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Visibility of Kolyvagin cohomology classes

It would also be interesting to study visibility at higher level of Kolyvagin cohomology classes. The following is a first ``test question'' in this direction.

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

$\displaystyle (\delta_1^* \pm \delta_\ell^*)_*(c_\ell) = 0 \in \H ^1(\mathbb{Q},J_0(Np))? $

When is

$\displaystyle {\mathrm{res}}_v((\delta_1^* \pm \delta_\ell^*)_*(c_\ell)) = 0 \in \H ^1(\mathbb{Q}_v,J_0(Np))? $


William Stein 2006-06-21