What is the significance of Coleman maps arising in Iwasawa theory?

Question

I have come across two instances of "Coleman map"

Let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $k_\infty$ be the unique $\mathbb{Z}_p$ extension of $\mathbb{Q}_p$ contained in $\mathbb{Q}_p(\mu_{p^\infty})$ with Galois group $\Gamma = 1+p\mathbb{Z}_p \cong \mathbb{Z}_p$. Let $k_n$ be the $n$-th layer in this tower. Let $T=T_p(E)$ be the $p$-adic Tate module of the elliptic curve $E$. Then the $\textit{Coleman map for E}$ is a map $$Col: \varprojlim_{n} H^1(k_n, T^*(1)) \rightarrow \Lambda=\mathbb{Z}_p[[\Gamma]] $$ I am referring to page 572 of this paper where I learn that the power series $Col_z(x)$ equals the $p$-adic L-function of the elliptic curve E when $z$ is a special element discovered by Prof. Kato.

There is another instance where I have come across a Coleman power series, as a power series that generates norm-coherent sequence of units in the tower $\mathbb{Q}_p(\mu_{p^ \infty})/\mathbb{Q}_p$. That is, if I have a sequence of units $\mathbb{u}=(u_n)_{n \geq 0} \in \varprojlim_{n \geq 0} \mathcal{O}^{\times}_{\mathbb{Q}_p(\mu_{p^n})}$ (the inverse limit on the right hand side is w.r.t. the norm map of fields $\mathbb{Q}_p(\mu_{p^m}) \rightarrow \mathbb{Q}_p(\mu_{p^n}), m \geq n$), then $\exists!$ unique power series $Col_{\mathbb{u}}(x) \in \mathbb{Z}_p[[x]]$ such that $Col_{\mathbb{u}}(\zeta_{p^{1+n}}-1)=u_n, \forall n\geq 0$. How does the first Coleman map relate to the second? Or even more generally, are there other instances where Coleman maps arise, and what is the general philosophy behind Coleman maps? Any thoughts and/or link to any articles/notes are welcome. Thank you!!

Answer 1

A huge chunk of the literature of p-adic Iwasawa theory is devoted to this question.

• One of the milestones in the subject is Perrin-Riou's book "P-adic L-functions and p-adic representations", and reading that would be one excellent way to learn the topic.
• There's also Colmez's Bourbaki seminar "Fonctions L p-adiques", if you're confident reading French.
• For a generalisation to non-commutative Iwasawa theory (and much more besides) there's a wonderful, but very hard, article by Fukaya and Kato "A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory".

To answer your direct question about how the two "Coleman maps" you describe are related: they are both special cases of a single more general construction. You can make this construction (at least) for any crystalline p-adic representation of the absolute Galois group of $\mathbf{Q}_p$. One example of such a representation is the 1-dimensional trivial representation, and that gives you Coleman's original map for norm-compatible systems of units; another example of such a representation is the Tate module of an elliptic curve, and then we get the first example from your question.

(PS: This is not quite true as stated; I should have said that applying Perrin-Riou's general construction to the trivial representation gives the logarithm of Coleman's power series for norm-compatible units. But that's a minor point.)