Torsion points on elliptic curves and Galois module structure
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 \textit{A. Srivastav} and \textit{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.
scientific article; zbMATH DE number 871765
scientific article