Kato's Euler System for Isogenous Elliptic Curves

Question

Let $E,E^\prime$ be elliptic curves over $\mathbb{Q}$ and also suppose they are $p$-isogenous. How are the Euler systems corresponding to the two isogenous elliptic curves related, if at all?

Answer 1

They are related because they come from the same element for $X_1(N)$. Suppose $E$ is the one of the two with the smaller degree of the modular parametrisattion $X_1(N)\to E$ of minimal degree. Then the isogeny $E\to E'$ sends the zeta elements from $E$ to the corresponding zeta elements for $E'$. This is by definition essentially. Conjecturally $E\to E'$ extends to an étale morphism on the Néron models over $\mathbb{Z}_p$. See Stevens' "Stickelberger elements and modular parametrizations of elliptic curves". Of course the dual isogeny $E'\to E$ maps to the elements of $E$ multiplied with the degree of the isogeny.

I spent some time thinking about the integrality of Kato's elements under isogenies, see also the question Are Kato's zeta elements integral?. One can deduce that Kato's divisbility for the main conjecture also holds for all curves in the isogeny class when $E[p]$ is reducible and $p$ is odd and semistable.