Elliptic curves with exceptionally large analytic order of the Tate–Shafarevich groups

Abstract

We exhibit 88 examples of rank zero elliptic curves over the rationals with |Ш(E)| > 63408 2 , which was the largest previously known value for any explicit curve. Our record is an elliptic curve E with |Ш(E)| = 1029212 2 = 2 4 • 79 2 • 3257 2. We use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of Ш. For instance, 410536 2 is the true order of Ш(E) for E = E4(21, −233) from the table in Section 2.3.