How to prove that the $\lambda$-invariant is constant for isogenous elliptic curves $?$



$\lambda$-invariant is constant for isogenous elliptic curves
Given the brevity of the question, I am not sure about the precise setup, so let me assume that you are looking at some $\mathbf{Z}\_p$-extension and that you assume (or know in the situation at hand) that the appearing Iwasawa modules are torsion, so that the $\lambda$-invariant is defined in the first place. For an Iwasawa module $X$, this invariant is just $\dim\_{\mathbf{Q}\_p} X \otimes\_{\mathbf{Z}\_p} \mathbf{Q}\_p$. Now if you have an isogeny $\phi$ of elliptic curves, looking at its dual isogeny $\psi$ you see that both $\phi \circ \psi$ and $\psi \circ \phi$ are just multiplication by $\deg \phi$. Consequently, $\phi$ induces an isomorphism of $X \otimes\_{\mathbf{Z}\_p} \mathbf{Q}\_p$ with the corresponding object for the isogenous curve, which yields the claim.


