Computing the Hilbert Class Polynomial Using p-adic Lifting

The Hilbert Class Polynomial of a quadratic imaginary field K describes the maximal unramified abelian extension of K. It also can be used to construct elliptic curves with a prescribed number of points over a finite field, which is important for cryptographic applications.  Classically,  this polynomial is computed in the complex analytic setting, but more recently, p-adic and multi-prime methods to compute it have been developed. In this talk, we'll introduce the Hilbert Class Polynomial, discuss the current state of methods to compute it and describe a p-adic lifting algorithm using a prime p inert in  K. This approach requires working with supersingular curves modulo p and many interesting complications ensue.