Computation of Hilbert class polynomials and modular polynomials from supersingular elliptic curves.

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $P$. For that, we revisit the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on ordinary curves, due to the supposed inefficiency of the supersingular case. While this was true a decade ago, it is not anymore due to the recent advances in the study of supersingular curves. Our main ingredients are two new heuristic algorithms to compute the $j$-invariants of supersingular curves having an endomorphism ring contained in some set of isomorphism class of maximal orders.