Modular polynomials via Kani's theorem

The classical modular polynomial $\phi_\ell(X,Y) \in \mathbb{Z}[X,Y]$ parametrizes pairs of elliptic curves connected by an isogeny of degree $\ell$. They are an important tool in algorithmic number theory, for instance modular polynomials are used in point counting algorithms or for the computation of endomorphism rings.
This talk will be about a new method for computing the modular polynomial. It has the same asymptotic time complexity as the currently best known algorithms, but does not rely on any heuristics. One of the main ingredients is the use of Kani's theorem to represent isogenies of elliptic curves as smooth-degree isogenies in higher dimensions. This is based on joint work with Damien Robert.