Constructing Kolyvagin Classes: Kolyvagin’s Conjecture and Nontrivial Elements in the Shafarevich-Tate Group

The standard references for this section are [Gro91], [Kol90] and [McC91]. Let E be an elliptic curve over Q of conductor N . Let K = Q( √ −D), where −D is a fundamental discriminant, D #= 3, 4, and all prime factors of N are split in K, i.e. (N) = NN̄ for an ideal N of the ring of integers OK of K with OK/N $ Z/NZ. We call such a discriminant a Heegner discriminant for E/Q. By the modularity theorem [BCDT01], there is a modular parameterization φ : X0(N) → E. We view OK and N as Z-lattices of rank two in C and observe that C/OK → C/N−1 is a cyclic isogeny of degree N between the elliptic curves C/OK and C/N−1. Here N−1 denotes the fractional ideal of OK for which NN−1 = OK . This isogeny corresponds to a complex point x1 ∈ X0(N). According to the theory of complex multiplication [Sil94, Ch.II], the point x1 is defined over the Hilbert class field K[1] of K. More generally, let On = Z + nOK be the order of index n in OK and let Nn = N ∩On. Then On/Nn $ Z/NZ and the map C/On → C/N−1 n is a cyclic isogeny of degree N and thus it defines a point xn ∈ X0(N)(C). Again, by the theory of complex multiplication, this point is defined over the ring class field K[n] of conductor n over K. One can use the parameterization φ : X0(N) → E to obtain points yn = φ(xn) on E. Define yK to be the point yK = TrK1/K(y1). If N ′ is another ideal with