,Y)$ as an equation cutting out the image of a certain modular correspondence $X_0(\ell) \rightarrow X_0(1) \times X_0(1)$ in the product $X_0(1) \times X_0(1)$. Let $g$ be a positive integer and $\overn \in \N^g$. We are interested in the moduli space that we denote by $\Mn$ of abelian varieties of dimension $g$ over a field $k$ together with an ample symmetric line bundle $\pol$ and a symmetric theta structure of type $\overn$. If $\ell$ is a prime and let $\overl=(\ell, \ldots , \ell)$, there exists a modular correspondence $\Mln \rightarrow \Mn \times \Mn$. We give a system of algebraic equations defining the image of this modular correspondence. We describe an algorithm to solve this system of algebraic equations which is much more efficient than a general purpose Gr¨obner basis algorithm. As an application, we explain how this algorithm can be used to speed up the initialisation phase of a point counting algorithm.
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Journal of Algebra, Elsevier, 2011, 343 (1), pp.248-277. <10.1016/j.jalgebra.2011.06.031>
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Jean-Charles Faugère, David Lubicz, Damien Robert. Computing modular correspondences for abelian varieties. Journal of Algebra, Elsevier, 2011, 343 (1), pp.248-277. <10.1016/j.jalgebra.2011.06.031>. <hal-00426338v2>

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