The Tate-Shafarevich group for elliptic curves with complex multiplication

where v ranges over all places of Q and Qv is the completion of Q at v, denote its TateShafarevich group. As usual, L(E/Q, s) is the complex L-function of E over Q. Since E is now known to be modular, Kolyvagin’s work [11] shows that X(E/Q) is finite if L(E/Q, s) has a zero at s = 1 of order ≤ 1, and that gE/Q is equal to the order of the zero of L(E/Q, s) at s = 1. His proof relies heavily on the theory of Heegner points and the work of Gross and Zagier. However, when L(E/Q, s) has a zero at s = 1 of order ≥ 2, all is shrouded in mystery. It is unknown whether or not L(E/Q, s) has a zero at s = 1 of order ≥ gE/Q, and no link between L(E/Q, s) and X(E/Q) has ever been proven. In particular, the finiteness of X(E/Q) is unknown for a single elliptic curve E/Q with gE/Q ≥ 2. This state of affairs is particularly galling for number theorists, since the conjecture of Birch and Swinnerton-Dyer even gives an exact formula for the order of X(E/Q), which predicts that in the vast majority of numerical examples X(E/Q) is zero when gE/Q ≥ 2. We also stress that in complete contrast to the situation for finding gE/Q, it is impossible to calculate X(E/Q) by classical descent methods, except for its p-primary subgroup for small primes p, usually with p ≤ 5.