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Elliptic curves with complex multiplication and Galois module structure. (English) Zbl 0705.14031
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
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##### References:
 [1] [CN-T] Cassou-Noguès, Ph., Taylor, M.J.: Rings of integers and elliptic functions. Progr. Math. vol. 66, Boston: Birkhäuser 1987 · Zbl 0621.12012 [2] [Ch] Chan, S.P.: Modular functions, elliptic functions and Galois module structure, J. Reine Angew. Math.375-376, 67–82 (1987) · Zbl 0613.12007 · doi:10.1515/crll.1987.375-376.67 [3] [C-H] Childs, L., Hurley, S.: Local normal bases for objects of finite commutative, co-commutative Hopf algebras (Preprint) [4] [L] Lang, S.: Elliptic functions. Reading MA: Addison-Wesley, 1973 · Zbl 0316.14001 [5] [Si] Silverman, J.H.: The arithmetic of elliptic curves. Springer Graduate Text vol. 106, 1985 [6] [Sr] Srivastav, A.: Swan modules and elliptic functions. Ill. J. Math.32, 462–483 (1988) · Zbl 0708.11062 [7] [T1] Taylor, M.J.: Relative Galois module structure of rings of integers and elliptic functions, Math. Proc. Cam. Phil. Soc.94, 389–397 (1983) · Zbl 0532.12009 · doi:10.1017/S0305004100000773 [8] [T2] Taylor, M.J.: Relative Galois module structure of rings of integers and elliptic functions. II. Ann. Math.121, 519–535 (1985) · Zbl 0594.12008 · doi:10.2307/1971208 [9] [T3] Taylor, M.J.: Relative Galois module structure of rings of integers and elliptic functions. III. Proc. Lond. Math. Soc.51, 415–431 (1985) · Zbl 0594.12009 · doi:10.1112/plms/s3-51.3.415 [10] [T4] Taylor, M.J.: On Fröhlich’s conjecture for rings of integers of tame extensions. Invent. Math.63, 41–79 (1981) · Zbl 0469.12003 · doi:10.1007/BF01389193 [11] [T5] Taylor, M.J.: Mordell-Weil groups and the Galois module structure of rings of integers, III. J. Math.32, 428–452 (1988) · Zbl 0631.14033 [12] [T6] Taylor, M.J.: Galois module structure of rings of integers in Kummer extensions. Bull. Lond. Math. Soc.12, 96–98 (1980) · Zbl 0422.12002 · doi:10.1112/blms/12.2.96 [13] [W] Waterhouse, W.C.: Trans. Am. Math. Soc.153, 181–189 (1971) · doi:10.1090/S0002-9947-1971-0269659-2
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