Computing modular polynomials by deformation

We present an unconditional CRT algorithm to compute the modular polynomial Phi(l)(X,Y) in quasi-linear time. The main ingredients of our algorithm are: the embedding of l-isogenies in smooth-degree isogenies in higher dimension, and the computation of m-th order deformations of isogenies. We provide a proof-of concept implementation of a heuristic version of the algorithm demonstrating the practicality of our approach. Our algorithm can also be used to compute the reduction of Phi(l )modulo p in quasi-linear time (with respect to l) O(l(2)(logp +logl)(O)).