Six unlikely intersection problems in search of effectivity

Publisher: Cambridge University Press

E-ISSN: 1469-8064|162|3|447-477

ISSN: 0305-0041

Source: Mathematical Proceedings of the Cambridge Philosophical Society, Vol.162, Iss.3, 2017-05, pp. : 447-477

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Abstract

We investigate four properties related to an elliptic curve Et in Legendre form with parameter t: the curve Et has complex multiplication, E−t has complex multiplication, a point on Et 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. Using a variety of techniques we solve these problems with varying degrees of effectivity (and for three of them we even present the list of all possible t).