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.