Heuristics on Tate-Shafarevitch Groups of Elliptic Curves Defined over Q

In a well-known paper, Cohen and Lenstra gave conjectures on class groups of number fields. We give here similar conjectures for Tate-Shafarevitch groups of elliptic curves defined over Q. For such groups (if they are finite), there exists a nondegenerate, alternating, bilinear pairing. We give some properties of such groups and then formulate heuristics which allow us to give precise conjectures.