Question 7.3.1
Suppose
![$ E\subset J_0(N)$](img581.png)
is an elliptic curve with conductor
![$ N$](img13.png)
,
and fix a prime
![$ \ell$](img104.png)
such that
![$ \overline{\rho}_{E,\ell}$](img278.png)
is surjective.
Fix a quadratic imaginary field
![$ K$](img6.png)
that satisfies the Heegner
hypothesis for
![$ E$](img271.png)
. For any prime
![$ p$](img302.png)
satisfying the
conditions of [
Rub89, Prop. 5], let
![$ c_{p} \in \H ^1(\mathbb{Q},E)[\ell]$](img582.png)
be the corresponding Kolyvagin
cohomology class.
There are two natural homomorphisms
![$ \delta_1^*, \delta_p^*:E \to J_0(Np)$](img583.png)
.
When is
When is