CM liftings of Supersingular Elliptic Curves

TL;DR

This paper introduces an algorithm, assuming GRH, to identify fundamental discriminants related to supersingular elliptic curves over finite fields, enhancing understanding of CM liftings in elliptic curve theory.

Contribution

The paper presents a novel algorithm that, under GRH, explicitly determines discriminants for supersingular elliptic curves, including a method to compute cases below a certain bound.

Findings

Algorithm under GRH for discriminant identification

Explicit constant D_p for surjectivity of the reduction map

Complete enumeration of discriminants below D_p

Abstract

Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D<0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $\order_{D}$ to supersingular elliptic curves in characteristic $p$. In the algorithm we first determine an explicit constant $D_p$ so that $|D|> D_p$ implies that the map is necessarily surjective and then we compute explicitly the cases $|D|<D_p$.