In this talk I 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 \overline{n} \in \mathbb{N}^g. We are interested in the moduli space that we denote by \mathcal{M}_{\overline{n}} of abelian varieties of dimension g over a field k together with an ample symmetric line bundle L and a theta structure of type \overline{n}. If \ell is a prime and let \overline{\ell}=(\ell, \ldots , \ell), there exists a modular correspondence \mathcal{M}_{\overline{\ell n}} \rightarrow \mathcal{M}_{\overline{n}} \times \mathcal{M}_{\overline{n}}. 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 Groebner basis algorithms. As an application, we explain how this algorithm can be used to speed up the initialisation phase of a point counting algorithm.

days10/DavidLubiczTalk (last edited 2008-11-14 13:42:09 by localhost)