An analogue for elliptic curves of the Grunwald–Wang example. (English) Zbl 1035.14007

Dvornicich, Roberto; Zannier, Umberto

C. R., Math., Acad. Sci. Paris 338, No. 1, 47-50 (2004).

Summary: We give examples of elliptic curves \(\mathcal E/\mathbb Q\) and rational points \(P \in \mathcal E(\mathbb Q)\) such that \(P\) is divisible by 4 in \(\mathcal E(\mathbb Q_v)\) for each rational place \(v\) but \(P\) is not divisible by 4 in \(\mathcal E(\mathbb Q)\). This is an analogue of a well-known example, with \(\mathbb G_m\) in place of \(\mathcal E\): namely, \(P=16\) is a rational 8th power locally almost everywhere, but not globally in \(\mathbb Q^{\ast} = \mathbb G_m(\mathbb Q)\) [E. Artin and J. Tate, “Class field theory” (1986; Zbl 0176.33504), chapter X, theorem 1].

MSC:

14G05	Rational points
14H52	Elliptic curves
11G05	Elliptic curves over global fields

Keywords:

elliptic curves; rational points

Citations:

Zbl 0176.33504

References:

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