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Simultaneous torsion in the Legendre family

Abstract

We improve a result due to Masser and Zannier, who showed that the set {λC{0,1}:(2,2(2λ)),(3,6(3λ))(Eλ)tors} \{\lambda \in {\mathbb C} \setminus \{0,1\} : (2,\sqrt{2(2-\lambda)}), (3,\sqrt{6(3-\lambda)}) \in (E_\lambda)_{\text{tors}}\} is finite, where Eλ ⁣:y2=x(x1)(xλ)E_\lambda \colon y^2 = x(x-1)(x-\lambda) is the Legendre family of elliptic curves. More generally, denote by T(α,β)T(\alpha, \beta), for α,βC{0,1}\alpha, \beta \in {\mathbb C} \setminus \{0,1\}, αβ\alpha \neq \beta, the set of λC{0,1}\lambda \in {\mathbb C} \setminus \{0,1\} such that all points with xx-coordinate α\alpha or β\beta are torsion on EλE_\lambda. 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)T(2,3) in question is actually empty. More generally, we obtain an explicit description of the set of parameters λ\lambda such that the points with xx-coordinate α\alpha and β\beta are simultaneously torsion, in the case that α\alpha and β\beta are algebraic numbers that not 2-adically close. We also improve another result due to Masser and Zannier dealing with the case that Q(α,β){\mathbb Q}(\alpha, \beta) has transcendence degree 1. In this case we show that #T(α,β)1\#T(\alpha, \beta) \le 1 and that we can decide whether the set is empty or not, if we know the irreducible polynomial relating α\alpha and β\beta. This leads to a more precise description of T(α,β)T(\alpha, \beta) also in the case when both α\alpha and β\beta are algebraic. We performed extensive computations that support several conjectures, for example that there should be only finitely many pairs (α,β)(\alpha, \beta) such that #T(α,β)3\#T(\alpha, \beta) \ge 3.Comment: 24 pages. v2: Improved 2-adic results, leading to more cases that can be treated explicitly. Used this to solve a problem considered in arXiv:1509.06573. Added some reference

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