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)


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
Full Text: DOI EuDML


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