Simultaneous Torsion in the Legendre Family

ABSTRACT

We improve a result due to Masser and Zannier, who showed that the set is finite, where E_λ: y² = x(x − 1)(x − λ) is the Legendre family of elliptic curves. More generally, denote by T(α, β), for , α ≠ β, the set of such that all points with x-coordinate α or β are torsion on E_λ. By further results of Masser and Zannier, all these sets are finite. We present a fairly elementary argument showing that the set T(2, 3) in question is actually empty. More generally, we obtain an explicit description of the set of parameters λ such that the points with x-coordinate α and β are simultaneously torsion, in the case that α and β are algebraic numbers that are not 2-adically close. We also improve another result due to Masser and Zannier dealing with the case that has transcendence degree 1. In this case, we show that and that we can decide whether the set is empty or not, if we know the irreducible polynomial relating α and β. This leads to a more precise description of T(α, β) also in the case when both α and β are algebraic. We performed extensive computations that support several conjectures, for example, that there should be only finitely many pairs (α, β) such that .

KEYWORDS:

• elliptic curves
• torsion points
• unlikely intersections

2000 AMS SUBJECT CLASSIFICATION:

• 14H52
• 14Q05

Acknowledgments

I would like to thank the organizers of the Second ERC Research Period on Diophantine Geometry for inviting me to attend this event; the first result presented here was obtained during the meeting. I would also like to thank David Masser and Umberto Zannier for fruitful discussion.