Elliptic curves with complex multiplication and Galois module structure. (English) Zbl 0705.14031

Srivastav, Anupam; Taylor, Martin J.

Invent. Math. 99, No. 1, 165-184 (1990).

The authors consider the Galois module structure of principal homogeneous spaces arising from the division of torsion points on an elliptic curve with complex multiplication. The main result is to show that, provided the division is carried by an endomorphism which is coprime to 6, then such principal homogeneous spaces are always free Galois modules; more precisely they are free over the Cartier dual of the group scheme \(\text{Ker}(e)\), defined over the spectrum of the ring of integers of the base field.
Classically the division of torsion points is used to generate abelian extensions of quadratic imaginary number fields. In this way, the above result can be used to describe the Galois module structure of rings of integers of such abelian extensions.

Reviewer: M. J. Taylor

MSC:

11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers
11G05	Elliptic curves over global fields
11G15	Complex multiplication and moduli of abelian varieties

Keywords:

Galois module structure; torsion points on an elliptic curve with complex multiplication; rings of integers

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