Which elements of $1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]^\times$ are in $\langle 1+x_1,1+x_2\rangle$?

Question

It's a classical fact that the commutative power series ring $\mathbb{Z}_p[[x_1,x_2]]$ is isomorphic to the completed group algebra $\mathbb{Z}_p[[\mathbb{Z}_p\times\mathbb{Z}_p]]$, the isomorphism sending generators $a_1,a_2$ of $\mathbb{Z}_p\times\mathbb{Z}_p$ to $1+x_1,1+x_2$.

In $\mathbb{Z}_p[[x_1,x_2]]$, we have the coset $1+(x_1,x_2)$ of the ideal $(x_1,x_2)$, which is a closed and open subgroup of its group of units, which certainly contain $1+x_1,1+x_2$, and hence contains the closed subgroup $\langle 1+x_1,1+x_2\rangle$ generated by $1+x_1,1+x_2$, which is isomorphic to $\mathbb{Z}_p\times\mathbb{Z}_p$, but is strictly smaller than $1+(x_1,x_2)$.

Is it possible to describe the image of the closed subgroup $\langle 1+x_1,1+x_2\rangle$ in $\mathbb{Z}_p[[x_1,x_2]]$?

Ie, suppose we are given a power series $f\in 1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]$. How can we determine if $f$ lies in the closed subgroup $\langle 1+x_1,1+x_2\rangle$?

(I know this question isn't really well-defined. I'm just looking for certain criteria, ideally linked to algebraic properties of the power series, which can be used to identify the image of $\langle 1+x_1,1+x_2\rangle$).

It would also be nice if one could describe the quotient $(1+( x_1,x_2))/\langle 1+x_1,1+x_2\rangle$.

Answer 1

The usual power series for $\log(1+x)$ determines an injection from $1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]$ into $\mathbb{Q}_p[[x_1,x_2]]$. A power series $f\in 1+(x_1,x_2)$ is in $\langle 1+x_1,1+x_2\rangle$ if and only if there are $\alpha_1$, $\alpha_2\in\mathbb{Z}_p$ such that \begin{align*} \log f(x)&=\alpha_1\log(1+x_1)+\alpha_2\log(1+x_2)\\ &=\alpha_1\sum_{n\geq 1}\frac{(-1)^{n-1}}{n}x_1^n+\alpha_2\sum_{n\geq 1}\frac{(-1)^{n-1}}{n}x_2^n. \end{align*} Note that $\alpha_1$ and $\alpha_2$ can be read off from $\log f(x)$: they are the coefficients of $x_1$ and $x_2$, respectively. So if $$ \log f(x)=\sum_{i,j\geq 0}a_{ij}x_1^{i}x_2^{j}, $$ a necessary and sufficient condition for $f(x)\in\langle 1+x_1,1+x_2\rangle$ is that $a_{ij}=0$ if $i$, $j\geq 1$, and $$ a_{n,0}=\frac{(-1)^{n-1}}{n}a_{1,0}, $$ $$ a_{0,n}=\frac{(-1)^{n-1}}{n}a_{0,1}. $$

Answer 2

The group algebra $\mathbb{Z}_p[[\mathbb{Z}_p\times \mathbb{Z}_p]]$ has the structure of a complete Hopf algebra, where $\mathbb{Z}_p\times \mathbb{Z}_p$ consists of precisely the group-like elements. Via the given isomorphism, we get a comultiplication on $\mathbb{Z}_p[[x_1,x_2]]$, where $1+x_1$ and $1+x_2$ are group-like. The formula for the comultiplication is $\Delta x_i=x_i\otimes 1+1\otimes x_i+x_i\otimes x_i$ for $i\in\{1,2\}$. A power series $f\in 1+(x_1,x_2)$ is in $\langle 1+x_1,1+x_2\rangle$ if and only if $f$ is group-like, i.e. if and only if there is an equality of formal power series $$ f(x_1+y_1+x_1y_1,x_2+y_2+x_2y_2) = f(x_1,x_2)f(y_1,y_2)\in\mathbb{Z}_p[[x_1,x_2,y_1,y_2]]. $$