Logarithm function on the Robba ring $\mathcal{R}_L$ and some properties of $\mathcal{R}_L$

Question

I am reading some of Colmez papers about $(\varphi,\Gamma)$-modules over the Robba ring $\mathcal{R}_L$, where $L$ is a finite extension of $\mathbb{Q}_p$. Now I would like to understand this ring better.

In some proofs he uses the logarithm of an element of $\mathcal{R}_L$. I could not yet find a source where a definition of the logarithm on this ring is given. My guess is that you define it with the usual power series. My first question is whether this is the right way to do so. Furthermore on what elements is this logarithm defined and how does it change the annulus of convergence of a series.

The second question is more general about the Robba ring. If we only observe the bounded elements $x\in (\mathcal{E}^t)^*$ we apparently have a unique decomposition $x=\alpha T^k x^+x^-$ with $\alpha \in L^*$, $k\in \mathbb{Z}$, $x^+\in 1+T\mathcal{O}_L\left[\!\left[ T \right]\!\right]$ and $x^-\in 1+\mathfrak{m}_L[\![ T^{-1} ]\!]\cap \mathcal{E}^t$. Does someone know where I can find a proof for this?

I would also be really happy about a source, where I can read about some basic properties of the Robba ring in general and especially about the two things I just named.

Answer 1

I don't think that Colmez uses the log of a general element of $\mathcal{R}_L$. The ring $\mathcal{R}_L$ is some localization/completion of the ring of coordinates on a Lubin-Tate formal group, and the log that appears in Colmez' papers is the formal group log, ie $\log_{LT}(X)$ if you choose a coordinate $X$ on your formal group (ie if you choose a variable for $\mathcal{R}_L$). If your group is $\mathbf{G}_m$ and $X$ is the usual variable, then $\log_{\mathbf{G}_m}(X) = \log(1+X)$.

As for the factorisation property, see for example Kedlaya's "A p-adic local monodromy theorem", section 6.2 (in particular prop 6.5), where a corresponding statement for matrices, not just elements, is proved.