Isolated and sporadic points on modular curves

 

A point of degree d on a curve is sporadic if the curve has only finitely many points of degree at most d. It is isolated if it doesn't lie in a geometric family of points of the same degree. On modular curves, these points correspond to elliptic curves with unexpected level structure over number fields of small degree. I will survey general results on isolated and sporadic points on modular curves, then present recent work on sporadic points on  the modular curves X_0(n) and X_1(n).