## Torsion points on elliptic curves and Galois module structure.(English)Zbl 0864.11055

Let $$E$$ be an elliptic curve defined over a number field $$F$$ with everywhere good reduction. For a prime $$p>3$$ let $${\mathfrak B}_i(F)$$ denote the $${\mathcal O}_F$$-Hopf algebra which belongs to the $${\mathcal O}_F$$-group scheme of $$p^i$$-torsion points on $$E$$. Let furthermore $${\mathfrak A}_i(F)$$ be its Cartier dual. For any point $$Q\in E(F)$$ let $$G_Q(i)= \{Q'\in E (\overline F) \mid p^iQ'=Q\}$$ and the Kummer algebra $$F_Q(i)= \text{Map} (G_Q(i), \overline F)^{\text{Gal} (\overline F/F)}$$. One can now consider the $${\mathfrak A}_i(F)$$-module $${\mathfrak C}_Q(i)$$ which is the “modified” integral closure of $${\mathcal O}_F$$ in $$F_Q(i)$$.
The author proves that $${\mathfrak C}_Q(i)$$ is a globally free $${\mathfrak A}_i(F)$$-module if $$Q$$ is a torsion point in $$E(F)$$. This generalizes a result of A. Srivastav and M. J. Taylor [Invent. Math. 99, 165-184 (1990; Zbl 0705.14031)] who proved the same result under the additional assumption that $$E$$ has complex multiplication. The proof depends upon the use of intersection theory on arithmetic surfaces.
Reviewer: H.-G.Rück (Essen)

### MSC:

 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11G05 Elliptic curves over global fields 14G25 Global ground fields in algebraic geometry 14H25 Arithmetic ground fields for curves

Zbl 0705.14031
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### References:

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