Global construction of associated orders in complex multiplication

Let E/L be an elliptic curve defined over a number field L with complex multiplication by the ring of integers of a quadratic imaginary number field K⊆L. We fix a prime ideal p in K and assume E to have good reduction modulo p. We consider cosets of the form P+G_m, where G_m denotes the group of points of order p^m and P a point of order p^r+m, , which can be viewed as the set of all “p^m-th roots” of some point in G_r.In analogy to the Kummer theory such a coset gives rise to the definition of an order in an algebra M_P and to an associated order A defined below by its local components. These objects naturally come into play in the Galois module structure of rings of integers in ray class fields over K. It is the aim of this article to construct global generators both for and A as algebras over the ring of integers of L. For the convenience of the reader we also include from (J. Number Theory 77 (1999) 97) a simple construction of Galois generators for over A. We thereby show that these generators also fit in the setting of this article that is more general than in Sch4]. The main results in Theorems 3, 6 and 7 are obtained assuming the base field L to contain “enough” torsion points of E.