Global construction of associated orders in complex multiplication

AbstractLet 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+Gm, where Gm denotes the group of points of order pm and P a point of order pr+m, r⩾0,m⩾1, which can be viewed as the set of all “pm-th roots” of some point in Gr.In analogy to the Kummer theory such a coset gives rise to the definition of an order O˜P in an algebra MP 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 O˜P 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 O˜P 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