MOMENTS OF THE ORDERS OF TATE–SHAFAREVICH GROUPS

Abstract

We give some conjectures for the moments of the orders of the Tate–Shafarevich groups of elliptic curves belonging to a family of quadratic twists. These conjectures follow from the predictions on L-functions given by the random matrix theory [12,5] and from the Birch and Swinnerton–Dyer conjecture. Furthermore, including the Cohen–Lenstra type heuristics for Tate–Shafarevich groups, we obtain some conjectural estimates for the regulator of rank 1 elliptic curves in a family of quadratic twists.