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The non-existence of abelian varieties of type (K) with everywhere good reduction over imaginary quadratic fields. (English) Zbl 0705.14040

The author generalizes his result of non-existence of elliptic curves with good reduction defined over an imaginary quadratic field k to the case of abelian varieties of some type. Let K be a totally real algebraic number field. Let (A,\(\theta\)) be an abelian variety of type \((K)\) defined over k, in a usual sense. Further we call (A,\(\theta\)) an abelian variety of type \((K)\) over \({\mathfrak o}_ k\), the ring of all integers in k, when the Néron model of A over \({\mathfrak o}_ k\) is an abelian scheme over \({\mathfrak o}_ k.\)
Then the main theorem asserts that there is no abelian variety of type \((K)\) over \({\mathfrak o}_ k\) when the class number of k is prime to 6 and there exists an element of \({\mathfrak o}_ K\), the ring of all integers in K, of norm 3.
Reviewer: K.Katayama

MSC:

14K05 Algebraic theory of abelian varieties
14G25 Global ground fields in algebraic geometry
11R11 Quadratic extensions
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