$\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2

Question

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 $?$

Answer 1

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 $X(E)\sim \Lambda / p^2$.