HEURISTICS ON TATE–SHAFAREVITCH GROUPS OF ELLIPTIC CURVES DEFINED OVER ℚ

CHRISTOPHE DELAUNAY

Abstract. 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.