On the projective invariants of collections of torsion points

In this lecture I introduce and discuss geometry of subsets of points in projective line $P^1$ obtained as projections of torsion points of elliptic curves. We consider standard degree two projections on $P^1$. If two elliptic curves have different ramification sets in $P^1$ then the intersections of the images of torsion points is finite number. In fact for many pairs of elliptic curves such intersection is trivial or consists of just one point. We conjecture that there is universal constant bounding such intersections independently of the curves involved (universal boundedness) However it is possible to get rather big intersection. The maximal intersection obtained so far is $22$. I explain in my lecture both the main tools to construct examples of pairs of elliptic curves where the intersection is big and the reasons for existing of a realistic universal bound for such an intersection.