# Impossible intersections for elliptic curves

# Impossible intersections for elliptic curves

We proved with Umberto Zannier that there are at most finitely many complex numbers $\lambda \neq {0,1}$ such that two points on the Legendre elliptic curve $y^2 = x(x-1)(x-\lambda)$ with coordinates $x=2$ and $x=3$ both have finite order. However we still do not know how to find these $\lambda$ effectively (there are probably none). We discuss various extensions of this result, some of them effective. For example some comments of Serre encouraged us to show that there are no $\lambda$ at all for coordinates $x=u$ and $x=v$ when the complex numbers $u,v$ are not both algebraic with $1,u,v$ linearly dependent over the field of rationals and $uv(u-1)(v-1)(u-v) \neq 0$. The proof needs a close study of what may be called "bicyclotomic polynomials."