Simultaneous torsion in the Legendre family

TL;DR

This paper refines results on torsion points in the Legendre family of elliptic curves, providing explicit descriptions and elementary proofs for certain parameter sets, and supports conjectures on finiteness of special pairs.

Contribution

It offers an elementary proof that the set T(2,3) is empty and characterizes parameters where points with given x-coordinates are torsion, advancing understanding of torsion in elliptic curves.

Findings

The set T(2,3) is empty.

Explicit description of parameters with torsion points for algebraic α, β.

Bound of at most one element in T(α, β) when transcendence degree is 1.

Abstract

We improve a result due to Masser and Zannier, who showed that the set $$ \{\lambda \in {\mathbb C} \setminus \{0,1\} : (2,\sqrt{2(2-\lambda)}), (3,\sqrt{6(3-\lambda)}) \in (E_\lambda)_{\text{tors}}\} $$ is finite, where $E_\lambda \colon y^2 = x(x-1)(x-\lambda)$ is the Legendre family of elliptic curves. More generally, denote by $T(\alpha, \beta)$, for $\alpha, \beta \in {\mathbb C} \setminus \{0,1\}$, $\alpha \neq \beta$, the set of $\lambda \in {\mathbb C} \setminus \{0,1\}$ such that all points with $x$-coordinate $\alpha$ or $\beta$ are torsion on $E_\lambda$. By further results of Masser and Zannier, all these sets are finite. We present a fairly elementary argument showing that the set $T(2,3)$ in question is actually empty. More generally, we obtain an explicit description of the set of parameters $\lambda$ such that the points with $x$-coordinate $\alpha$ and $\beta$ are simultaneously torsion, in the case that $\alpha$ and $\beta$ are algebraic numbers that not 2-adically close. We also improve another result due to Masser and Zannier dealing with the case that ${\mathbb Q}(\alpha, \beta)$ has transcendence degree 1. In this case we show that $\#T(\alpha, \beta) \le 1$ and that we can decide whether the set is empty or not, if we know the irreducible polynomial relating $\alpha$ and $\beta$. This leads to a more precise description of $T(\alpha, \beta)$ also in the case when both $\alpha$ and $\beta$ are algebraic. We performed extensive computations that support several conjectures, for example that there should be only finitely many pairs $(\alpha, \beta)$ such that $\#T(\alpha, \beta) \ge 3$.