Submission date: 22 September 2015

Abstract:

We investigate four properties related to an elliptic curve E_t in Legendre form with parameter t: the curve E_{t} has complex multiplication, E_{-t} has complex multiplication, a point on E_t with abscissa 2 is of finite order, and t is a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. We solve these problems effectively and in certain cases also explicitly.

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Full text arXiv 1509.06573: pdf, ps.