Order of vanishing of $p$-adic $L$-function In Greenberg's online notes "Introduction to Iwasawa Theory for Elliptic Curves", Exercise 4.7 asks the following:
Suppose that $E/\mathbb{Q}$ is an elliptic curve and $p$ is a prime of good ordinary reduction. Suppose the Iwasawa main conjecture is true for $E$; that is, suppose that the characteristic ideal of the $p$-primary Selmer group of $E$ over $\mathbb{Q}_{\text{cyc}}$ is generated by the $p$-adic $L$-function of $E$, denoted $L_p(E/\mathbb{Q}, T)$. Show that
$$ \text{ord}_{T=0} L_p(E/\mathbb{Q}, T) \geq \text{rank }(E/\mathbb{Q}).$$
(Here $\mathbb{Q}_{\text{cyc}}$ is the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$.)
I'm not too sure how to attack this problem. The main conjecture allows us to relate the $p$-adic $L$-function to the characteristic power series of the Selmer group of $E$. But how do we relate the characteristic power series of the Selmer group to the rank of $E$ over $\mathbb{Q}$? Are there any tips anyone has to attack this?
 A: So, as I said, there’s more than one way to make this argument, depending on the technical details. For the sake of simplicity, I’ll assume that $p>2$, that you’re talking of the $p$-adic Selmer group $S$, endowed with a continuous action of $G=1+p\mathbb{Z}_p=Gal(\mathbb{Q}_{cyc}/\mathbb{Q})$, which makes it a module over $\Lambda=\mathbb{Z}_p[[1+p\mathbb{Z}_p]]$.
Note that $\Lambda \cong \mathbb{Z}_p[[T]]$ with $T$ corresponding to $\varpi=[1+p]-1$.
Assume that this module is finitely generated and torsion (otherwise, I don’t even know what the characteristic ideal is – but given the right definitions, we should still get a similar result if $S$ is not torsion).
By the structure theorem, we have an exact sequence of $\Lambda$-modules $0 \rightarrow \bigoplus_i{\Lambda/(f_i)} \rightarrow S \rightarrow C \rightarrow 0$ with irreducible $f_i$ and $C$ finite, so that the $p$-adic $L$-function is a unit times the product of the $f_i$. So if $r$ is the vanishing order of your $p$-adic $L$-function at $T=0$, then exactly $r$ of the $f_i$ are (associated to) $\varpi$.
Now take the $\varpi$-torsion (you may need to consider $S/\varpi S$ instead): if $f_i \notin \Lambda^{\times}\varpi$, then $\Lambda/(f_i)[\varpi]=0$. It follows that we have an exact sequence of $\Lambda/(\varpi) =\mathbb{Z}_p$-modules, given by $0 \rightarrow (\Lambda/(\varpi))^r \rightarrow S[\varpi] \rightarrow C[\varpi]$. As $C[\varpi]$ is finite, the $\mathbb{Z}_p$-rank of $S[\varpi]$ is exactly $r$.
You may need to consider $S/\varpi S$ instead, and the argument is similar if you consider the exact sequence $C[\varpi] \rightarrow \bigoplus_i{\Lambda/(f_i,\varpi)} \rightarrow S/\varpi S \rightarrow C/\varpi C \rightarrow 0$, and note that if $f_i \notin \Lambda^{\times}\varpi$, then $\Lambda/(f_i,\varpi)$ is finite.
