Elliptic binomial diophantine equations

The complete sets of solutions of the equation $\binom{n}{k} = \binom{m}{\ell}$ are determined for the cases $(k,\ell) = (2,3)$, $(2,4)$, $(2,6)$, $(2,8)$, $(3,4)$, $(3,6)$, $(4,6)$, $(4,8)$. In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case ($(k,\ell) = (3,6)$) we had to devise a new variant of the method.