Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$. Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ with $\Gamma=\text{Gal}(\mathbb{Q}^{cyc}/\mathbb{Q}) \cong \mathbb{Z}_{p}$. Suppose $ \Sigma $ denotes any finite set of primes containing $p, \infty $, and the primes of bad reduction for $E$; $ \mathbb{Q}_{\Sigma} $ is the compositum of all finite extensions of $ \mathbb{Q} $ unramified outside $ \Sigma $. The Selmer group $Sel(E[p^{\infty}]/\mathbb{Q}^{cyc}) $ of $ E $ over $ \mathbb{Q}^{cyc} $ can be defined as the kernel of the following "global-to-local" map \begin{equation} \xi: H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}^{cyc},E[p^{\infty}])\longrightarrow \prod_{l\in\Sigma}H_{l}(\mathbb{Q}^{cyc},E[p^{\infty}])\end{equation} For $l\neq p $, $ H_{l}(\mathbb{Q}^{cyc},E[p^{\infty}]):=\prod_{\eta \mid l} H^{1}((\mathbb{Q}^{cyc})_{\eta},E[p^{\infty}]) $ with $ \eta $ running over the primes of $ \mathbb{Q}^{cyc} $ lying over $ l $, and $$H_{p}(\mathbb{Q}^{cyc},E[p^{\infty}]):=H^{1}((\mathbb{Q}^{cyc})_{\eta_{p}},E[p^{\infty}])/L_{\eta_{p}}$$ where $ \eta_{p} $ is the unique prime of $ \mathbb{Q}^{cyc} $ lying over $ p $, $ I_{\eta_{p}} $ is the inertia subgroup of $ G_{(\mathbb{Q}^{cyc})_{\eta_{p}}}, $ and $L_{\eta_{p}}=\text{Ker}\left(H^{1}((\mathbb{Q}^{cyc})_{\eta_{p}},E[p^{\infty}])\longrightarrow H^{1}(I_{\eta_{p}},\widetilde{E}[p^{\infty}])\right), \sim $ is reduction modulo $ p $.

Kato has proved that the Pontryagin dual $ X(E/\mathbb{Q}^{cyc}) $ of $ Sel(E[p^{\infty}]/\mathbb{Q}^{cyc}) $ is a finitely generated torsion $ \Lambda $-module where $\Lambda =\mathbb{Z}_{p}[[\Gamma)]] \cong \mathbb{Z}_{p}[[T]]$ by identifying $T=\gamma - 1 $ for a fixed topological generator $\gamma$ of $\Lambda$. Hence, by the classification of finitely generated $ \Lambda $-modules one has a pseudo-isomorphism $$X(E/\mathbb{Q}^{cyc}) \sim (\oplus_{i=1}^{s}\Lambda/(f_{i}(T)^{a_{i}}))\oplus(\oplus_{j=1}^{t}\Lambda/(p^{\mu^{j}_{E}}))$$ where $s,t,a_{i},\mu_{j} \in \mathbb{N}$, $f_{i}$ is distinguished and irreducible for all $i$. Since, the $ a_{i} $'s and the $ \mu^{j}_{E} $'s are positive integers, one can define the algebraic Iwasawa invariants $ \lambda_{E}^{alg} $ and $ \mu^{alg}_{E} $ by $$\lambda_{E}^{alg} = \sum_{i=1}^{s}a_i.deg(f_{i}(T)), \hspace{.3cm} \mu_{E}^{alg} = \sum_{j=1}^{t} \mu^{j}_{E}$$

Now I have the following questions regarding the above definition -

1) Why are we considering only 'the cyclotomic $\mathbb{Z}_{p}$-extensions of $ \mathbb{Q} $' and 'the finite extensions of $ \mathbb{Q} $ unramified outside $ \Sigma $' $?$

2) Why do we define $ H_{l}(\mathbb{Q}^{cyc},E[p^{\infty}])$ in the above manner $?$

3) What is the importance of the Iwasawa invariants of elliptic curves i.e what informations do we get about the rank and torsion of elliptic curves from these invariants $?$


1 Answer 1


1) Iwasawa theory, as practiced by K.Iwasawa, is concerned with $\mathbb Z_{p}$-extensions. There is only one $\mathbb Z_{p}$-extension of $\mathbb Q$. Over more generally number fields, and in more general context, it is emphatically not true that only the cyclotomic extension is considered. The reason why we consider the finite extensions of $\mathbb Q$ unramified outside $\Sigma$ is much more fundamental: we do so because the group cohomology of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ is in fact very pathological: only if one restricts to $\operatorname{Gal}(\mathbb Q_{S}/\mathbb Q)$ do we obtain a well-behaved cohomology theory. I recommend you learn this before attacking the hard questions of Iwasawa theory.

2) The short answer is that one recovers in this way through cohomological means the usual Selmer groups defined by geometrical means. The longer answer is that Selmer groups should encode extension of geometric objects in the so called category of motives. Putting the condition you described ensures that these extensions are no more ramified than your original object, but in order to see this, you need to remark that the image of projective systems of cohomology classes in the cyclotomic $\mathbb{Z}_{p}$-extension are unramified.

3) I am sure there are dozens of references discussing this question. With which are you familiar?

For any of these quite general questions, it will be more profitable to you to read original articles on the topic than asking on Mathoverflow, in my opinion.


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