Consider the elliptic curves - 


$ E\_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $ 


$ E\_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $ 


with both good ordinary reduction and $\mu\_{E\_i}$-invariants equal to $2$ at $p= 3$ for $i=1,2$. Let $\Lambda = \mathbb{Z}\_{p}[[T]] $ and $ K=\mathbb{Q}\_{\infty} $ be the cyclotomic $ \mathbb{Z}\_{p} $-extension of $ \mathbb{Q} $. Then the Pontrjagin dual $ X\_{E}(\mathbb{Q}\_{\infty}) $ of $ Sel\_{E}(\mathbb{Q}\_{\infty})\_{p} $ is a finitely generated torsion $ \Lambda $-module and one has a pseudo-isomorphism


$$
X\_{E}(\mathbb{Q}\_{\infty}) \sim (\bigoplus\_{i=1}^{s}\Lambda/f\_{i}(T)^{a\_{i}}) \bigoplus (\bigoplus\_{j=1}^{t}\Lambda/p^{\mu^j\_E}) $$ and $$\mu\_E=\sum\_{j=1}^t \mu^j\_E$$ 


How to compute whether $ \mu^1\_E=\mu^2\_E=1$ or $\mu^1\_E=2$ in the above decomposition for the given two elliptic curves $?$



$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2
It took me a while to realise that this is an interesting question. The formulation above makes it sound like a computational problem for a specific curve, so let me first reformulate it:


Let $E/k$ be an elliptic curve over a number field $k$ and let $\varphi:E \to E'$ be a cyclic isogeny of degree $p^2$ for some prime $p$. To determine how the $\mu$-invariant (with respect to some $\mathbb{Z}\_p$-extension $k\_{\infty}$, say the cyclotomic one) of $E$ changes one has to look at how the complex conjugations act on the kernel $C$ of $\varphi$. So if it changes by $\mu\_E =\mu\_{E'} + 2$, how does the $\Lambda$-module change? By two additional $\Lambda/p$ or by one additional $\Lambda/p^2$ or otherwise?


The analytic side of the main conjecture would not help us with that. On the algebraic side, we get a exact sequence of $\Lambda$-modules $$ \to H^2(C)^\* \to X(E') \to X(E) \to H^1(C)^\* \to $$ where $X(E)$ and $X(E')$ are the Pontryagin duals of the Selmer groups of $E$ and $E'$ over $k\_{\infty}$ and $H^i(C)^\*$ are the Pontryagin duals of the "Selmer groups" of $C$; i.e., $H^i(C)$ is the kernel of global-to-local map on the $H^i(k\_{\infty}, C)$. So the question is what is the structure of the $\Lambda$-modules $H^i(C)^\*$.


In the particular cases, asked above, $k=\mathbb{Q}$ and $p=3$, and the kernel $C$ is in a short exact sequence $$0\to \mu\_p \to C \to \mu\_p \to 0$$ of Galois-modules, but it is not $\mu\_{p^2}$. It is possible to see that $H^1(\mu\_p)^\* $ is pseudo-isomorphic to $\Lambda/p$ and $H^2(\mu\_p)^\*\sim 0$. From this we find that $H^1(C)$ sits between two $H^1(\mu\_p)$, but also that $H^1(C)[p] \sim H^1(\mu\_p)$. Hence $H^1(C)^\* \sim \Lambda/{p^2}$. The two examples have moreover $X(E') \sim 0$, so $X(E)\sim \Lambda / p^2$.


