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On Taylor’s conjecture for Kummer orders. (English) Zbl 0726.11038

Let \(K\) be an imaginary quadratic number field, \(L\) a finite extension of \(K\), \((E/L)\) an elliptic curve, defined over \(L\), with everywhere good reduction and admitting complex multiplication by \(O_ K\). Let \({\mathfrak A}\) be a nonzero integral \(O_ K\)-ideal. M. J. Taylor has associated to this set-up a certain order \(\Lambda\) and a homomorphism \(\psi : E(L)\to \text{Cl}(\Lambda)\) and he has conjectured that \(E(L)_{\text{tor}}\subset\ker\psi\). This has been proved under a certain hypothesis.
In the present paper the author’s consider a case where this hypothesis is not satisfied and there they prove the following weak version: \(E(L)_{\text{tor}}\subset\ker\psi'\) where \(\psi'\) is the composition of \(\psi\) with the natural homomorphism \(\text{Cl}(\Lambda)\to\text{Cl}(\Lambda')\) where \(\Lambda'\) is the maximal order containing \(\Lambda\).

MSC:

11G05 Elliptic curves over global fields
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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References:

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