The Tate–Shafarevich group for elliptic curves with complex multiplication

The Tate-Shafarevich group for elliptic curves with complex multiplication

where v ranges over all places of Q and Qv is the completion of Q at v, denote its TateShafarevich group. As usual, L(E/Q, s) is the complex L-function of E over Q. Since E is now known to be modular, Kolyvagin’s work [11] shows that X(E/Q) is finite if L(E/Q, s) has a zero at s = 1 of order ≤ 1, and that gE/Q is equal to the order of the zero of L(E/Q, s) at s = 1. His proof relies heavily on ...

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.

Aliquot Cycles for Elliptic Curves with Complex Multiplication

We review the history of elliptic curves and show that it is possible to form a group law using the points on an elliptic curve over some field L. We review various methods for computing the order of this group when L is finite, including the complex multiplication method. We then define and examine the properties of elliptic pairs, lists, and cycles, which are related to the notions of amicabl...

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...