Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: ABSTRACT improve a result due to Masser and Zannier, who shod that the set is finite, where Eλ: 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 Eλ. By further results of Masser and Zannier, all these sets are finite.  present a fairly elementary argument showing that the set T(2, 3) in question is actually empty. More generally,  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.  also improve another result due to Masser and Zannier dealing with the case that has transcendence degree 1. In this case,  show that and that  can decide whether the set is empty or not, if  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 a...
['Michael Stoll']
I am writing a research paper on a idea Simultaneous Torsion in the Legendre Family
[{"from": "human", "value": "I am writing a research paper on a idea Simultaneous Torsion in the Legendre Family"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: ABSTRACT improve a result due to Masser and Zannier, who shod that the set is finite, where E\u03bb: y2 = x(x \u2212 1)(x \u2212 \u03bb) is the Legendre family of elliptic curves. More generally, denote by T(\u03b1, \u03b2), for , \u03b1 \u2260 \u03b2, the set of such that all points with x-coordinate \u03b1 or \u03b2 are torsion on E\u03bb. By further results of Masser and Zannier, all these sets are finite.  present a fairly elementary argument showing that the set T(2, 3) in question is actually empty. More generally,  obtain an explicit description of the set of parameters \u03bb such that the points with x-coordinate \u03b1 and \u03b2 are simultaneously torsion, in the case that \u03b1 and \u03b2 are algebraic numbers that are not 2-adically close.  also improve another result due to Masser and Zannier dealing with the case that has transcendence degree 1. In this case,  show that and that  can decide whether the set is empty or not, if  know the irreducible polynomial relating \u03b1 and \u03b2. This leads to a more precise description of T(\u03b1, \u03b2) also in the case when both \u03b1 a..."}]