Frobenius lifts and elliptic curves with complex multiplication

Elliptic Curves and Complex Multiplication

Complex Multiplication Tests for Elliptic Curves

We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized algorithm can be adapted to yield the discriminant of the endomorphism ring of the curve.

Complex Multiplication Structure of Elliptic Curves

Let k be a finite field and let E be an elliptic curve over k. In this paper we describe, for each finite extension l of k, the structure of the group E(l) of points of E over l as a module over the ring R of endomorphisms of E that are defined over k. If the Frobenius endomorphism ? of E over k does not belong to the subring Z of R, then we find that E(l)$R R(?&1), where n is the degree of l o...

Complex multiplication and canonical lifts

The problem of constructing CM invariants of higher dimensional abelian varieties presents significant new challenges relative to CM constructions in dimension 1. Algorithms for p-adic canonical lifts give rise to very efficient means of constructing high-precision approximations to CM points on moduli spaces of abelian varieties. In particular, algorithms for 2-adic and 3-adic lifting of Frobe...

On Primitive Points of Elliptic Curves with Complex Multiplication

Let E be an elliptic curve defined over Q and P ∈ E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the quadratic imaginary field k. Denote by ME,P the set of rational primes ` such that ` splits in k, E has good reduction at `, and P is a primitive point modulo `. Under the generalized Riemann hypothesis, we can determine the positivity of the de...