In this talk I give a higher dimensional equivalent of the classical modular polynomials Φℓ(X,Y). If j is the j-invariant associated to an elliptic curve Ek over a field k then the roots of Φℓ(j,X) correspond to the j-invariants of the curves which are ℓ-isogeneous to Ek. Denote by X0(N) the modular curve which parametrizes the set of elliptic curves together with a N-torsion subgroup. It is possible to interpret Φℓ(X,Y) as an equation cutting out the image of a certain modular correspondence X0(ℓ)→X0(1)×X0(1) in the product X0(1)×X0(1).
Let g be a positive integer and n∈Ng. 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 L and a theta structure of type n. If ℓ is a prime and let ℓ=(ℓ,...,ℓ), there exists a modular correspondence Mℓn→Mn×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 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.
