Simultaneous Torsion in the Legendre Family. Issue 4 (2nd October 2017)

• Record Type: Journal Article
• Authors: Stoll, Michael
• Is Part Of: Experimental mathematics. Volume 26:Issue 4(2017)
• Page Start: 446
• Page End: 459
• Publication Date: 2017-10-02
• Subjects: elliptic curves -- torsion points -- unlikely intersections; 14H52 -- 14Q05; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2016.1201443
• Languages: English
• ISSNs: 1058-6458

Abstract

ABSTRACT: We improve a result due to Masser and Zannier, who showed that the set is finite, where E λ : y 2 = 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 .